Report on Sensitivity Analysis

Radu Serban

Keith Grant, Alan Hindmarsh, Steven Lee, Carol Woodward

Center for Applied Scientific Computing, LLNL

Work performed under the auspices of the U.S. Department of EnergyBy Lawrence Livermore National Laboratory under Contract W-7405-Eng-48

TOPS SciDAC Project All-hands Meeting

2CASC

Differential and Nonlinear Solvers @ CASC

CVODE – explicit ODE solver

IDA – implicit DAE solver

KINSOL – Krylov Inexact Newton solver

User main routineUser problem-defining functionUser preconditioner function

CVODEODE

Integrator

IDADAE

Integrator

KINSOLNonlinear

Solver

BandLinearSolver

PreconditionedGMRES

Linear Solver

GeneralPreconditioner

Modules

VectorKernels

DenseLinearSolver

3CASC

Sensitivity Analysis: What for?

• Model evaluation• Most and/or least influential parameters

• Model reduction

• Uncertainty quantification

• Optimization • design optimization• optimal control• parameter estimation• …

• …

4CASC

Forward Sensitivity Analysis

• Explicit ODE (CVODE)

• i-th sensitivity equation

• Gradient of a derived function

pyy

pyfy

0

0

,,

t

t

ii

iii

tp

y

p

y

p

f

p

y

y

f

p

y

0

d

d

d

d

d

d

d

d

0

p

g

p

y

y

g

p

g

pyg

d

d

d

d

),,(t

yp

py

gpy

pys

gpy

NN

NN

NNN

1

d/d

,,

R

RRR

Computational effort:

Remarks:

• Sensitivity r.h.s. can be user-defined, AD-generated, or FD-approximated

• Sensitivity equations are independent of g!

5CASC

Adjoint Sensitivity Analysis

• Explicit ODE (CVODE)

• For a derived function

• Adjoint ODE

• Gradient of derived function

pyy

pyfy

0

0

,,

t

t

ft

ttt

0)d,,()( pygpG

0λ

y

gλ

y

fλ

ft

TT

yg

y

gpy

λ

gpy

NN

N

NNN

1

,,

R

RRR

Computational effort:

ft

t p tt0

00TT )(d

d

dpp yλfλg

p

G

Remarks:

• Formulation can be extended to find gradients of g(tf,y,p)

• No FD approximation of the adjoint r.h.s.

• Adjoint equations are independent of p!

6CASC

Forward Sensitivity Variants of CASC Solvers

• CVODES currently available

User main routineSpecification of problem parametersActivation of sensitivity computationUser problem-defining functionUser preconditioner function

CVODESODE

Integrator

IDASDAE

Integrator

KINSOLSNonlinear

Solver

BandLinearSolver

PreconditionedGMRES

Linear Solver

GeneralPreconditioner

Modules

VectorKernels

DenseLinearSolver

Options- sensitivity approach (simultaneous or staggered)- user-defined, FD, or AD-generated sensitivity r.h.s.- error control on sensitivity variables- user-defined tolerances for sensitivity variables

7CASC

Adjoint Sensitivity Variants of CASC Solvers

• CVODEA currently available

User main routineActivation of sensitivity computationUser problem-defining functionUser reverse functionUser preconditioner functionUser reverse preconditioner function

CVODEAODE

Integrator

IDAADAE

Integrator

KINSOLANonlinear

Solver

BandLinearSolver

PreconditionedGMRES

Linear Solver

GeneralPreconditioner

Modules

ModifiedVectorKernels

DenseLinearSolver

Implementation- check point approach; total cost is 2 forward solutions + 1 backward solution - integrate any system backwards in time- may require modifications to some user-defined vector kernels

8CASC

Effects of Aerosols on Cloud Properties*

Problem dimensions• Ny 300

• Np=2

Problem description

• Condensation-evaporation eqs. coupled with eqs. of parcel motion and properties Implicit ODEs

Sensitivity of cloud liquid water to

temperature and water vapor profiles

*K. Grant, C. Chuang, S. Lee, C. Woodward

9CASC

Groundwater Flow

Problem description

• Variably saturated flow nonlinear elliptic PDEs Nonlinear eqs.

• Study influence of permeability field on solution (pressure)

• Quantify uncertainty in solution due to uncertainty in relative permeability and saturation curves

Problem dimensions• Ny=19000

• Np=3

*C. Woodward, K. Grant, R. Maxwell

10CASC

2-D Advection-Diffusion

u0

G for u0=(x-x’,y-y’)

10)(

)(0)(

123

t,x,y,tλ

Ωx,y,x,y,tλ

λλ/λλ yyxxxt

0)()(

)(0)(

23

0

t,x,yux,y,tu

Ωx,y,x,y,tu

uu/uu yyxxxt

dxdydtx,y,tuG )( dxdyx,yδux,y,λδGx,y

)()0( 0

]10[]10[]20[)( ,,,Ωx,y,t

Problem dimensions• Nu=800

• Np=800

Problem description

• 2-D time-dependent PDEs with homogeneous Dirichlet B.C. Explicit ODEs

11CASC

CVD of Superconducting Thin Films (YBaCuO)*

Problem description

• Compressible, chemically reacting, stagnation-flow equations

• 1-D time-varying PDEs Hessenberg index-2 DAEs

• Control film stoichiometry through inlet composition

Problem dimensions• Ny 500

• Np=24

*L. Raja, R. Kee, R. Serban, L. Petzold

12CASC

Future Developments

• Code development:• IDAS and IDAA• KINSOLA

• SciDAC collaboration: • Terrascale Supernova Initiative: Sensitivity analysis for radiation

hydrodynamics (CVODES/IDAS)• Other?

• TOPS collaborations?• Time-dependent DE constrained optimization• Other?