ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2012

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 970

Efficient and Reliable Simulationof Quantum MolecularDynamics

KATHARINA KORMANN

ISSN 1651-6214ISBN 978-91-554-8466-8urn:nbn:se:uu:diva-180251

Dissertation presented at Uppsala University to be publicly examined in 2446, Polacksbacken,Lägerhyddsvägen 2, Uppsala, Friday, October 19, 2012 at 10:00 for the degree of Doctor ofPhilosophy. The examination will be conducted in English.

AbstractKormann, K. 2012. Efficient and Reliable Simulation of Quantum Molecular Dynamics. ActaUniversitatis Upsaliensis. Digital Comprehensive Summaries of Uppsala Dissertations fromthe Faculty of Science and Technology 970. 52 pp. Uppsala. ISBN 978-91-554-8466-8.

The time-dependent Schrödinger equation (TDSE) models the quantum nature of molecularprocesses. Numerical simulations based on the TDSE help in understanding and predicting theoutcome of chemical reactions. This thesis is dedicated to the derivation and analysis of efficientand reliable simulation tools for the TDSE, with a particular focus on models for the interactionof molecules with time-dependent electromagnetic fields.

Various time propagators are compared for this setting and an efficient fourth-ordercommutator-free Magnus-Lanczos propagator is derived. For the Lanczos method, severalcommunication-reducing variants are studied for an implementation on clusters of multi-coreprocessors. Global error estimation for the Magnus propagator is devised using a posteriorierror estimation theory. In doing so, the self-adjointness of the linear Schrödinger equation isexploited to avoid solving an adjoint equation. Efficiency and effectiveness of the estimate aredemonstrated for both bounded and unbounded states. The temporal approximation is combinedwith adaptive spectral elements in space. Lagrange elements based on Gauss-Lobatto nodes areemployed to avoid nondiagonal mass matrices and ill-conditioning at high order. A matrix-freeimplementation for the evaluation of the spectral element operators is presented. The frameworkuses hybrid parallelism and enables significant computational speed-up as well as the solutionof larger problems compared to traditional implementations relying on sparse matrices.

As an alternative to grid-based methods, radial basis functions in a Galerkin setting areproposed and analyzed. It is found that considerably higher accuracy can be obtained with thesame number of basis functions compared to the Fourier method. Another direction of researchpresented in this thesis is a new algorithm for quantum optimal control: The field is optimizedin the frequency domain where the dimensionality of the optimization problem can drasticallybe reduced. In this way, it becomes feasible to use a quasi-Newton method to solve the problem.

Keywords: time-dependent Schrödinger equation, quantum optimal control, exponentialintegrators, spectral elements, radial basis functions, global error control and adaptivity, high-performance computing implementation

Katharina Kormann, Uppsala University, Department of Information Technology, Division ofScientific Computing, Box 337, SE-751 05 Uppsala, Sweden.

© Katharina Kormann 2012

ISSN 1651-6214ISBN 978-91-554-8466-8urn:nbn:se:uu:diva-180251 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-180251)

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I K. Kormann, S. Holmgren, H. O. Karlsson. Accurate time propagationfor the Schrödinger equation with an explicitly time-dependentHamiltonian. J. Chem. Phys., 128:184101, 2008.1

Contributions: The ideas were developed in close cooperation with the co-authors.The author of this thesis implemented the methods and performed analysis as well ascomputations. She prepared a draft of the manuscript and finished it with assistancefrom the co-authors.

II K. Kormann, S. Holmgren, H. O. Karlsson. Global error control of thetime-propagation for the Schrödinger equation with a time-dependentHamiltonian. J. Comput. Sci., 2:178–187, 2011.2

Contributions: The ideas were developed in consultation with the co-authors. Theauthor of this thesis implemented the methods and performed analysis as well ascomputations. She prepared a draft of the manuscript and finished it with assistancefrom the co-authors.

III K. Kormann. A time-space adaptive method for the Schrödingerequation. Technical Report 2012-023, Department of InformationTechnology, Uppsala University, 2012. (Submitted)

Contributions: The author of this thesis is the sole author of this paper.

IV K. Kormann, A. Nissen. Error control for simulations of a dissociativequantum system. In G. Kreiss, P. Lötstedt, A. Målqvist, M. Neytcheva,editors, Numerical Mathematics and Advanced Applications 2009,pages 523–531, Springer, Berlin, 2010.3

Contributions: The ideas were developed, the methods implemented, and themanuscript written in close collaboration between the authors of the paper. The authorof this thesis had the main responsibility for the parts regarding discretization errors andthe error balancing.

1Reprinted with permission. Copyright 2008, American Institute of Physics.2Reprinted with permission from Elsevier.

V M. Gustafsson, K. Kormann, S. Holmgren. Communication-efficientalgorithms for numerical quantum dynamics. In K. Jónasson, editor,Applied Parallel and Scientific Computing, volume 7134 of LectureNotes in Computer Science, pages 368–378, Springer, Berlin, 2012.3

Contributions: The ideas were developed in close collaboration between the authorsof the paper and the manuscript was prepared in cooperation with the first author, indiscussions with the third author. The author of this thesis was responsible for the errorcontrol.

VI K. Kormann, M. Kronbichler. Parallel finite element operatorapplication: Graph partitioning and coloring. In 2011 Seventh IEEEInternational Conference on eScience, pages 332–339, 2011.4

Contributions: The ideas and the implementation framework were developed and themanuscript drafted in close collaboration between the authors of the paper. The authorof this thesis had the main responsibility for the partitioning and coloring algorithm andthe application to the Schrödinger equation.

VII K. Kormann, E. Larsson. An RBF-Galerkin approach to thetime-dependent Schrödinger equation. Technical Report 2012-024,Department of Information Technology, Uppsala University, 2012.Contributions: The ideas were developed in discussion between the authors. Theauthor of this thesis performed the analysis, implemented the method, and carried outthe numerical experiments. She prepared a draft of the manuscript and finished it incollaboration with the second author.

VIII K. Kormann, S. Holmgren, H. O. Karlsson. A Fourier-coefficient basedsolution of an optimal control problem in quantum chemistry.J. Optim. Theory Appl., 147:491–506, 2010.3

Contributions: The ideas were developed by the author in discussion with the co-authors. The author of this thesis implemented the methods and performed analysisas well as computations. She prepared a draft of the manuscript and finished it inconsultation with the co-authors.

3Reprints were made with kind permission from Springer Science and Business Media.4 c! 2011, IEEE. Reprinted with permission.

Related work

Although not explicitly discussed in the comprehensive summary, the follow-ing papers are related to the contents of this thesis.

• M. Kronbichler and K. Kormann. A generic interface for parallelcell-based finite element operator application. Comp. Fluids,63:135–147, 2012.

• K. Kormann and E. Larsson. Radial basis functions for thetime-dependent Schrödinger equation. In Numerical Analysis andApplied Mathematics: ICNAAM 2011, number 1389 in AIP ConferenceProceedings, pages 1323–1326, 2011.

• M. Gustafsson, A. Nissen, and K. Kormann. Stable difference methodsfor block-structured adaptive grids. Technical Report 2011-022,Department of Information Technology, Uppsala University, 2011.

The framework for cell-based finite element operator application has beenpublished as a part of the deal.II library. There are also two tutorial programspublished as part of the deal.II project, step-37 (multigrid solver based on theframework) and step-48 (nonlinear example explaining the parallelization ofthe framework).

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Time-Dependent Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Born–Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Interactions with Time-Dependent Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Physical and Mathematical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Numerical Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Discretization of the Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 Pseudospectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.2 Localized Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.3 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.1 Suitable Integrators – a Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.2 Rotating Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Error Estimation and Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1 Temporal Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Spatial Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Residual-Based Error Estimation and Global Error Control . . . . . . . 27

5 High-Performance Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.1 Communication-Avoiding Time Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Stencil-Based Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3 Hybrid Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Quantum Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8 Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1. Introduction

Quantum mechanics describes how fundamental particles and aggregationsthereof interact. While quantum effects are usually not apparent in macro-scopic structures, they govern how chemical products are formed and reac-tants are consumed. In experimental quantum dynamics, scientists measurethe effects of the wave nature of particles as, for instance, reaction rates orscattering cross-sections. Such experiments are complemented by a study ofthe mathematical models. These theoretical solutions reveal the dynamics ofthe particles and thereby provide understanding of the observables in termsof the underlying molecular properties. For most practical problems, analyt-ical expressions of the solution are unknown, making computer simulationsessential.

With the aim of controlling reactions, quantum chemists put efforts in prob-ing and manipulating the dynamics of electrons and atomic nuclei. The emer-gence of femtosecond lasers in the late 1980s made it possible to “watch” themotion of nuclei. The development of laser technologies has continued andaround the turn of the millennium it even became possible to follow the motionof the electrons, now using attosecond pulses. The possibility to control reac-tions on the molecular level opens new opportunities for chemical processes,for instance in the semiconductor or catalysis industries [98]. Furthermore, ithas potential impact to electronics where scientists try to design miniaturizeddevices whose structures are made up of very few atoms only [26].

The number of degrees of freedom in the quantum model of a moleculeis generally very large. For instance, in the full description of the quantumnature of the comparably small water molecule (H2O), 39 spatial degrees offreedom are involved. Moreover, the motion within a molecule ranges overscales of many orders of magnitude in both time and physical space. Theso-called Born–Oppenheimer approximation is a useful simplification whichallows for splitting electronic and nuclear motion. This way the 30 elec-tronic and the nine nuclear degrees of freedom in the H2O molecule modelare separated. Still, the description of the dynamics of the nuclei remainshigh-dimensional and the corresponding numerical simulation extremely chal-lenging (see Sec. 2.4). This thesis concentrates on such nuclear Schrödingerequations. There are many successful examples of further modeling—mostlyinvolving relations from classical mechanics—that facilitate simulations oflarger molecules. It is the aim of this thesis to exploit algorithmic efficiencyand modern computer power to provide a simulation tool for the dynamics ofas large molecules as possible without any classical modeling. Firstly, such a

9

tool for fully quantum dynamical simulations can be used to validate furtherapproximations and enable error analysis and control. Moreover, there are stilleffects like tunneling that are very difficult to cover when classical approxima-tions are applied. Quantum effects are also often restricted to a small part ofthe molecule. In this case, an efficient quantum solver for such a confined partof the molecule could be coupled to a more classical model of the rest of themolecule.

The building blocks of an efficient solver as proposed in this thesis are er-ror control to get reliable results and to adapt the discretization to the solution(see Ch. 4) and a memory-lean and parallel implementation (see Ch. 5). Spec-tral elements are suited for this purpose since they are of high order whilestill being localized enough to enable parallelization. Moreover, they are flex-ible when it comes to mesh refinement and the dual-weighted residual methodprovides a theory for global error control. For time-stepping, an exponentialMagnus–Lanczos propagator offers good stability and accuracy without theneed of solving systems of equations.

A Galerkin discretization based on radial basis functions as analyzed in thisthesis has the potential to further improve efficiency and compatibility withsemiclassical methods. Finally, it is proposed to consider pulse optimizationin Fourier space to solve quantum optimal control problems and to apply asuperlinearly converging optimization routine (cf. Ch. 6).

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2. Time-Dependent Quantum Dynamics

The Schrödinger equation was formulated for the first time by Erwin Schrödin-ger in a series of papers [121, 122, 123, 124]. His first article [121] consideredthe case of the hydrogen atom and the Schrödinger eigenvalue problem wasdevised starting from a classical Hamiltonian differential equation, taking intoaccount several observations on the quantum nature of particles. Schrödinger’swork is still fundamental for contemporary quantum mechanics.

The general time-dependent form of the Schrödinger equation, derived inSchrödinger’s fourth article [124], reads

ih!! t

"(x,t) = !H"(x,t), (2.1)

where h is Planck’s constant divided by 2# and H is the (quantum) Hamilto-nian consisting of the kinetic and the potential energy operators of the studiedsystem. The spatial coordinates x ! Rd represent the spatial position of eachparticle in the system and t denotes time. The wave function " is found inL2(H1(Rd),(0,t f ]) and has a temporal derivative in L2(H"1(Rd),(0,t f ]). Thesquare of its modulus, $(x,t) = |"(x,t)|2, gives the probability density of thesystem. The partial differential equation (2.1) needs to be closed with an initialvalue, "(x,0) = "0.

2.1 Born–Oppenheimer ApproximationWhen describing the full dynamics of a molecule, the Hamiltonian is estab-lished by a sum of the kinetic and the potential energies of each nucleus andeach electron as well as the electron-nuclear potential energy. In this section,we will consider the time-independent Schrödinger equation,

H"(x) = E"(x).

Let us denote by Ri the coordinates of nucleus Ni and by Zi and Mi its chargeand mass, respectively. For electron ei, we denote its coordinates by ri, andby e and m we denote electronic charge and mass, respectively. With thisnotation, the full molecular Hamiltonian is given by (cf. Refs. [125, 135]),

!H = "%i

h2

2Mi&Ni + %

i> j

ZiZ je2

|Ri "R j|"%

i

h2

2m&ei + %

i> j

e2

|ri " r j|"%

i, j

Z je2

|ri "R j|

= !TN + !VN + !Te + !Ve + !VeN ,

11

where &# denotes the Laplacian with respect to the coordinates of particle (*).Since the model includes the position of each particle as (three) degrees offreedom, the full system rapidly becomes extremely high-dimensional. There-fore it is necessary to separate electronic and nuclear coordinates already forsmall molecules.

Since the mass of a nucleus is a factor 103 to 105 larger than the mass ofan electron, the nuclei move much slower than the electrons. On the timescale of the vibration of electrons, the nuclei are usually almost stationary. Inthe Born–Oppenheimer approximation, it is assumed that the wave function"(r,R) can be split into an electronic and a nuclear part as

"(r,R) = '(r;R)((R).

Then the electronic Schrödinger equation is solved separately for fixed nuclearcoordinates,

!He'(r;R) = Eel(R)'(r;R),

with the electronic Hamiltonian being !He = !Te + !Ve + !VeN . Substituting thisansatz and the electronic Schrödinger equation into the full equation yields

!H ('(r;R)((R)) ="%i

h2

2Mi('(r;R)&Ni((R)+2)N"i'(r;R) ·)Ni((R)

+&Ni'(r;R)((R))+(Eel(R)+VN)'(r;R)((R).(2.2)

The terms 2)N"i'(r;R) ·)Ni((R) and &Ni'(r;R)((R) in (2.2) hamper aseparation of the nuclear part. For most configurations, they are howeverof the same magnitude as the mass ratio between electrons and nuclei tosome power. Therefore, they are dismissed in the Born–Oppenheimer ap-proximation. However, these terms become relevant at nuclear configura-tions where several electronic states have the same potential energy. To im-prove the Born–Oppenheimer approximation it is common to couple severalnuclear Schrödinger equations for different electronic states. The nuclearwave functions for each electronic state are collected in a vector ((R) =(( 1(R), . . . ,( n(R))T . Usually, one applies a so-called diabatic transforma-tion and models the coupling of the various electronic states by potential terms.For a two-state system, for instance, the Hamiltonian becomes the block ma-trix

H ="

T +V 1 VcV H

c T +V2

#,

where V1/2 is the diabatic potential energy surface (PES) of state 1 or 2, re-spectively, and Vc models the coupling of the two PES in the diabatic frame-work. A PES for a certain state is the sum of the internuclear repulsion andthe eigenenergy.

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A typical feature of a PES is that it goes to infinity as the internuclear dis-tance approaches zero due to nuclear-nuclear repulsion. Moreover, as the dis-tance becomes large and the nuclei become too far from each other to interact,the PES approaches an asymptotic value. Many PES have a minimum valueand there exist one or more vibrational eigenstates that are bound, that is themolecular bond is stable, whereas others are monotonically decreasing forincreasing distance and all eigenstates are unstable so that dissociation even-tually occurs.

2.2 Interactions with Time-Dependent FieldsGiven precomputed values for the electronic spectrum Eel(R) in the diabaticrepresentation, we can form the molecular Hamiltonian describing the dynam-ics of the nuclei under the Born–Oppenheimer approximation. In addition, weoften do not only want to consider the dynamics of an isolated molecule but itsinteraction with a time-dependent field. Applications we have in mind are, forinstance, multiple pulse optical spectroscopy or control of molecular dynamicsby laser fields (cf. Ref. [135]). Optical spectroscopy can be applied to analyzethe structure of complex molecules like biopolymers, and the manipulation ofmolecular dynamics can initiate chemical reactions.

The period of nuclear vibrations in a molecule is on the time scale of tens orhundreds of femtoseconds. It was thus with the introduction of femtosecondlasers in the mid-1980s that it became experimentally possible to track andeven manipulate the dynamics of nuclei in molecules [101]. Zewail and hisgroup probed various molecules with femtosecond pulse pairs [151], and lateralso demonstrated how molecular states could be controlled with the help oftime-delayed pulses [110].1 Fleming and co-workers [120] used time-delayedpulses with controlled phase for manipulating molecules. The groups of Crim[130] and Zare [22] attempted to use tunable laser-pulses where the wave-length could be varied for steering chemical reactions. Advances in pulseshaping techniques based on grating and filtering by, amongst others, Weinerand co-workers [63, 141], spurred the development of femtosecond chemistry.The recent development of attosecond lasers also facilitated physicists to probethe motion of electrons, taking place on the atto-scale, cf. Ref. [80].

To model interactions of a molecule with a femtosecond laser field, themolecular Hamiltonian (denoted !H0 in the following) is augmented by an ad-ditional term representing the interaction with the electromagnetic field. Aweak field can be modeled classically using the dipole approximation. Given

1A. H. Zewail was rewarded the 1999 Nobel price in Chemistry “for his pioneer-ing investigations of chemical reactions on the time-scale they really occur.” (quotefrom the extended version of the press release of the Royal Swedish Academy ofSciences, see www.nobelprize.org/nobel_prizes/chemistry/laureates/1999/advanced-chemistryprize1999.pdf, September 4, 2012.)

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internuclear distance

pulse

target state

initial state

Figure 2.1. Schematic configuration of the interaction of a molecule with a time-dependent field *(t). The ground state (dashed line) is coupled to an excited state(dash-dot line) with a laser field. The second excited state (dotted line) is coupled tothe first excited state by a static crossing of the PES.

the laser field *(t) and the transition dipole moment µ , the (nuclear) time-dependent Schrödinger equation (TDSE) reads

ih!! t

((R,t) =$H0(R)+ µ(R)*(t)

%((R,t). (2.3)

Fig. 2.1 shows a typical configuration where we have a molecule in its groundstate and want to excite it to a target state with the help of a femtosecond pulse.

2.3 Physical and Mathematical PropertiesIn this section, we discuss some fundamental properties of the physical modelthat also play an important role for the design of numerical methods. Firstof all, total probability in a closed quantum mechanical system is conserved.This means physically that no matter is destroyed or created. Probability con-servation can be expressed as

&|((R,t)|2 dR = 1 for all t,

i.e., the L2 norm of the wave function is conserved.An important property of the evolution of the TDSE is time reversibility.

No information is destroyed, as it would be if, e.g., diffusion were involved.Hence, if the Hamiltonian is known for the studied time interval, we can re-construct the initial state from the final state. In the mathematical formulation

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of the TDSE, the self-adjointness of the Hamiltonian (together with the imag-inary unit) assures norm conservation and time reversibility.

Since classical Hamiltonian systems are well-studied, it is of interest thatthe TDSE can be rewritten as a classical system by defining the configurationvariable q(R,t) =

$2hRe(((R,t)) and the canonically conjugate momentum

p(R,t) =$

2hIm (((R,t)) (cf. Ref. [53]). An important conservation law forclassical Hamiltonian systems is area conservation for the flow. Such a flow iscalled symplectic. This property in the classical version of the TDSE assuresnorm conservation of ( (cf. Ref. [89]).

2.4 Numerical ChallengesThe simulation of molecular processes based on the chemical model discussedin the previous chapter entails several challenges. Most standard numericalmethods for discretization of the spatial variables of a partial differential equa-tion are grid-based, i.e., one distributes nodes in a more or less regular patternover the computational domain and evaluates the solution in those points. Oneintricacy with such a discretization is the fact that the number of independentvariables increases linearly with the number N of atoms in the system. Moreprecisely, the total number of degrees of freedom is 3N, i.e., three coordinatesfor each nucleus. Six degrees of freedom are external2 (three translational andthree rotational), giving 3N " 6 internal degrees of freedom [135, Sec. 12.4].Only the internal degrees of freedom are of importance for a quantum descrip-tion of an isolated molecule. For a grid based model, the number of grid pointsgrows exponentially with the dimensionality: When n grid points per dimen-sion are used, the number of grid points becomes n3N"6. Hence, only verysmall molecules can be handled using such numerical schemes.

Adaptive mesh refinement (cf. [33], Paper III), sparse grids (cf. [54, 50]),and parallelization (cf. [15, 57], Papers V and VI) are options to increase theproblem size that can be simulated slightly. Also, using radial basis functionsas proposed in Paper VII is an option to reduce the size of the discrete system.An alternative to full quantum simulations is to use semiclassical methods thatcontain further modeling. As opposed to purely classical models, the particlesare still represented by a wave function. The basis functions are not definedby some abstract approximation space but their parameters are determined by(classical properties of) the problem. Heller [62] introduced the concept offrozen Gaussians where the initial wave packet is represented by several com-plex Gaussians with fixed width, so-called coherent states, whose centers andphases are propagated according to classical trajectories. This method hasbeen refined in many ways. Within the MP/SOFT framework [149] more co-herent states are successively added and their parameters are optimized. The

2Note that the number of internal degrees of freedom is 3N "5 for linear molecules where therotation along the molecular axis collapses.

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group of Martínez has refined semiclassical methods to also cover more com-plicated quantum effects [11] by a multiple spawning technique. In [37] semi-classical propagation with Hagedorn wave packets was proposed where thepropagation according to the harmonic oscillator is corrected by a perturba-tion term. The method of coupled coherent states [126] uses Monte-Carlosampling of trajectories. A different ansatz is provided by the multi configu-ration time-dependent Hartree (MCTDH) method [100] which is based on adecomposition of the wave packet into a product of single particle functionsand the high-dimensional linear PDE is reduced to a number of low dimen-sional nonlinear PDEs. In this way, larger molecules can be studied if theyexhibit a certain structure. Despite much progress in this area, difficulties canstill arise when e.g. tunneling occurs or static or dynamic couplings are con-tained in the model.

The wave function does not only depend on the spatial coordinates but alsoon time. Since multiple time scales are usually present in the model, a hugenumber of time steps is often necessary. High-frequency oscillations have tobe resolved or modeled. High order and adaptive integrators or multi-scalealgorithms can help in reducing the number of time steps. There are vari-ous sources for high-frequency oscillations depending on the particular typeof application. In the semiclassical scaling, oscillations occur due to highfrequencies in the spatial variables. For simulations of interaction with a time-dependent field, the oscillations of the laser field pose limitations to the stepsize. A simple multi-scale model, allowing for larger time steps, is the so-called rotating wave approximation (see Sec. 3.2.2). A review of numericalintegrators for highly oscillatory Hamiltonian systems was provided by Co-hen et al. [28].

A third potential difficulty when constructing numerical methods is the un-boundedness of the spatial domain in which the TDSE is posed. For a wideclass of problems the wave function stays within a certain domain, i.e., themolecule stays intact. In such cases, the computational domain can be trun-cated and the PDE is usually closed by periodic or homogeneous Dirichletboundary conditions. However, the modeling of dissociative processes posesdifficulties when the domain is truncated and parts of the wave packet leavesthe computational domain. Standard methods to handle dissociation from thecomputational domain are complex absorbing potentials [95, 55] and perfectlymatched layers [12, 1, 103]. In both cases, artificial damping is introduced tothe system and physical properties like norm conservation and time reversibil-ity are no longer valid in the computational domain. Therefore, numericalmethods designed for bound states can be difficult to generalize to dissocia-tive problems. On the other hand, particular pseudospectral methods havebeen designed for unbounded domains (see e.g. [140, 102]). A similar alter-native discussed in Paper VII is to use globally defined radial basis functionswithout any boundary conditions. However, these methods still need to resolvethe whole wave packet to some extent.

16

The challenges discussed so far concern solving one single TDSE prob-lem. When this equation appears within an optimization loop, e.g., for findingthe optimal shape of an interacting laser pulse, additional complexity is intro-duced. The Schrödinger equation has to be solved many times, and moreoveran adjoint equation might have to be computed to calculate optimality condi-tions. This introduces extra demands on both computing power and memorysize.

17

3. Discretization of the Schrödinger Equation

For a computer simulation of the TDSE, we have to discretize spatial andtemporal variables. Preferably, the discretization should satisfy the same con-servation properties as the continuous problem (cf. Sec. 2.3). Moreover, it isdesirable that the method can easily and efficiently be generalized to multipledimensions. In this chapter, we review common methods from the literaturewith a special emphasis on the methods applied in the attached papers.

3.1 Spatial DiscretizationSince the solutions of the time-dependent Schrödinger equation are generallyof class C+, high-order methods are more efficient considering both comput-ing time and memory requirements. Therefore, pseudospectral methods arecommonly used in the field. While these methods work very well in a stan-dard setting, stencil-based methods and radial basis functions are more flexiblealternatives.

3.1.1 Pseudospectral MethodsThe idea of spectral methods is to represent the approximate solution as aweighted sum of basis functions with global support. When a collocationapproach based on this weighted sum is used, one is talking about a pseu-dospectral method [48, 44]. Pseudospectral methods have been introduced asdiscrete variable representation (DVR) to quantum dynamics [87, 86]. Colbertand Miller [29] introduced the sinc-DVR and Feit & Fleck [41] and Kosloff &Kosloff [77] proposed the dynamic Fourier method where the Hamiltonian isnot computed explicitly but the action of the kinetic energy operator is eval-uated via the Fast Fourier Transform. While the Fourier method is limited tobound-state computations where the computational domain can be truncated,pseudospectral discretizations based on Hermite functions [14, 36] operate ininfinite domains. Especially, the dynamic Fourier method is very popular inthe quantum dynamics community due to its high accuracy and fast evalu-ation. Moreover, it can nicely be combined with the so-called split operatortime-evolution method (see Sec. 3.2.1) yielding an easy-to-implement full dis-cretization.

Pseudospectral methods are generalized to higher dimensions by tensorproducts which makes adaptive refinement difficult. However, one does not

18

need to use the full grid. Instead one can use sparse representations basedon the hyperbolic cross. Hallatschek [61] devised a sparse grid algorithm forthe Fourier method which was applied to the Schrödinger equation by Gradi-naru [49]. Also, Griebel and Hamaekers [54] considered sparse grids for theSchrödinger equation. Attempts have been made to use the Fourier methodmore efficiently by transforming the coordinates according to the specific po-tential. This procedure is called the mapped Fourier method [39] and wasextended to time-dependent problems by Kleinekathöfer and Tannor [75].

3.1.2 Localized MethodsLocalized stencils as in finite differences or finite elements allow for moregeneral grids with refinement adapted to the solution (see Ch. 4). Moreover,localized methods reduce the data dependencies for derivative computations toa number of neighboring points compared to the global dependency in pseu-dospectral derivative approximation. For a given number of grid points, local-ized methods are, of course, less accurate, but their superior properties whenit comes to parallelization and adaptivity can nonetheless make high orderstencils competitive. Particularly finite differences have attracted attention forthe solution of the TDSE. A drawback of finite difference methods comparedto the Fourier method is the fact that the dispersion relation is only approxi-mately recovered. Gray and Goldfield [51] have therefore devised dispersionfitted finite differences.

The convenience of finite differences is their ease of implementation andthe good conditioning of high-order stencils [56]. Finite element methods, onthe other hand, are mostly used with comparably low order. However, thereis a variant—referred to as spectral elements—that is suited for use with highorders [73]. When using high order elements, the nodes have to be clusteredtowards the element boundary in order to achieve a well-conditioned approx-imations and avoid effects from the Runge phenomenon. One such type ofelements is the Lagrange element with support points distributed according tothe nodes of the Gauss–Lobatto quadrature rule. This type of element alsocomes with a second advantage: One can use Gauss–Lobatto quadrature withnodes corresponding to the interpolation points when evaluating the mass andstiffness integrals which gives a diagonal mass matrix. In this way, the inte-grals are only approximated but the approximation order of the quadrature ruleis equal to or higher than the approximation order of the finite element. Thistype of elements was used for the solution of quantum scattering problems byManolopoulos & Wyatt [96] and Rescigno & McCurdy [112]. With the ef-ficient implementation framework proposed in Paper VI and [82], high-orderfinite elements can be almost as narrow as finite differences on equi-spacedgrids while at the same time allowing for mesh refinement without loss in theapproximation order (cf. Chs 4 and 5).

19

3.1.3 Radial Basis FunctionsApproximation using radial basis functions (RBF) [144, 38, 24] is an alter-native to grid-based methods. In this method, the basis functions are radialfunctions centered around a set of points that can be arbitrarily distributedover the computational domain. The solution is approximated by a weightedsum of the basis functions and the partial differential equation can be solvedby either choosing a collocation or a Galerkin ansatz.

There are various types of radial functions that are commonly used for RBF-based approximations. Choosing a globally supported and globally smoothfunction, one gets a spectrally convergent method [144, 114]. Radial basisfunction discretization based on a Galerkin formulation of the partial differen-tial equation was analyzed in [143]. Compared to RBF-collocation [72], how-ever, RBF-Galerkin has received much less attention so far. This can primarilybe attributed to the fact that a Galerkin formulation requires the evaluation ofintegrals. In Galerkin formulations for finite elements this is usually done bynumerical quadrature. For spectral methods, however, the use of numericalquadrature would spoil the approximation order. In our special setting forthe TDSE, however, analytical evaluation is a viable alternative: When usingGaussian basis functions the computational domain does not have to be trun-cated and there are closed-form expressions for the integrals involved in thesesettings—at least as long as the potentials are polynomials or Gaussians.

Collocation, on the other hand, is easier to implement, but finding a stablediscretization is more intricate. Problems with eigenvalue stability related toboundary treatment were pointed out by Platte and Driscoll [108]. For theTDSE, we do not have to bother with boundary conditions. Nevertheless, wehave encountered stability problems due to the combination of second deriva-tives and the potential term. Fornberg and Lehto [45] described a stabilizationprocedure based on artificial diffusion. The problem with this remedy, how-ever, is that damping is introduced to the whole spectrum. Hence the eigenval-ues still do not represent the correct physics and we prefer an RBF-Galerkinansatz for the TDSE. In Paper VII, we analyze the convergence properties ofour Galerkin ansatz without explicit boundary treatment. When reducing thefill distance between the nodes in a fixed domain, exponential convergence isshown up to a point where the error flattens out due to the systematic errorfrom the part of the domain where no basis functions are centered. Numer-ical experiments show that RBF discretizations are more accurate than thedynamic Fourier method already for an equidistant node distribution. On theother hand, dense matrices have to be handled for RBF computations, andthe efficiency must be improved to actually outperform the dynamic Fouriermethod. Conceivable approaches are truncation or some other type of local-ization or the application of fast transforms.

20

3.2 Time EvolutionAs discussed in Sec. 2.4, the temporal dimension has to be treated with extracare since the number of time steps needed is usually much larger than thenumber of grid points per dimension. Throughout this section, we considerthe semi-discretized TDSE,

ddt

u(t) = " ih

H(t)u(t), (3.1)

where H denotes a matrix that represents the Hamiltonian in the approxima-tion space and u = (ui) is the coefficient vector representing the wave function.

There is a large spectrum of methods designed for time-evolution of ODEsystems. For our purpose, an explicit method1 is preferable since solving alinear system can become very costly and memory consuming when high-order spatial methods in high dimensions are involved. The monographs [59,85, 117] present methods specially suited for Hamiltonian systems. There aretwo families of methods that are most often used for the TDSE, exponentialintegrators and partitioned Runge–Kutta (PRK) methods [91] which we bothwill discuss in more detail below.

Note that, in this thesis, we only consider the case where short time steps arerequired because we assume that the Hamiltonian is explicitly time-dependent.The situation with a time-independent Hamiltonian has been studied exten-sively, see the comparative study by Leforestier et al. [84].

3.2.1 Suitable Integrators – a ComparisonExponential integratorsFor a linear ODE with a time-independent right-hand side, i.e., for the semi-discretized TDSE with time-independent Hamiltonian, the evolution operatoris analytically given as the exponential of the Hamiltonian matrix times thetime span. The situation becomes more complicated for time-dependent right-hand sides but, for sufficiently small time intervals, the Magnus expansion[18, 94] provides an expression of the evolution operator U(t +&t,t), definedby u(t + &t) = U(t + &t,t)u(t), in the form of the exponential of a series ex-pansion,

U(t +&t,t) = exp

'

%l%0

,l

(.

Each term ,l contains l + 1 integrals over l commutators of the Hamiltonianat different points in time. Since the lth (l > 0) term decays as (&t)2& l

2 '+3 forsufficiently small time steps, a truncated version of this expansion provides anatural starting point for numerical evolution methods.

1A method that does not involve the solution of large linear equation systems, e.g., throughfactorization or iterative solvers.

21

After truncating the Magnus expansion, the matrix exponential has to beevaluated. The simplest approach is the split operator [40]. The method isa second-order accurate exponential Strang splitting where the Hamiltonianis split into potential and kinetic energy operators and the time-dependenceis evaluated at the mid-point. This form is appealing if we have a fast trans-formation between the diagonal representations of the potential and the kineticenergy operators. This is the case for a Fourier spectral approximation in spacewhere derivative computations are diagonal operations in Fourier space andwhere the Fast Fourier Transform represents a computationally cheap trans-formation from coordinate to Fourier space.

A more flexible alternative is to use a Krylov subspace method for computa-tions of the action of the matrix exponential. For a symmetric Hamiltonian ma-trix, coming from an arbitrary spatial discretization method, the Lanczos algo-rithm can be used and for non-symmetric Hamiltonians, that can for instancearise if absorbing boundaries are applied, the computationally more expensiveArnoldi method. Krylov methods can also be combined with a higher-ordertruncation of the Magnus expansion to get more accurate integrators. Evaluat-ing higher Magnus terms is quite costly, though, since an increasing number ofcommutators has to be computed. Several attempts have been made to simplifythe terms. Blanes et al. [19] rewrite the expansion reducing the total numberof commutators for truncation up to a certain order and Blanes & Moan [21]split the matrix exponential to completely avoid commutators (with the priceof several exponentials to be computed per time step).

However, one can use the special structure of the Hamiltonian in the caseof a matter-field-interaction problem where the spatial dependence in the tran-sition dipole moment is usually rather weak. In this case the second orderMagnus term simplifies to a block-diagonal matrix. Hence, increasing the or-der of the truncation from order two to four can be achieved with very littleadditional computational effort. This is proposed in Paper I.

Runge–Kutta methodsA second class of integrators relies on the fact that the complex Hamilto-nian system of the TDSE can be transformed into a real Hamiltonian systemas pointed out in Sec. 2.3. So-called partitioned Runge–Kutta methods aresymplectic methods of Runge–Kutta type that are designed to mimic the con-servation laws exhibited by classical Hamiltonian systems. However, mostPRK methods are implicit. Explicit PRK methods are available for separableHamiltonian functions where H(p,q) = H(p) + H(q). If the Hamiltonian isexplicitly time-dependent, this separation cannot be made. However, Gray andVerosky [53] exploited the Magnus expansion to keep the method explicit fortime-dependent Hamiltonians. Sanz-Serna and Portillo [118] proposed a moreelegant procedure, by introducing an additional conjugate pair of variables torepresent the time dimension. The separation also fails in case the Hamilto-

22

nian is complex-valued. This is, e.g., the case when modeling dissociation byadding complex terms to the potential or the kinetic energy.

By varying the number of stages, it is possible to design optimal Runge–Kutta coefficients for special problems in terms of accuracy and stability.Suzuki [132, 133, 134] and Yoshida [150] demonstrated how to construct highorder symplectic methods. Later McLachlan [99] came up with the idea toconstruct methods with an optimal error constant for given order and Blanes& Moan [20] refined this concept. In Refs. [17] and [52], optimal coefficientsfor TDSE problems have been devised, even though both articles only provideexamples with time-independent Hamiltonians.

In Paper I, the numerical propagation for a model of the rubidium diatom(Rb2) is studied. In these calculations, the methods proposed in Ref. [20]perform better than those tailored to the TDSE [17, 52]. However, exponentialintegrators are even more efficient for low to moderate accuracy requirements.

3.2.2 Rotating Wave ApproximationThe simulation of matter-field interaction requires the resolution of the oscil-lations of the laser pulse. This necessitates small time steps. For computationswith a low accuracy requirement, this shortage can be overcome with the helpof the rotating wave approximation (RWA) [111]. Consider a two state systemwith Hamiltonian

H ="

T +Vg(R) f (t)cos(-t)f (t)cos(-t) T +Ve(R)

#,

where - is the frequency of the laser field and f (·) a slowly varying envelopefunction. Define the transformation

W :="

I 00 e"i-t · I

#,

of the wave packet. Then, the TDSE for the transformed wave packet . :=W"1( reads

ih!.! t

="

T +Vg(R) 12 f (t)

$1 + e"i2-t%

12 f (t)

$1 + ei2-t% T +Ve(R)" h-

#. .

If the laser frequency is very high compared to the variations of the envelope,the time-averaged influence of the terms e±i2-t is insignificant. In the RWA,this term is dismissed. When propagating the solution of the RWA-TDSE, theoscillatory frequency of the laser field need not be resolved. This correspondsto a separation of scales where effects on the scale of #/- and below areneglected. A detailed discussion of the modeling error introduced by the RWAcan be found in Paper I.

23

10 2 100 102 10410 10

10 5

100

105

step size [fs]

erro

r

GaussFilonRWAFilon RWT

Figure 3.1. Matter-field interaction for the rubidium diatom solved with Magnus–Lanczos propagator of order four. The integrals in the Magnus expansion are inte-grated with Gauss or Filon-type quadrature, respectively. The methods start converg-ing once &t is below the period of the oscillation of the laser field (verticle line). TheRWA gives good results first but flattens out. Filon-type integration applied after therotating wave transformation (Filon-RWT) yields slightly improved results comparedto the RWA.

The RWA is suitable for low accuracy computations where we do not needto resolve the oscillations of the laser pulse with the time step. Clearly astraight-forward discretization with a standard method for numerical quadra-ture fails to give a meaningful approximation if the oscillations are underre-solved. On the other hand, there are so-called Filon-type methods for highlyoscillatory quadrature [69] that are more suitable to use in this regime. Still,the RWA is considerably more accurate but Filon-type integration can slightlyimprove the result when applied after transformation with W (see Figure 3.1).

24

4. Error Estimation and Adaptivity

In the previous chapter, we have discussed several methods that are suitablefor the numerical solution of the time-dependent Schrödinger equation. Onemajor goal of the research discussed in this thesis is to provide an efficientsolver that adapts the discretization to the shape of the solution and providesthe solution with a certain prescribed accuracy. In order to design such asolver one needs to have an error estimate at hand and a way to discretize theHamiltonian for unevenly distributed resolution.

4.1 Temporal AdaptivityIn order to design an algorithm that automates the choice of the time stepand is capable of meeting a given error tolerance, adaptive step size control isdesirable. A theoretical study of accuracy and convergence rates for severalstandard time-marching methods for the TDSE was provided by Lubich [92].However, those estimates are not easily computable within a step size controlalgorithm.

Within the framework of Runge–Kutta methods, efficient techniques forlocal error control have been designed based on embedded methods of twoconsecutive orders. Admittedly, a straight-forward implementation of PRKmethods with variable step size suffers from the fact that symplecticity is lost(cf. Ref. [117]). Sophisticated techniques have been developed to circumventthis problem (cf., e.g., Ref. [16, 60]) which are, however, strongly dependenton a good choice of a parameter function.

For the Magnus expansion, on the other hand, an easy-to-compute errorestimate based on extrapolation is available [19] and has been successfullyapplied to the TDSE (cf. Ref. [145] and Paper II). This can be combined witha Lanczos algorithm that chooses the size of the Krylov subspace to meet thesame tolerance. The Lanczos error can be estimated according to Ref. [67],similar to what is used to stop iterative solvers like conjugate gradients [115].

4.2 Spatial AdaptivityWhile pseudospectral methods can only be adapted to a specific problem bytransforming the coordinates (cf. Sec. 3.1.1), finite elements and finite differ-ences allow for more general grid adaptation. Although finite elements with

25

tetrahedral meshes allow for arbitrary mesh refinement on unstructured gridswithout hanging nodes, we prefer to use structured refinement to benefit fromthe reduced memory consumption. Also, the additional degrees of freedom in-side each tetrahedron for high order exhibit some structure anyway. The meshis split into patches of different refinement level. Interfaces between thesepatches can either be treated continuously with constraints on degrees of free-dom that are only active on one side of the interface [113] or discontinuouslywith penalty terms. The latter procedure is usually referred to as discontin-uous Galerkin [65] and used in conjunction with penalization over all faces.However, this comes with computational and memory overhead compared tothe methods used in Papers III and VI.

Finite differences, on the other hand, are very easy to implement on equidis-tant grids, but adaptive mesh refinement is more intricate to achieve in a stablemanner. For the treatment of domain boundaries, the so-called summation-by-parts technique was proposed by Kreiss and Scherer [81] to achieve stablediscretizations. This construction can also be used to treat interfaces betweenblocks with different refinement levels both in a continuous [79] and a dis-continuous setting [97, 104]. However, order reduction around interfaces andespecially corner points is problematic.

While adaptivity calls for elaborate treatment of irregularities in the meshwhen using grid-based methods, radial basis functions are mesh tolerant. Inprinciple, one can distribute the centering points in an arbitrary pattern and thetheory is naturally formulated for scattered data (cf. [24, Ch. 5] and [38, Ch.1]). However, ill-conditioning can appear when the nodes are clustered tooclosely. This calls for scaling of the shape parameters which in turn deterio-rates the convergence. Optimal scattering of nodes in radial basis function dis-cretization is an active area of research and many questions are still open. Anidea in one dimension is to equidistribute the points according to the arclengthof the solution [119]. This is, however, difficult to generalize to higher dimen-sions. A mathematically rigorous but computationally costly framework is thegreedy algorithm by Ling and Schaback [88] which phrases the problem ofscattering the nodes as an optimization problem. Driscoll and Heryudono [34]propose a subsampling method where the shape parameter is proportional tothe distance to the nearest neighbor despite the loss of spectral convergence.In the setting of the Schrödinger eigenvalue problem, Degani [31] proposesan alternative strategy that places the points according to the potential in theHamiltonian, i.e., relying on properties of the equation instead of the solution.On the sphere, where no boundaries have to be taken into account, it has beenproposed to redistribute the points based on an electro-static repulsion modelof the shape of the solution [43].

26

4.3 Residual-Based Error Estimation and Global ErrorControl

For both temporal and spatial discretizations, much research has been on de-signing error estimates. For spatial discretization, various techniques for errorestimation appear in the literature. There are strategies that are only basedon the value of the solution, some on derivative information of the solution,and others on the residual. Multiresolution analysis provides a tool to analyzethe properties of the solution and can be used to identify regions where thesolution contains more information and regions where it contains less. Suchwavelet analysis was applied by Jameson [70] to adapt finite difference stencilsto the solution. Error estimation based on derivatives in an a priori estimatewas analyzed by Eriksson & Johnson [35].

All of these strategies examine errors at a given point in time only, butfor time-dependent problems one wants to know how these errors influencethe result at the final simulation time. A posteriori error estimates that arebased not only on the residuals but also incorporate information from a dualproblem fill this gap. A duality argument was first proposed by Babu!ka andMiller [3] and refined by Becker and Rannacher [9, 10]. While an a posteriorierror estimation can in principle be derived for any discretization technique,the built-in Galerkin orthogonality of the finite element method makes it apowerful tool. Since the error is perpendicular to the approximation, it ispossible to estimate the error with which each cell contributes to the globalerror. For finite differences, for instance, such a posteriori estimates can onlymake a meaningful prediction of the global error. Paper IV employs the globalestimation. Duality-based error control can also be formulated for ordinarydifferential equations, see [27].

Papers II and III are devoted to the design of a reliable simulation methodfor the TDSE. For the numerical solution u of the TDSE at the final time t f ,we consider a functional, represented by ' , of the total error, e(t f ) := ((t f )"u(t f ). Defining the dual TDSE

ih!! t

/(t) = H/(t), /(t f ) = ' , 0 ( t < t f ,

we can express the error functional as

)' ,e(t f )* =& t f

0

")/(t),

!! t

e(t)*+ ) !! t

/(t),e(t)*#

dt, (4.1)

where )·, ·* denotes the L2 inner product. For a finite element semi-discretiza-tion satisfying a Galerkin orthogonality relation, equation (4.1) becomes

)' ,e(t f )* = (4.2)& t f

0

"ih))/+(t),

h2

2M)u(t)*+ i

h)/+,V (t)u(t)*+ )/+(t),

!! t

u(t)*#

dt,

27

Figure 4.1. Final solution on an adaptively refined mesh. Simulation of matter-fieldinteraction in OClO with Gauss–Lobatto elements of order 7.

where /+ denotes the part of the dual solution in the orthogonal complementof the approximation space. As mentioned in Sec. 3.1.2, we use a specialtype of elements, Gauss–Lobatto elements, in Paper III, and a discretizationbased on these elements does not satisfy the weak equation exactly in theapproximation space. However, the truncation due to inexact quadrature is notof leading order (for elements of order greater than three) and can therefore beneglected in a first-order error estimate.

When discretizing in time as well, we get an additional term in the errorexpression (4.2) accounting for the perturbation due to numerical time prop-agation. Since there are very efficient methods for temporal propagation ofthe TDSE and we expect the number of time steps to exceed the number ofgrid points per dimension for the application we have in mind, we do not usefinite elements for discretization of the time variable. Instead, we choose aMagnus–Lanczos propagator. When combining temporal and spatial adaptiv-ity in such a mixed discretization method, care has to be taken because refine-ment in space can influence the temporal error (cf. [66]). Therefore, we pro-pose an adaptive algorithm that handles the temporal adaptivity independently.We consider the spatially semi-discretized TDSE as a system of ordinary dif-ferential equations and apply the theory developed by Cao and Petzold [27].Exploiting the linearity and the norm conservation of the TDSE, we derive ana posteriori estimate that is independent of the solution of the dual problem,see Paper II. Figure 4.1 shows the final solution and the adapted mesh of asimulation of the OClO molecule with the algorithm described in Paper III.

When designing an adaptive algorithm that should control errors from var-ious sources, it is important to balance the different error terms. Paper IV

28

considers such error balancing for the error due to numerical approximation intime and space and an additional error from domain truncation that becomesimportant when dissociative configurations are considered. The dissociativepart of the wave packet is handled by a perfectly matched layer as describedin [103]. An argumentation as in Paper II is used to dismiss the dual problemeven though the norm of the dual is no longer conserved due to the artificialdissipation in the layer. However, it is argued that the error estimate is still ef-ficient if one makes sure that the dual problem and the computational domainare chosen such that the interesting features of the problem take place withinthe computational domain.

While residual-based error estimation has proved to be quite effective forlocalized methods, it is more intricate in a setting with spectral methods: Sincelocal changes exhibit a global effect, it is difficult to predict the effect of abasis change. Also boundary effects have to be taken into account. We havetested straight-forward residual-based error estimation for radial basis functionapproximation with little success.

29

5. High-Performance Implementation

Since applications in quantum dynamics are large-scale problems, an efficientand parallel implementation is essential. In most cases, memory is the limit-ing factor. A common strategy is therefore to decompose the spatial domaininto partitions, distributed between nodes in a computer cluster and to let eachnode perform the simulation on its share. However, the simulation on the dif-ferent parts of the domain are not independent since derivative computationsuse information from neighboring cells. Therefore, the nodes have to com-municate and exchange information. Since communication is generally moreexpensive than computation on contemporary computer architectures with in-terconnected nodes (clusters), care has to be taken when implementing dataexchange. It is preferable to overlap data exchange with computation when-ever possible. In order to optimally exploit parallelism on all levels, hybridparallelization has to be considered in addition. The implementations con-sidered in this thesis include three levels of parallelism: Distributed-memoryparallelization with the Message Passing Interface (MPI), shared-memory par-allelization with Intel Threading Building Blocks (TBB) or OpenMP, and ex-plicit vectorization.

Two examples of software projects that are targeted to computing the solu-tion of the full time-dependent Schrödinger equation in a parallel setting arePyProp [15] and HAParaNDA [57]. The former was developed by Birkelandand is maintained at the Bergen Computational Quantum Mechanics Group.The code is based on a tensor product discretization which allows for combi-nations of different types of approximations—spectral and localized—for theindividual independent variables of the problem. The HAParaNDA project byGustafsson, on the other hand, is focussed on finite difference approximations.Currently, it supports time-stepping based on the Lanczos propagation meth-ods including the step-size control devised in Paper II. The spatial discretiza-tion is built on equally sized blocks with ghost layers. The parallelization is ahybrid of OpenMP and MPI. Paper V relies on the HAParaNDA framework.It analyzes the impact of communication in the Lanczos time-stepping algo-rithm. Paper VI discusses a stencil-based implementation of finite elements(published in the deal.II library [7]) that is similar to finite difference stencils.The focus of the paper is on strategies for shared-memory parallelization. Theexperiments reported in Paper III are based on this implementation.

30

5.1 Communication-Avoiding Time SteppingOne way to reduce the overhead due to communication is to use a temporalpropagation method where several matrix-vector products can be computed in-dependently from each other. Most common numerical propagators, however,construct the approximation of the new time step with a number of stages thatare constructed in an iteration where each stage depends on the previous ones.In the Lanczos algorithm, we moreover have to compute inner products thatrequire all-to-all communication which is considerably more expensive thanthe nearest-neighbor communication needed in sparse matrix-vector products.

Each stage of the Lanczos algorithm in its standard formulation requiresone matrix-vector product as well as two inner products that require commu-nication. Paige [106] has considered rearrangements of the algorithm in a waythat is less efficient for serial implementations (since one extra vector update isnecessary) but avoids one global synchronization point per stage by clusteringthe two inner products. The so-called s-step Lanczos [74] method goes onestep further: It really modifies the algorithm with the aim of clustering blocksof s stages of the Lanczos algorithm between synchronization points. Thisreduction in communication comes with the price of one extra matrix-vectorproduct as well as the fact that the resulting algorithm is unstable in floating-point arithmetics when the Krylov subspace becomes too large. It is thereforeessential to control the error and to choose the Krylov subspace adaptively.

In Paper V, we have tested these communication-reducing Lanczos vari-ants in the setting of a temporal propagator with adaptively chosen size of theKrylov subspace. However, we do not see a clear advantage of the communi-cation reduction in our experiments. The result was rather similar run-timesfor all three Lanczos variants with a slight advantage for the few-synchroniza-tion Lanczos. An enhancement of the s-step Lanczos method is the so-calledcommunication-avoiding Lanczos method proposed by Hoemmen [68]. Thenumerical features of this algorithm were further investigated by Gustafssonet al. [58].

5.2 Stencil-Based ImplementationHigh-order methods are preferable for the TDSE since they keep the numberof variables needed low. Also, the number of arithmetic operations involvingeach degree of freedom (DoF) increases with the order which reduces memorytransfer in relation to the number of computation. This comes, however, withthe price that each DoF couples to a larger number of other DoFs for derivativeapproximations. Therefore, a sparse-matrix representation of the Hamiltonianwill contain many entries per row. Storing this matrix requires much morememory than a vector storing the solution data. Hence, memory consumptionwould be dominated by the matrix, and it is essential to rely on an implemen-tation that does not form the matrix explicitly. While it is relatively common

31

1 2 3 4 5 6 8 1010

1

102

103

104

degree of finite element

flops

per

DoF

sparsematrix free

(a) Computational complexity.

1 2 3 4 5 6 8 1010

2

101

100

101

degree of finite element

wal

l clo

ck ti

me

sparsematrix free

(b) Wall clock time on Nehalem EP.

Figure 5.1. Comparison of complexity per degree of freedom (DoF) and wall timein seconds per matrix-vector product for a 3D Laplacian (problem size fixed at 1.77million DoFs) for Gauss–Lobatto elements with matrix-free and sparse-matrix imple-mentations.

to use stencil-based implementations for finite difference methods, finite ele-ments are mostly implemented by assembling a matrix in the beginning of thecomputation and then solving the problem using this matrix. This formulationis chosen in most common open-source projects for generic finite element pro-gramming like, e.g., deal.II [7] or FEniCS [90]. Specialized software targetedat spectral elements, on the other hand, choose a matrix-free implementation.Examples are SPECFEM 3D [76, 8] and Nektar++ [137, 138]. However, thesepackages do not support refined meshes with hanging nodes. In [82], datastructures for matrix-free implementation of finite elements within the deal.IIframework were proposed. These structures provide the framework for theexperiments with adaptive finite elements reported in this thesis.

For high order finite elements in high dimensions, a stencil-based imple-mentation does not only reduce memory consumption but the computationalcomplexity, too. This is specially relevant for Gauss–Lobatto elements wherethe complexity per degree of freedom is reduced from O(pd) in a sparse-matrix setting to O(d p) for d dimensions and convergence order p. Thespeedup in practical computations is even higher: Sparse matrix-vector prod-ucts are memory bandwidth bound so that higher arithmetic intensity can beachieved for a matrix-free implementation where data is reused. Figure 5.1 il-lustrates complexity and run time for the evaluation of the action of the Lapla-cian in three dimensions. In these simulations, the matrix-free version reachesabout 50% of the peak performance for order ten while the correspondingnumber for the sparse matrix is only about 10%.

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5.3 Hybrid ParallelizationWhen solving partial differential equations on a distributed-memory system,there are two issues that have to be dealt with: the parallelization of the gridstructure and data types that enable communication of vector entries at theboundary of the processor’s domain. The more general the grid structure, themore complicated and memory-consuming becomes the data structure to holdthe grid information. To simplify the grid structure in adaptive mesh refine-ment, Berger and Colella proposed a structured version [13]. For instance,the software projects Chombo [30], SAMRAI [147], and AMROC [32] pro-vide solvers with finite differences on structured adaptively refined meshes.The p4est package [25] handles parallel meshes by a global coarse mesh andprocessor-local hierarchically refined cells. It can be combined with generic(finite element) PDE solver software and there is an interface [6] to the deal.IIpackage.

When the grid structure of the problem is set up, computations boil downto the computation of matrix-vector products or the solution of linear equationsystems. It is very common to rely on linear algebra packages like PETSc[5, 4] or Trilinos [64] to handle communication between remote nodes. Whilethis generally works very well for implementations relying on matrices, thecorresponding data structures suffer from a relatively large overhead in casethe computations are broken down into cell-wise evaluations of the differentialoperator. In this case, it is favorable to explicitly implement communicationusing MPI commands.

In an operator evaluation, computations on the various cells are not inde-pendent since more than one cell contributes to the results for DoFs at elementboundaries. For shared memory computations, we therefore have to preventdata loss which can arise when processes try to write into the same data fieldsimultaneously. In order to avoid this, coloring strategies [115, Sec. 12.4] arecommonly used. Simple coloring suffers from explicit synchronization pointsand poor data reuse in processor caches. In Paper VI, we therefore proposea strategy of partitioning and coloring on two levels which avoids explicitsynchronization. The dependencies form a graph of tasks that are dynamicallyscheduled using Intel TBB. Compared to pure coloring, the cache performanceis also improved.

On a single processing unit, instruction-level parallelism is possible throughvectorized data types. Explicit vectorization is heavily used in the BLAS pack-ages, but is more rare in user code. Matrix-vector products that are evaluatedcell-wise can be vectorized by clustering the instructions for two (or more,depending on the processor’s instruction set) cells [82].

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6. Quantum Optimal Control

In the laboratory, physicists use laser pulses to manipulate the energetic stateof molecules and thereby initiate chemical reactions. They design the laserfields that yield the desired outcome. This chapter is devoted to the questionof how to use simulation and numerical optimization to find a suitably shapedlaser field for a given purpose.

A generic objective would be to find a pulse * that minimizes the function

J1(* ,() =1

t f " t0

& t f

t0)((R,t)|!O|((R,t)*dt, (6.1)

where !O = !O1 ·0 (t " t f )+ !O2(t) and ( is the wave packet solving the TDSE(2.3) for the interaction with the laser field * . The operator !O1 defines sometarget state at final time t f and !O2(t) allows us to include a time-dependentobjective, such as the penalization of a special molecular state (that is unde-sired).

In this form, the problem is ill-posed since there is no restriction on thestrength of the pulse. A strategy to resolve this issue is to introduce a so-calledTikhonov regularization, i.e., to add the term

J2(*) = 1& t f

t0*2(t)dt, 1 > 0,

to the objective function. The constant 1 indirectly controls the total energyof the optimal pulse and has to be chosen in a way so the resulting pulse has astrength that can be achieved with the experimental equipment. The completeoptimization problem now reads

min*!L2([t0,t f ])

J (* ,() = J1(* ,()+J2(*),

subject to ih!! t

( =)

!H0 + µ*(t)*

( , ((R,t0) = (0.(6.2)

The optimality system for (6.2) is given by

ih!! t

( =)

!H0 + µ*(t)*

( , ((R,0) = (0, (state equation)

!! t

/ = " ih

$H0 + µ*(t)

%/ " 1

t f " t0!O1(t)((r,t),

/(x,t f ) = !O2((r,t f ), (adjoint equation)1h

Im ()/ |µ|(*)+2* = 0, (complementarity)

(6.3)

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where / is the adjoint variable. It was shown by Peirce et al. [107] that (6.2)has a solution in the special case where

J1(* ,() = )((t f )"' ,((t f )"'*.

The essential property of J1 in the proof is weak lower semi-continuity, andhence the reasoning in [107] can also be adopted to more general objectives.

The first studies of quantum optimal control problems included only finaltime objectives. In the late 1980s, the pulse shaping problem was formulatedas an optimal control problem and solved with the conjugate gradient (CG)method [78, 107, 128]. The CG algorithm is a standard first order methodwhich may suffer from slow convergence [46]. The efficiency of the numericaloptimization was improved in a paper by Somlói and co-workers [131] whoadopted the Krotov method [83] for quantum control. The Krotov iterationwas refined by Zhu and co-workers [152, 153] and by Maday and Turinici[93]. The idea of this method is to solve the first-order optimality condition(6.3) for * , yielding

*(t) = " 1h2 Im ()/ |µ|(*) .

Since / and ( depend on * , a fixed point iteration is applied to find the optimalfield. Within the quantum control literature, the method is referred to as themonotonic algorithm.

Ohtsuki et al. [105] showed how to include time-dependent targets of thegeneral form (6.1) to the optimal control formulation. This time-dependentobjective problem was also tackled with the monotonic algorithm (cf. also thetutorial [146]).

All of these methods are based on the first order optimality condition. Usu-ally, convergence can be improved by using a quasi-Newton method that in-cludes information of an approximate Hessian. Among those methods, theBFGS algorithm [23, 42, 47, 127] is the most successful one. A disadvantageof a quasi-Newton method for the quantum optimal control problem is that thenumber of control parameters is the number of time steps, which in general isvery large. Therefore, memory shortage will make it impossible to store thefull approximate Hessian. Hence, only low-memory variants are possible.

Judson and Rabitz [71] proposed an alternative way to design laser pulses:feedback control. This offers a possibility to design laser pulses also for sys-tems where the Hamiltonian is unknown or for molecules that are too large tobe simulated in a computer. The computer only successively generates pulses(usually based on some global optimization strategy) and then the fitness ofthe pulse is determined by an experiment. In the experiment, the incominglaser pulse is grated to spatially disperse light of different frequencies whichthen can be delayed independently from each other through a spatial filter[142, 148]. This technique is also-called Fourier transform femtosecond pulseshaping. Assion et al. [2] point out that the optimization procedure should

35

consider the spectral phase as the experimental shaper does. In Ref. [129], aspecial global optimization strategy is applied to find optimal Fourier coeffi-cients with computer simulation.

Using the Fourier coefficients as control variables like in feedback controlalgorithms, one can drastically reduce the dimensionality of the optimizationproblem as demonstrated in Paper VIII. In this way, it becomes feasible touse quasi-Newton methods even for long-time quantum optimal control prob-lems. Moreover, one can make sure that the theoretically found pulses can berealized in practice.

Note that a quasi-Newton method was also applied to quantum optimal con-trol of Bose–Einstein condensate in the context of von Winckel and Borzì’sstudy on a suitable norm for minimization [136].

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7. Summary and Outlook

This thesis focuses on the numerical solution of the time-dependent Schrödin-ger equation modeling matter-field interaction on the femtosecond time scale.This phenomenon is fundamentally quantum mechanical in nature. Therefore,semiclassical methods have difficulties to fully describe the action and there isa need to develop efficient direct methods for the solution of the TDSE. Theunderlying partial differential equation is characterized by a highly-oscillatorytemporal dependence of the Hamiltonian operator. In Paper I, a Magnus–Lanczos propagator of order four is adapted for this special problem. In acomparison of various propagation methods its efficiency has been underlined.This propagator is put into an adaptive setting in Paper II. The adaptivity isenhanced in Paper III where mesh refinement in a spectral element setting isincluded. Paper VI describes an efficient implementation framework for spec-tral elements that uses the data types for matrix-free implementation intro-duced in [82] and describes the parallelization strategies with focus on a novelcoloring-type algorithm for shared-memory architectures. Another aspect ofparallelization is considered in Paper V which analyzes communication in theLanczos algorithm. Paper IV considers error estimation for a dissociative sys-tem and combines the results for the time evolution in Paper II with errorsfrom spatial discretization and domain truncation.

With the proposed implementation and adaptivity, three-particle problemsin three dimensions can be solved on a multi-core processor. An implemen-tation of the framework that can handle high-dimensional grids would enablethe study of larger molecules on computer clusters. Supposedly the algorithmdeveloped for the linear Schrödinger equation could be applied to simulatequantum fluids and quantum turbulence. However, this requires the modelsand the theory to be augmented to include nonlinearities.

Paper VII proposes a radial basis function approximation of the TDSE ina Galerkin setting. We show convergence and stability for this method andreport experiments that demonstrate the high accuracy of the method. Thereare two features of this ansatz that are promising: the high accuracy combinedwith more locality in the method compared to pseudospectral methods as wellas the resemblance to coherent states in semiclassical methods. On the otherhand, the present implementation of the method is both memory and CPU-timeintense. Therefore, the inherent locality of the method needs to be exploitedin order to facilitate parallelization and to save memory.

Another direction of further research is to devise a strategy to scatter thenodes of the radial basis functions. It would be interesting to analyze how the

37

error estimation theory proposed for more traditional discretization methodsin this thesis should be modified to fit in the RBF setting. One challenge in thisresearch is how to model boundary effects: At the boundary, nodes need to beclustered to countervail ill-conditioning of the interpolation [109], an effectthat cannot directly be measured by the size of the solution or the residual inthis region.

The resemblance of the radial basis functions to wave packets used in semi-classical methods suggests a combination of the two for simulations where asmall number of independent variables models truly quantum dynamical pro-cesses while a larger number can be treated almost classically.

Finally, a new algorithm to solve the quantum optimal control problem isdevised in Paper VIII. The pulse is optimized in Fourier space and a quasi-Newton method is applied. The optimal pulse can be found in fewer iterationscompared to the well-established monotonic algorithm and the control vari-ables in Fourier space give the possibility to restrict the pulse to a certain bandwidth. The latter fact could be explored further in combination with physi-cal requirements. For instance, the resolution of the discrete Fourier compo-nents could be coupled to the resolution of the pulse shaper. In the presentimplementation of the algorithm, the envelope of the field is also modeledwith the Fourier components. This might be reasonable to change by apply-ing more local transforms [116]. Another direction of research is to considermore complicated objectives like revival patterns [139] or problems includingdissociative processes.

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8. Effektiva och tillförlitliga simuleringar avkvantmolekyldynamik

Kvantmekaniken beskriver hur materiens minsta byggstenar växelverkar. Se-dan laser, med frekvenser i samma storleksordning som vibrationerna i atom-kärnorna, uppfanns har en hel disciplin utvecklats som försöker förstå – ochäven manipulera – molekylernas struktur med hjälp av laserns elektromagne-tiska fält. Kvanteffekterna uppenbarar sig inte i de flesta makroskopiska struk-turer, men blir avgörande när man vill designa så kallade nanokomponentersom består av enbart ett fåtal molekyler. I experimentell kvantmekanik mäterman effekter av partikelrörelser vilka ger indikation på hur partiklarnas våg-funktion ser ut. Därför bidrar det till förståelsen av mätningarna att lösa Schrö-dingerekvationen som är den matematiska beskrivningen av vågfunktionerna.Oftast är man tvungen att lösa ekvationerna med hjälp av datorer då analytis-ka lösningar inte är kända. När det gäller lösning av Schrödingerekvationenmed hjälp av datorer är huvudutmaningen dimensionalitetens förbannelse: ef-tersom antalet rumsdimensioner i ekvationen är proportionerligt mot antaletpartiklar växer antalet noder i en nätbaserad beräkningsmetod exponentielltmed antalet partiklar.

Denna doktorsavhandling beaktar frågan hur man kan utnyttja moderna nu-meriska metoder och datorkraft för att simulera hur molekyler beter sig i väx-elverkan med elektromagnetiska fält på ett effektivt och tillförlitligt sätt. Hu-vudfokus ligger på utveckling av en lösare som beräknar resultatet med efter-strävad noggrannhet. Lösaren utnyttjar adaptivitet i rum såväl som i tid och ärparallelliserad och implementerad på ett minneseffektivt sätt.

Spektrala metoder ger mycket effektiv rumsdiskretisering i enkla lösare förSchrödingerekvationen, men de är mycket svåra att parallellisera och anpas-sa till lösningens utseende. Denna avhandling föreslår istället användandet avspektralelement, finita element av hög noggrannhetsordning. Fördelen medspektralelement är tvåfaldig: De kombinerar förhållandevis hög noggrannhetmed relativt lokaliserade stenciler som går att parallellisera över datorkluster.Därutöver finns det genom metoden för dualviktade residualer en teori för hurde lokala diskretiseringsfelen påverkar slutsresultatets noggrannhet.

Efter diskretiseringen i rummet uppstår ett system av ordinära differentia-lekvationer som löses med en numerisk tidsstegningsmetod. Idealiskt ska me-toden bevara lösningens norm på samma sätt som den är bevarad i den konti-nuerliga modellen. Dessutom vill man använda sig av en explicit metod somundviker lösning av linjära system vilket är mycket kostsamt och minneskrä-vande i högre dimensioner. Det finns två typer av tidsstegare som har dessa

39

egenskaper: exponentialintegratorer och partitionerade Runge-Kutta-metoder.I avhandlingen jämförs olika utformningar av dessa metoder för simuleringav växelverkan mellan ljus och materia. För denna tillämpning härleds enfjärde ordningens exponentialintegrator baserad på Magnusutvecklingen vil-ken undviker evaluering av matriskommutatorer. Denna är annars nödvändigti högre ordningens termer i Magnusutvecklingen. Matrisexponentialen beräk-nas slutligen approximativt med hjälp av Lanczosalgoritmen. Denna metod vi-sar sig vara mycket effektivt och vidareutvecklas till att inkludera felkontroll.Eftersom Schrödingerekvationen är självadjungerande blir det lokala trunke-ringsfelet varken dämpat eller förstärkt vilket utnyttjas i den globala felupp-skattningen. Det visar sig dock att man kan applicera teorin även för dissocia-tiva system där modellen inte är självadjungerande.

Kombinerar man en rumsdiskretisering baserad på spektralelement med enexponentialintegrator, är massmatrisen som vanligen uppstår i finita element-diskretiseringar besvärande eftersom den innebär att man är tvungen att lösaett linjärt ekvationssystem trots att tidsstegningsmetoden är explicit. Därföranvänds Lagrangepolynom som centreras i noderna hos Gauss-Lobatto-kva-draturformeln, och sedan approximeras integralerna i mass- och styvhetsma-triserna med just denna kvadraturformel. På så sätt blir massmatrisen diagonaloch kan enkelt inverteras. Eftersom Gauss-Lobatto-kvadraturformeln fördelarnoderna så att de ligger tätare närmare randen kringgår man också Rungefeno-menet för högre ordningens polynomapproximation. Därför är dessa elementmycket lämpade som spektralelement.

I vanliga fall implementeras finita element baserat på en gles matris som in-nehåller all information om differentialoperatorn projicerat på finita element-rummet. När man använder basfunktioner av högre ordning är dock matriseninte speciellt gles och en sådan implementation kräver mycket minne. Somen del av denna avhandling presenteras ett ramverk inom biblioteket deal.IIför att applicera finita element-operatorer cellbaserat. Idén är att för varje celllagra hur den kan transformeras till en enhetscell. Funktionen transformerassedan elementvis till enhetscellen där differentialoperatorn appliceras. Dennaimplementation gör det möjligt att utnyttja tensorproduktstrukturen som finnsin Gauss-Lobatto-elementen för att minska beräkningskomplexiteten. Dess-utom reduceras minnensanvändningen – och därigenom minnesflödet underberäkningarna – för högre ordningar. Ramverket är parallelliserat på tre ni-våer: MPI kan användas på kluster med distribuerat minne, Intel ThreadingBuilding Blocks används för att hantera parallellism på ett system med delatminne, och operationer på två eller flera celler klumpas ihop för att utnyttjaberäkningskärnornas vektorenheter. Denna avhandling fokuserar på parallel-lisering på system med delat minne. När man parallelliserar matris-vektor-multiplikation och vektorerna sparas på delat minne måste man undvika attolika processorer räknar på celler som delar frihetsgrader. Detta görs vanligengenom att dela upp domänen i ”olika färger” så att celler som tillhör sammafärg inte delar frihetsgrader. Sedan bearbetar man en färg i taget. Problemet är

40

att man får en synkroniseringspunkt efter varje färg och att man kan få ganskamånga färger på adaptiva nåt. Därför föreslås en uppdelning på två nivåer somundviker explicita synkroniseringspunkter. Istället läggs beräkningarna ut dy-namiskt enligt denna uppdelning så att två beräkningsenheter aldrig samtidigtskriver i samma element.

En annan aspekt av parallelliseringen som behandlas är kommunikationenmellan olika datorer inom ett kluster. Medan matris-vektor-multiplikation van-ligen bara kräver utbyte av data mellan grannar kräver skalärprodukter att al-la beräkningsnoder kommunicerar med varandra. Detta försvårar parallellise-ringen. I en vanlig implemention av Lanczosalgoritmen finns det två ställeni varje iteration där en skalärprodukt beräknas. I denna avhandling undersöksolika varianter av Lanczosalgoritmen som är designade för att minska kom-munikationen till priset av lite fler beräkningar. Det visar sig att effekterna avminskad kommunikation och ökat beräkningsarbete ungefär tar ut varandra påde system som det har räknats på.

Som ett sidospår undersöks också möjligheten att använda radiella basfunk-tioner (RBF) som diskretiseringsmetod för Schrödingerekvationen. Olika bas-funktioner omnämns i litteraturen, men Gaussianer visar sig vara mest lämp-liga för Schrödingerekvationen. RBF är en modern metod som är mycket lo-vande av två anledningar: metoden ger spektral noggrannhet men till skillnadfrån vanliga spektralmetoder kan man sprida basfunktionernas centrum god-tyckligt. Dessutom liknar basfunktionerna de funktioner som används i vissasemiklassiska metoder. Det finns alltså goda förutsättningar för att kombineraen RBF-diskretisering med semiklassika metoder. Eftersom basfunktionernaär definierade på hela rummet och avtar mot oändligheten på samma sätt somlösningen av Schrödingerekvationen är man inte tvungen att applicera någraartificiella randvillkor utan kan betrakta problemet på hela rummet på sammasätt som i det kontinuerliga problemet. Inom RBF-litteraturen är det vanligastmed kollokation vid lösning av differentialekvationer, men denna avhandlingföreslår istället en Galerkinansats för Schrödingerekvationen. Detta eftersomdet uppstår stabilitetsproblem med en kollokationsansats samtidigt som inte-gralerna i mass- och styvhetsmatriserna kan beräknas analytiskt på ett relativtenkelt sätt. Både stabilitet och konvergens visas. Metoden konvergerar expo-nentiellt upp till en viss nivå, vilken beror på storleken av domänen som bas-funktionerna är fördelade på. Metoden visar sig vara mycket noggrann, mendet finns behov att öka effektiviteten när det gäller lösningen av det linjärasystemet som uppstår vid diskretiseringen.

Slutligen behandlar avhandlingen även problemet att optimera ett elektro-magnetiskt fält som manipulerar en molekyl, t. ex. med målet att initiera enkemiskt reaktion. Trots att laserpulserna tas fram i experiment genom att för-ändra parametrarna i frekvensrummet, är det vanligt att hitta det optimala fäl-tet inom tidsrummet. I denna avhandling föreslås istället att betrakta pulsen ifrekvensrummet även i optimeringsproblemet. På detta sätt blir det lättare attsäkerställa att pulsen som hittas teoretiskt också kan framställas i praktiken

41

samtidigt som man oftast kan koncentrera sig på färre parametrar. Därigenomblir det också möjligt att använda en kvasi-Newtonmetod (som kräver att detlagras en approximativ Hessian) istället för den monotona algoritmen, en linjäralgoritm som vanligen används i detta sammanhang. Effekten är en avsevärduppsnabbning.

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Acknowledgments

First of all, I want to express my gratitude to my advisor Sverker Holmgrenfor introducing me to this exciting subject and countless stimulating discus-sions. I have appreciated the freedom and responsibility he gave me to developmy own ideas and interests. Invaluable for this thesis was the help that HansO. Karlsson offered me to understand the chemistry behind the computationsand to acquire knowledge on new challenges in chemistry. Many thanks toElisabeth Larsson for sharing her broad knowledge on radial basis functionsand for our great collaboration.

I have very much enjoyed doing research together with Magnus Gustafsson,Martin Kronbichler, and Anna Nissen. I want to thank Martin Berggren andEddie Wadbro for sharing their expertise on numerical optimization and AxelMålqvist for sharing his on finite elements. Also, I gratefully acknowledgediscussions on various topics of programming with Wolfgang Bangerth, Mag-nus Gustafsson, Martin Kronbichler, Emanuel Rubensson, and Elias Rudberg.I am thankful to Arieh Iserles as well as Gabriel Turinici, Claude LeBries, andJulien Salomon for inviting me to discuss my research. Many thanks to VasileGradinaru for discussing my research as an opponent at the occasion of mylicentiate defense. Finally, I appreciated Emil Kieri’s and Martin Kronbich-ler’s detailed comments to this comprehensive summary and other parts of mywork.

Over the years the colloquia on numerical quantum dynamics, radial basisfunctions, high dimensional problems, and high performance computing at theDivision of Scientific Computing have been excellent platforms for exchang-ing ideas.

This work was supported by the Graduate School in Mathematics and Com-puting (FMB).

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