Slide 1

Materials Process Design and Control Laboratory 1 An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations Nicholas Zabaras and Xiang Ma Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 101 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/ Symposium on `Numerical Methods for Stochastic Computation and Uncertainty Quantification' in the 2008 SIAM Annual Meeting, San Diego CA, July 7-11, 2008 Slide 2 Materials Process Design and Control Laboratory 2 Outline of the presentation o Stochastic analysis o Issues with gPC and ME-gPc o Collocation based approach o Basics of stochastic collocation o Issues with existing stochastic sparse grid collocation method (Smolyak algorithm) o Adaptive sparse grid collocation method o Some numerical examples Slide 3 Materials Process Design and Control Laboratory 3 Motivation All physical systems have an inherent associated randomness SOURCES OF UNCERTAINTIES Multiscale material information inherently statistical in nature. Uncertainties in process conditions Input data Model formulation approximations, assumptions. Why uncertainty modeling ? Assess product and process reliability. Estimate confidence level in model predictions. Identify relative sources of randomness. Provide robust design solutions. Engineering component Heterogeneous random Microstructural features ProcessControl? Slide 4 Materials Process Design and Control Laboratory 4 Problem definition Define a complete probability space. We are interested to find a stochastic function such that for P-almost everywhere (a.e.), the following equation hold: where are the coordinates in, L is (linear/nonlinear) differential operator, and B is a boundary operators. In the most general case, the operator L and B as well as the driving terms f and g, can be assumed random. In general, we require an infinite number of random variables to completely characterize a stochastic process. This poses a numerical challenge in modeling uncertainty in physical quantities that have spatio-temporal variations, hence necessitating the need for a reduced-order representation. Slide 5 Materials Process Design and Control Laboratory 5 Representing Randomness S Sample space of elementary events Real line Random variable MAP Collection of all possible outcomes Each outcome is mapped to a corresponding real value Interpreting random variables as functions A general stochastic process is a random field with variations along space and time A function with domain (, , S) 1. Interpreting random variables 2. Distribution of the random variable E.g. inlet velocity, inlet temperature 3. Correlated data Ex. Presence of impurities, porosity Usually represented with a correlation function We specifically concentrate on this. Slide 6 Materials Process Design and Control Laboratory 6 Representing Randomness 1.Representation of random process - Karhunen-Loeve, Polynomial Chaos expansions 2. Infinite dimensions to finite dimensions - depends on the covariance Karhunen-Lov expansion Based on the spectral decomposition of the covariance kernel of the stochastic process Random process Mean Set of random variables to be found Eigenpairs of covariance kernel Need to know covariance Converges uniformly to any second order process Set the number of stochastic dimensions, N Dependence of variables Pose the (N+d) dimensional problem Slide 7 Materials Process Design and Control Laboratory 7 Karhunen-Loeve expansion Stochastic process Mean function ON random variables Deterministic functions Deterministic functions ~ eigen-values, eigenvectors of the covariance function Orthonormal random variables ~ type of stochastic process In practice, we truncate (KL) to first N terms Slide 8 Materials Process Design and Control Laboratory 8 The finite-dimensional noise assumption By using the Doob-Dynkin lemma, the solution of the problem can be described by the same set of random variables, i.e. So the original problem can be restated as: Find the stochastic function such that In this work, we assume that are independent random variables with probability density function. Let be the image of. Then is the joint probability density of with support Slide 9 Materials Process Design and Control Laboratory 9 Uncertainty analysis techniques Monte-Carlo : Simple to implement, computationally expensive Perturbation, Neumann expansions : Limited to small fluctuations, tedious for higher order statistics Sensitivity analysis, method of moments : Probabilistic information is indirect, small fluctuations Spectral stochastic uncertainty representation: Basis in probability and functional analysis, can address second order stochastic processes, can handle large fluctuations, derivations are general Stochastic collocation: Results in decoupled equations Slide 10 Materials Process Design and Control Laboratory 10 Spectral stochastic presentation A stochastic process = spatially, temporally varying random function CHOOSE APPROPRIATE BASIS FOR THE PROBABILITY SPACE HYPERGEOMETRIC ASKEY POLYNOMIALS PIECEWISE POLYNOMIALS (FE TYPE) SPECTRAL DECOMPOSITION COLLOCATION, MC (DELTA FUNCTIONS) GENERALIZED POLYNOMIAL CHAOS EXPANSION SUPPORT-SPACE REPRESENTATION KARHUNEN-LOVE EXPANSION SMOLYAK QUADRATURE, CUBATURE, LH Slide 11 Materials Process Design and Control Laboratory 11 Generalized polynomial chaos (gPC) Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input Stochastic output Askey polynomials in input Deterministic functions Stochastic input Askey polynomials ~ type of input stochastic process Usually, Hermite, Legendre, Jacobi etc. Slide 12 Materials Process Design and Control Laboratory 12 Generalized polynomial chaos (gPC) Advantages: 1)Fast convergence 1. 2)Very successful in solving various problems in solid and fluid mechanics well understood 2,3. 3)Applicable to most random distributions 4,5. 4)Theoretical basis easily extendable to other phenomena 1.P Spanos and R Ghanem, Stochastic finite elements: A spectral approach, Springer-Verlag (1991)Stochastic finite elements: A spectral approach 2.D. Xiu and G. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, Comput. Methods Appl. Mech. Engrg. 187 (2003) 137--167.Modeling uncertainty in flow simulations via generalized polynomial chaos 3.S. Acharjee and N. Zabaras, Uncertainty propagation in finite deformations -- A spectral stochastic Lagrangian approach, Comput. Methods Appl. Mech. Engrg. 195 (2006) 2289--2312.Uncertainty propagation in finite deformations -- A spectral stochastic Lagrangian approach 4.D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comp. 24 (2002) 619-644.The Wiener-Askey polynomial chaos for stochastic differential equations 5.X. Wan and G.E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs, SIAM J Sci Comp 28(3) (2006) 455-464.Beyond Wiener-Askey expansions: Handling arbitrary PDFs Slide 13 Materials Process Design and Control Laboratory 13 Generalized polynomial chaos (gPC) Disadvantages: 1)Coupled nature of the formulation makes implementation nontrivial 1. 2)Substantial effort into coding the stochastics: Intrusive 3)As the order of expansion (p) is increased, the number of unknowns (P) increases factorially with the stochastic dimension (N): combinatorial explosion 2,3. 4)Furthermore, the curse of dimensionality makes this approach particularly difficult for large stochastic dimensions 1.B. Ganapathysubramaniana and N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, J. Comp Physics, 2007, doi:10.1016/j.jcp.2006.12.014Sparse grid collocation schemes for stochastic natural convection problems 2.D Xiu, D Lucor, C.-H. Su, and G Karniadakis, Performance evaluation of generalized polynomial chaos, P.M.A. Sloot et al. (Eds.): ICCS 2003, LNCS 2660 (2003) 346-354Performance evaluation of generalized polynomial chaos 3.B. J. Debusschere, H. N. Najm, P. P. Pebray, O. M. Knio, R. G. Ghanem and O. P. Le Matre, Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes, SIAM Journal of Scientific Computing 26(2) (2004) 698-719. Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes Slide 14 Materials Process Design and Control Laboratory 14 Failure of gPC Gibbs phenomenon: Due to its global support, it fails to capture the stochastic discontinuity in the stochastic space. X-velocity Y-velocity Capture unsteady equilibrium in natural convection Therefore, we need some method which can resolve the discontinuity locally. This motivates the following Multi - element gPC method. Slide 15 Materials Process Design and Control Laboratory 15 Multi-element generalized polynomial chaos (ME-gPC) Basic concept: Decompose the space of random inputs into small elements. In each element of the support space, we generate a new random variable and apply gPCE. The global PDF is also simultaneously discretized. Thus, the global PC basis is not orthogonal with the local PDF in a random element. We need to reconstruct the orthogonal basis numerically in each random element. The only exception is the uniform distribution whose PDF is a constant. We can also decompose the random space adaptively to increase its efficiency. 1.X. Wan and G.E. Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, SIAM J Sci Comp 209 (2006) 901-928. Slide 16 Materials Process Design and Control Laboratory 16 Decomposition of the random space where We define a decomposition of as Based on the decomposition, we define the following indicator random variables: otherwise Then a local random vector is defined in each element subject to a conditional PDF where Slide 17 Materials Process Design and Control Laboratory 17 An adaptive procedure Let us assume that the gPC expansion of random field in element is From the orthogonality of gPC, the local variance approximated by PC with order is given by The approximate global mean and variance can be expressed by Let denote the error of local variance. We can write the exact global variance as Slide 18 Materials Process Design and Control Laboratory 18 An adaptive procedure We define the local decay rate of relative error of the gPC approximation in each element as follows: Based on and the scaled parameter, a random element will be selected for potential refinement when the following condition is satisfied where and are prescribed constants. Furthermore, in the selected random element, we use another threshold parameter to choose the most sensitive random dimension. We define the sensitivity of each random dimension as where we drop the subscript for clarity and the subscript denotes the mode consisting only of random dimension with polynomial order Slide 19 Materials Process Design and Control Laboratory 19 An adaptive procedure All random dimensions which satisfy will be split into two equal random elements in the next time step while all other random dimensions will remain unchanged. Through this adaptive procedure, we can reduce the total element number while gaining the same accuracy and efficiency. If corresponds to element in the original random space, the element and will be generated in the next level if the two criteria are both satisfied. Slide 20 Materials Process Design and Control Laboratory 20 An Example: K-O problem So it can successfully resolve the discontinuity in stochastic space. However, due to its gPC nature in each element and the tree like adaptive refinement, this method still suffers from the curse of dimensionality. Slide 21 Materials Process Design and Control Laboratory 21 First summary The traditional gPC method fails when there are steep gradients/finite discontinuities in stochastic space. gPC related method suffers from the so called curse of dimensionality when the input stochastic dimension is large. High CPU cost for large scale problems. Although the h-adaptive ME-gPC can successfully resolve discontinuity in stochastic space, the number of element grows fast at the order. So the method is still a dimension-dependent method. Therefore, an adaptive framework utilizing local basis functions that scales linearly with increasing dimensionality and has the decoupled nature of the MC method, offers greater promise in efficiently and accurately representing high-dimensional non-smooth stochastic functions. Slide 22 Materials Process Design and Control Laboratory 22 Collocation based framework The stochastic variable at a point in the domain is an N dimensional function. The gPC method approximates this N dimensional function using a spectral element discretization (which causes the coupled set of equations) Spectral descretization represents the function as a linear combination of sets of orthogonal polynomials. Instead of a spectral discretization, use other kinds of polynomial approximations Use simple interpolation! That is, sample the function at some points and construct a polynomial approximation Slide 23 Materials Process Design and Control Laboratory 23 Collocation based framework Function value at any point is simply Stochastic function in 2 dimensions Need to represent this function Sample the function at a finite set of points Use polynomials (Lagrange polynomials) to get a approximate representation Spatial domain is approximated using a FE, FD, or FV discretization. Stochastic domain is approximated using multidimensional interpolating functions Slide 24 Materials Process Design and Control Laboratory 24 From one-dimension to higher dimensions Denote the one dimensional interpolation formula as with the set of support nodes: In higher dimension, a simple case is the tensor product formula For instance, if M=10 dimensions and we use k points in each direction Number of points in each direction, k Total number of sampling points 21024 359049 4 1.05x10 6 5 9.76x10 6 10 1x10 10 This quickly becomes impossible to use. One idea is only to pick the most important points from the tensor product grid. Slide 25 Materials Process Design and Control Laboratory 25 Conventional sparse grid collocation (CSGC) 1D sampling 2D sampling In higher dimensions, not all the points are equally important. Some points contribute less to the accuracy of the solution (e.g. points where the function is very smooth, regions where the function is linear, etc.). Discard the points that contribute less: SPARSE GRID COLLOCATION Tensor product Slide 26 Materials Process Design and Control Laboratory 26 Conventional sparse grid collocation Both piecewise linear as well as polynomial functions, L, can be used to construct the full tensor product formula above. The Smolyak construction starts from this full tensor product representation and discards some multi-indices to construct the minimal representation of the function Define the differential interpolant, as the difference between two interpolating approximations with where, is the multi-index representation of the support nodes. N is the dimension of the function f and q is an integer (q>N). k = q-N is called the depth of the interpolation. As q is increased more and more points are sampled. The sparse grid interpolation of the function f is given by Slide 27 Materials Process Design and Control Laboratory 27 Conventional sparse grid collocation The interpolant can be represented in a recursive manner as with Can increase the accuracy of the interpolant based on Smolyaks construction WITHOUT having to discard previous results. Can build on previous interpolants due to the recursive formulation To compute the interpolant, one only needs the function values at the sparse grid point given by: One could select the sets, in a nested fashion such that Similar to the interpolant representation in a recursive form, the sparse grid points can be represented as: with and Slide 28 Materials Process Design and Control Laboratory 28 Choice of collocation points and nodal basis functions One of the choice is the Clenshaw-Curtis grid at non-equidistant extrema of the Chebyshev polynomials. ( D.Xiu etc, 2005, F. Nobile etc, 2006)D.Xiu etc, 2005F. Nobile etc, 2006 By doing so, the resulting sets are nested. The corresponding univariate nodal basis functions are Lagrangian characteristic polynomials: There are two problems with this grid: (1) The basis function is of global support, so it will fail when there is discontinuity in the stochastic space. (2) The nodes are pre-determined, so it is not suitable for implementation of adaptivity. Slide 29 Materials Process Design and Control Laboratory 29 Choice of collocation points and nodal basis functions Therefore, we propose to use the Newton-Cotes grid using equidistant support nodes and use the linear hat function as the univariate nodal basis. It is noted that the resulting grid points are also nested and the grid has the same number of points as that of the Clenshaw-Curtis grid. Comparing with the Lagrangian polynomial, the piecewise linear hat function has a local support, so it is expected to resolve discontinuities in the stochastic space. The N-dimensional multi-linear basis functions can be constructed using tensor products: Slide 30 Materials Process Design and Control Laboratory 30 Comparison of the two grids Clenshaw-Curtis grid Newton-Cotes grid Slide 31 Materials Process Design and Control Laboratory 31 From nodal basis to multivariate hierarchical basis We start from the 1D interpolating formula. By definition, we have withand we can obtain since we have For simplifying the notation, we consecutively number the elements in, and denote the j-th point of as hierarchical surplus Slide 32 Materials Process Design and Control Laboratory 32 Nodal basis vs. hierarchical basis Nodal basis Hierarchical basis Slide 33 Materials Process Design and Control Laboratory 33 Multi-dimensional hierarchical basis For the multi-dimensional case, we define a new multi-index set which leads to the following definition of hierarchical basis Now apply the 1D hierarchical interpolation formula to obtain the sparse grid interpolation formula for the multivariate case in a hierarchical form: Here we define the hierarchical surplus as: Slide 34 Materials Process Design and Control Laboratory 34 Error estimate 1.V. Barthelmann, E. Novak and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12 (2000), 273288High dimensional polynomial interpolation on sparse grids 2.E. Novak, K. Ritter, R. Schmitt and A. Steinbauer, On an interpolatory method for high dimensional integration, J. Comp. Appl. Math. 112 (1999), 215228On an interpolatory method for high dimensional integration Depending on the order of the one-dimensional interpolant, one can come up with error bounds on the approximating quality of the interpolant 1,2. If piecewise linear 1D interpolating functions are used to construct the sparse interpolant, the error estimate is given by: Where N is the number of sampling points, If 1D polynomial interpolation functions are used, the error bound is: assuming that f has k continuous derivatives Slide 35 Materials Process Design and Control Laboratory 35 Hierarchical integration So now, any function can now be approximated by the following reduced form It is just a simple weighted sum of the value of the basis for all collocation points in the current sparse grid. Therefore, we can easily extract the useful statistics of the solution from it. For example, we can sample independently N times from the uniform distribution [0,1] to obtain one random vector Y, then we can place this vector into the above expression to obtain one realization of the solution. In this way, it is easy to plot realizations of the solution as well as the PDF. The mean of the random solution can be evaluated as follows: where the probability density function is 1 since we have assumed uniform random variables on a unit hypercube. Slide 36 Materials Process Design and Control Laboratory 36 Hierarchical integration The 1D version of the integral in the equation above can be evaluated analytically: Since the random variables are assumed independent of each other, the value of the multi-dimensional integral is simply the product of the 1D integrals. Denoting we rewrite the mean as Thus, the mean is just an arithmetic sum of the hierarchical surpluses and the integral weights at each interpolation point. Slide 37 Materials Process Design and Control Laboratory 37 Hierarchical integration To obtain the variance of the solution, we need to first obtain an approximate expression for Then the variance of the solution can be computed as Therefore, it is easy to extract the mean and variance analytically, thus we only have the interpolation error. This is one of the advantages to use this interpolation based collocation method rather than quadrature based collocation method. Slide 38 Materials Process Design and Control Laboratory 38 Adaptive sparse grid collocation (ASGC) Let us first revisit the 1D hierarchical interpolation For smooth functions, the hierarchical surpluses tend to zero as the interpolation level increases. On the other hand, for non-smooth function, steep gradients/finite discontinuities are indicated by the magnitude of the hierarchical surplus. The bigger the magnitude is, the stronger the underlying discontinuity is. Therefore, the hierarchical surplus is a natural candidate for error control and implementation of adaptivity. If the hierarchical surplus is larger than a pre-defined value (threshold), we simply add the 2N neighbor points of the current point. Slide 39 Materials Process Design and Control Laboratory 39 Adaptive sparse grid collocation As we mentioned before, by using the equidistant nodes, it is easy to refine the grid locally around the non-smooth region. It is easy to see this if we consider the 1D equidistant points of the sparse grid as a tree-like data structure Then we can consider the interpolation level of a grid point Y as the depth of the tree D(Y). For example, the level of point 0.25 is 3. Denote the father of a grid point as F(Y), where the father of the root 0.5 is itself, i.e. F(0.5) = 0.5. Thus the conventional sparse grid in N- dimensional random space can be reconsidered as Slide 40 Materials Process Design and Control Laboratory 40 Adaptive sparse grid collocation Thus, we denote the sons of a grid point by From this definition, it is noted that, in general, for each grid point there are two sons in each dimension, therefore, for a grid point in a N- dimensional stochastic space, there are 2N sons. The sons are also the neighbor points of the father, which are just the support nodes of the hierarchical basis functions in the next interpolation level. By adding the neighbor points, we actually add the support nodes from the next interpolation level, i.e. we perform interpolation from level |i| to level |i|+1. Therefore, in this way, we refine the grid locally while not violating the developments of the Smolyak algorithm Slide 41 Materials Process Design and Control Laboratory 41 Adaptive sparse grid collocation To construct the sparse grid, we can either work through the multi-index or the hierarchical basis. However, in order to adaptively refine the grid, we need to work through the hierarchical basis. Here we first show the equivalence of these two ways in a two dimensional domain. We always start from the multi-index (1, 1), which means the interpolation level in first dimension is 1, in second dimension is also 1. For level 1, the associated point is only 0.5. So the tensor product of the coordinate is Thus, for interpolation level 0 of smolyak construction, we have Slide 42 Materials Process Design and Control Laboratory 42 Adaptive sparse grid collocation Recalling, we define the interpolation level k as. So, now we come to the next interpolation level, For this requirement, the following two multi-indices are satisfied: Let us look at the index (1,2) The associated set of points is 0.5 (j=1). The associated set of points is 0 (j=1) and 1 (j=2). So the tensor product is Then the associated sum is Slide 43 Materials Process Design and Control Laboratory 43 Adaptive sparse grid collocation In the same way, for multi-index (2,1), we have the corresponding collocation points So the associated sum is Recall So for Smolyak interpolation level 1, we have Slide 44 Materials Process Design and Control Laboratory 44 Adaptive sparse grid collocation So, from this equation, it is noted that we can store the points with the surplus sequentially. When calculating interpolating value, we only need to perform a simple sum of each term. We dont need to search for the point. Thus, it is very efficient to generate the realizations. Here, we give the first three level of multi-index and corresponding nodes. Slide 45 Materials Process Design and Control Laboratory 45 Adaptive sparse grid collocation In fact, the most important formula is the following From this equation, in order to calculate the surplus, we need the interpolation value at just previous level. That is why it requires that we construct the sparse grid level by level. Now, let us revisit the algorithm in a different view. we start from the hierarchical basis. We still use the two-dimension grid as an example. Recalling Slide 46 Materials Process Design and Control Laboratory 46 Adaptive sparse grid collocation As previous given, the first level in a sparse grid is always a point (0.5,0.5). There is a possibility that the function value at this point is 0 and thus the refinement terminates immediately. In order to avoid the early stop for the refinement process, we always refine the first level and keep a provision on the first few hierarchical surpluses. It is noted here, the associated multi-index with this point (0.5, 0.5) is (1,1),i.e. and it corresponds to level of interpolation k = 0. So the associated sum is still Now if the surplus is larger than the threshold, we add sons (neighbors) of this point to the grid. Slide 47 Materials Process Design and Control Laboratory 47 Adaptive sparse grid collocation In one-dimension, the sons of the point 0.5 are 0,1. According to the definition, the sons of the point (0.5,0.5) are which are just the points in the level 1 of sparse grid. Indeed, the associated multi-index is Recalling when we start from multi-index So actually, we can see they are equivalent. Now, let us summarize what we have So the associated sum is Slide 48 Materials Process Design and Control Laboratory 48 Adaptive sparse grid collocation After calculating the hierarchical surplus, now we need to refine. Here we assume we refine all of the points, i.e. threshold is 0. Notice here, all of the son points are actually from next interpolation level, so we still construct the sparse grid level by level. First, in one dimension, the sons of point 0 is 0.25, 1 is 0.75. So the sons of each point in this level are: Notice here, we have some redundant points. Therefore, it is important to keep track of the uniqueness of the newly generated (active) points. Slide 49 Materials Process Design and Control Laboratory 49 Adaptive sparse grid collocation After delete the same points, and arrange,we can have which are just all of the support nodes in k=2. Therefore, we can see by setting the threshold equal zero, conventional sparse grid is exactly recovered. We actually work on point by point instead of the multi-index. In this way, we can control the local refinement of the point. Here, we equally treat each dimension locally, i.e., we still add the two neighbor points in all N dimensions. So there are generally 2N son (neighbor) points. If the discontinuity is along one dimension, this method can still detect it. So in this way, we say this method can also detect the important dimension. Slide 50 Materials Process Design and Control Laboratory 50 Adaptive sparse grid collocation: implementation As shown before, we need to keep track of the uniqueness of the newly active points. To this end, we use the data structure from the standard template library in C++ to store all the active points and we refer to this as the active index set. Due to the sorted nature of the, the searching and inserting is always very efficient. Another advantage of using this data structure is that it is easy for a parallel code implementation. Since we store all of the active points from the next level in the, we can evaluate the surplus for each point in parallel, which increases the performance significantly. In addition, when the discontinuity is very strong, the hierarchical surpluses may decrease very slowly and the algorithm may not stop until a sufficiently high interpolation level. Therefore, a maximum interpolation level is always specified as another stopping criterion. Slide 51 Materials Process Design and Control Laboratory 51 Adaptive sparse grid collocation: Algorithms Let be the parameter for the adaptive refinement threshold. We propose the following iterative algorithm beginning with a coarsest possible sparse grid, i.e. with the N-dimensional multi-index i = (1,,1), which is just the point (0.5,,0.5). 1)Set level of Smolyak construction k = 0. 2)Construct the first level adaptive sparse grid. Calculate the function value at the point (0.5, , 0.5). Generate the 2N neighbor points and add them to the active index set. Set k = k + 1; 3)While and the active index set is not empty Copy the points in the active index set to an old index set and clear the active index set. Calculate in parallel the hierarchical surplus of each point in the old index set according to Here, we use all of the existing collocation points in the current adaptive sparse grid This allows us to evaluate the surplus for each point from the old index set in parallel. Slide 52 Materials Process Design and Control Laboratory 52 Adaptive sparse grid collocation: Algorithms For each point in the old index set, if Generate 2N neighbor points of the current active point according to the above equation; Add them to the active index set. Add the points in the old index set to the existing adaptive sparse grid Now the adaptive sparse grid becomes Set k = k + 1; 4)Calculate the mean and variance, the PDF and if needed realizations of the solution. In practice, instead of using surplus of the function value, it is sometimes preferable to use surplus of the square of the function value as the error indicator. This is because the hierarchical surplus is related to the calculation of the variance. In principle, we can consider it like a local variance. Thus, is more sensitive to the local variation of the stochastic function than. Error estimate of the adaptive interpolant Slide 53 Materials Process Design and Control Laboratory 53 NUMERICAL EXAMPLES Slide 54 Materials Process Design and Control Laboratory 54 Adaptive sparse grid interpolation Ability to detect and reconstruct steep gradients Slide 55 Materials Process Design and Control Laboratory 55 A dynamic system with discontinuity We consider the following governing equation for a particle moving under the influence of a potential field and friction force: If we set the potential field as, then the differential equation has two stable fixed points and an unstable fixed point at. The stochastic version of this problem with random initial position in the interval can be expressed as with stochastic initial condition where denotes the response of the stochastic system,, and is uniformly distributed random variable over. Slide 56 Materials Process Design and Control Laboratory 56 A dynamic system with discontinuity If we choose and a relatively large friction coefficient, a steady state solution is achieved in a short time. The analytical steady-state solution is given by This results in the following statistical moments In the following computations, the time integration of the ODE is performed using a fourth-order Runge-Kutta scheme and a time step was used. The analytic PDF is two dirac mass with unequal strength. Slide 57 Materials Process Design and Control Laboratory 57 Failure of gPC expansion Slide 58 Materials Process Design and Control Laboratory 58 Failure of gPC expansion Mean and variances exhibit oscillations with the increase of expansion order Slide 59 Materials Process Design and Control Laboratory 59 Failure of gPC expansion Therefore, the gPC expansion fails to converge the exact solutions with increasing expansion orders. It fails to capture the discontinuity in the stochastic space due to its global support, which is the well known Gibbs phenomenon when applying global spectral decomposition to problems with discontinuity in the random space. Slide 60 Materials Process Design and Control Laboratory 60 Failure of Lagrange polynomial interpolation Much better than gPC, but still exhibit oscillation near discontinuity point. Slide 61 Materials Process Design and Control Laboratory 61 Failure of Lagrange polynomial interpolation Slide 62 Materials Process Design and Control Laboratory 62 Failure of Lagrange polynomial interpolation Similarly, the CSGC method with Lagrange polynomials also fails to capture the discontinuity. However, better predictions of the low-order moments are obtained than the gPC method. Though the results indeed converge, the converged values are not accurate. Near the discontinuity point of the solution, there are still several wriggles, which will not disappear even for higher interpolation levels. Slide 63 Materials Process Design and Control Laboratory 63 Sparse grid collocation with linear basis function Slide 64 Materials Process Design and Control Laboratory 64 Sparse grid collocation with linear basis function Thus, it is correctly predicting the discontinuity behavior using linear basis function. And it is also noted that there are no oscillations in the solutions shown here. Slide 65 Materials Process Design and Control Laboratory 65 Adaptivity For CSGC method, a maximum interpolation level of 15 is used and 32769 points are need to reach accuracy of. For the ASGC method, a maximum interpolation level of 30 is used. Only 115 number of points are required to achieve accuracy of. Slide 66 Materials Process Design and Control Laboratory 66 Highly discontinuous solution Now we consider a more difficult problem. We set, and. All other conditions remain as before. The reduction of the friction coefficient, together with higher initial energy will result in the oscillation of the particle for several cycles between each of the two stable equilibrium points. Failure of gPC (left) and polynomial interpolation (right) at t = 250s. The failure of Lagrange polynomial interpolation is much more obvious. Slide 67 Materials Process Design and Control Laboratory 67 Highly discontinuous solution Results using the CSGC method with linear basis functions. Slide 68 Materials Process Design and Control Laboratory 68 The Kraichnan-Orszag three-mode problem The governing equations are (X.Wan, G. Karniadakis, 2005),2005 subject to stochastic initial conditions: There is a discontinuity when the random initial condition approaches (x 1 =x 2 ). So let us first look at the discontinuity in the deterministic solution. Slide 69 Materials Process Design and Control Laboratory 69 The Kraichnan-Orszag three-mode problem Singularity at plane Slide 70 Materials Process Design and Control Laboratory 70 The Kraichnan-Orszag three-mode problem For computational convenience and clarity we first perform the following transformation The discontinuity then occurs at the planes and. Thus, we first study the stochastic response subject to the following random 1D initial input: where. Since the random initial data can cross the plane we know that gPC will fail for this case. (X.Wan, G. Karniadakis, 2005)2005 The resulting governing equations become: Slide 71 Materials Process Design and Control Laboratory 71 The Kraichnan-Orszag three-mode problem Singularity at plane Slide 72 Materials Process Design and Control Laboratory 72 One-dimensional random input: variance gPC fails after t = 6s while the ASGC method converges even with a large threshold. The solid line is the result from MC-SOBOL sequence with 10 6 iterations. Slide 73 Materials Process Design and Control Laboratory 73 One-dimensional random input: adaptive sparse grid From the adaptive sparse grid, it is seen that most of the refinement after level 8 occurs around the discontinuity point Y = 0.0, which is consistent with previous discussion. The refinement stops at level 16, which corresponds to 425 number of points, while the conventional sparse grid requires 65537 points. Adaptive sparse grid with Slide 74 Materials Process Design and Control Laboratory 74 One-dimensional random input The exact solution is taken as the results given by MC- SOBOL The error level is defined as Slide 75 Materials Process Design and Control Laboratory 75 One-dimensional random input: long term behavior Slide 76 Materials Process Design and Control Laboratory 76 One-dimensional random input: Realizations These realizations are reconstructed using hierarchical surpluses. It is seen that at earlier time, the discontinuity has not yet been developed, which explains the reason that the gPC is accurate at earlier times. Slide 77 Materials Process Design and Control Laboratory 77 Two-dimensional random input Next, we study the K-O problem with 2D input: Now, instead of a point, the discontinuity region becomes a line. Slide 78 Materials Process Design and Control Laboratory 78 Two-dimensional random input It can be seen that even though the gPC fails at a larger time, the adaptive collocation method converges to the reference solution given by MC-SOBOL with 10 6 iterations. Slide 79 Materials Process Design and Control Laboratory 79 Two-dimensional random input The error level is defined as The number of points of the conventional grid with interpolation 20 is 12582913. Slide 80 Materials Process Design and Control Laboratory 80 Two-dimensional random input: Long term behavior Error in the mean (left) and the variance (right) of y 1 with different thresholds for the 2D K-O problem in [0, 50]. Slide 81 Materials Process Design and Control Laboratory 81 Three-dimensional random input Next, we study the K-O problem with 3D input: This problem is much more difficult than any of the other problems examined previously. This is due to the strong discontinuity ( the discontinuity region now consists of the planes Y 1 = 0 and Y 2 = 0) and the moderately-high dimension. It can be verified from comparison with the result obtained by MC-SOBOL that unlike the previous results, here 2 x 10 6 iterations are needed to correctly resolve the discontinuity. Due to the symmetry of y 1 and y 2 and the corresponding random input, the variances of y 1 and y 2 are the same. Finally, in order to show the accuracy of the current implementation of the h-adaptive ME-gPC, we provide the results for this problem. Slide 82 Materials Process Design and Control Laboratory 82 Three-dimensional random input:ASGC Evolution of the variance of y 1 =y 2 (left) and y 3 (right) for the 3D K-O problem using ASGC Slide 83 Materials Process Design and Control Laboratory 83 Three-dimensional random input: ME-gPC Evolution of the variance of y 1 =y 2 (left) and y 3 (right) for the 3D K-O problem using ME-gPC with and Slide 84 Materials Process Design and Control Laboratory 84 Three-dimensional random input Due to the strong discontinuity in this problem, it took much longer time for both methods to arrive at the same accuracy. Slide 85 Materials Process Design and Control Laboratory 85 Stochastic elliptic problem In this example, we compare the convergence rate of the CSGC and ASGC methods through a stochastic elliptic problem in two spatial dimensions. As shown in previous examples, the ASGC method can accurately capture the non-smooth region in the stochastic space. Therefore, when the non-smooth region is along a particular dimension (i.e. one dimension is more important than others), the ASGC method is expected to identify it and resolve it. In this example, we demonstrate the ability of the ASGC method to detect important dimensions when each dimension weigh unequally. This is similar to the dimension-adaptive method, especially in high stochastic dimension problems. Slide 86 Materials Process Design and Control Laboratory 86 Stochastic elliptic problem Here, we adopt the model problem with the physical domain. To avoid introducing errors of any significance from the domain discretization, we take a deterministic smooth load with homogeneous boundary conditions. The deterministic problem is solved using the finite element method with 900 bilinear quadrilateral elements. Furthermore, in order to eliminate the errors associated with a numerical K-L expansion solver and to keep the random diffusivity strictly positive, we construct the random diffusion coefficient with 1D spatial dependence as where are independent uniformly distributed random variables in the interval Slide 87 Materials Process Design and Control Laboratory 87 Stochastic elliptic problem This expansion is similar to a K-L expansion of a 1D random field with stationary covariance Small values of the correlation L correspond to a slow decay, i.e. each stochastic dimension weighs almost equally. On the other hand, large values of L results in fast decay rates, i.e., the first several stochastic dimensions corresponds to large eigenvalues weigh relatively more. By using this expansion, it is assumed that we are given an analytic stochastic input. Thus, there is no truncation error. This is different from the discretization of a random filed using the K-L expansion, where for different correlation lengths we keep different terms accordingly. In this example, we fix N and change L to adjust the importance of each stochastic dimension. In this way, we investigate the effects of L on the ability of the ASGC method to detect the important dimensions. Slide 88 Materials Process Design and Control Laboratory 88 Stochastic elliptic problem: N = 5 Slide 89 Materials Process Design and Control Laboratory 89 Stochastic elliptic problem: N = 5 We estimate the approximation error for the mean and variance. Specifically, to estimate the computation error in the q-th level, we fix N and compare the results at two consecutive levels, e.g. the error for the mean is The previous figures shows results for different correlation lengths at N=5. For small L, the convergence rates for the CSGC and ASGC are nearly the same. On the other hand, for large L, the ASGC method requires much less number of collocation points than the CSGC for the same accuracy. This is because more points are placed only along the important dimensions which are associated with large eigenvalues. Next, we study some higher-dimensional cases. Due to the rapid increase in the number of collocation points, we focus on moderate to higher correlation lengths so that the ASGC is effective. Slide 90 Materials Process Design and Control Laboratory 90 Stochastic elliptic problem: higher-dimensions #points:4193 Comparison: CSGC: 3.5991x10 11 Slide 91 Materials Process Design and Control Laboratory 91 Stochastic elliptic problem: higher-dimensions In order to further verify our results, we compare the mean and the variance when N = 75 using the AGSC method with with the exact solution obtained by MC-SOBOL simulation with 10 6 iterations. Computational time: ASGC : 0.5 hour MC: 2 hour Slide 92 Materials Process Design and Control Laboratory 92 Application to Rayleigh-Bnard instability Cold wall = -0.5 hot wall Insulated Filled with air (Pr = 0.7). Ra = 2500, which is larger than the critical Rayleigh number, so that convection can be initialized by varying the hot wall temperature. Thus, the hot wall temperature is assumed to be a random variable. Prior stochastic simulation, several deterministic computations were performed in order to find out the range where the critical point lies in. Slide 93 Materials Process Design and Control Laboratory 93 Rayleigh-Bnard instability: deterministic case Evolution of the average kinetic energy for different hot wall temperatures As we know, below the critical temperature, the flow vanishes, which corresponds to a decaying curve. On the other hand, the growing curve represents the existence of heat convection. Therefore, the critical temperature lies in the range [0.5,0.55]. Slide 94 Materials Process Design and Control Laboratory 94 Rayleigh-Bnard instability: deterministic formulation Steady state Nusselt number which denotes the rate of heat transfer: This again verifies the critical hot wall temperature lies in the range [0.5,0.55]. However, the exact critical value is not known to us. So, now we try to capture this unstable equilibrium using the ASGC method. Slide 95 Materials Process Design and Control Laboratory 95 Rayleigh-Bnard instability: stochastic formulation We assume the following stochastic boundary condition for the hot wall temperature: where Y is a uniform random variable in the interval [0,1]. Following the discussion before, both conductive and convective modes occur in this range. Slide 96 Materials Process Design and Control Laboratory 96 Rayleigh-Bnard instability: stochastic formulation Solution of the variables versus hot- wall temperature at point (0.1,0.5) Linear: conduction nonlinear: convection Slide 97 Materials Process Design and Control Laboratory 97 Rayleigh-Bnard instability: stochastic formulation Prediction of the temperature when using ASGC (left) and the solution of the deterministic problem using the same (right). Fluid flow vanishes, and the contour distribution of the temperature is characterized by parallel horizontal lines which is a typical distribution for heat conduction. Slide 98 Materials Process Design and Control Laboratory 98 Rayleigh-Bnard instability: stochastic formulation Prediction of the temperature when using ASGC (top) and the solution of the deterministic problem using the same (bottom). Slide 99 Materials Process Design and Control Laboratory 99 Rayleigh-Bnard instability: Mean ASGC (top) MC-SOBOL (bottom) Slide 100 Materials Process Design and Control Laboratory 100 Rayleigh-Bnard instability: Variance ASGC (top) MC-SOBOL (bottom) Slide 101 Materials Process Design and Control Laboratory 101 Conclusion An adaptive hierarchical sparse grid collocation method is developed based on the error control of local hierarchical surpluses. Besides the detection of important and unimportant dimensions as dimension-adaptive methods can do, additional singularity and local behavior of the solution can also be revealed. By employing adaptivity, great computational saving is also achieved than the conventional sparse grid method and traditional gPC related method. Solution statistics is easily extracted with a higher accuracy due to the analyticity of the linear basis function used. The current computational code is a highly efficient parallel program and has a wider application in various uncertainty quantification area.