Polynomial Chaos Approximation Research Articles

Stochastic virtual element methods for uncertainty propagation of stochastic linear elasticity

This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve uncertainties in material properties, external forces, boundary conditions, etc. A stochastic virtual element space that couples the deterministic virtual element space and the stochastic space is constructed for this purpose and used to approximate the unknown stochastic solution. Two numerical frameworks are then developed to solve the derived stochastic virtual element equations, including a Polynomial Chaos approximation based approach and a weakly intrusive approximation based approach. In the Polynomial Chaos based framework, the stochastic solution is approximated using the Polynomial Chaos basis and solved via an augmented deterministic virtual element equation that is generated by applying the stochastic Galerkin procedure to the original stochastic virtual element equation. In the weakly intrusive approximation based framework, the stochastic solution is approximated by a summation of a set of products of random variables and deterministic vectors, where the deterministic vectors are solved via converting the original stochastic problem to deterministic virtual element equations by the stochastic Galerkin approach, and the random variables are solved via converting the original stochastic problem to one-dimensional stochastic algebraic equations by the classical Galerkin procedure. This method avoids the curse of dimensionality in high-dimensional stochastic problems successfully since all random inputs are embedded into one-dimensional stochastic algebraic equations whose computational effort weakly depends on the stochastic dimension. Numerical results on 2D and 3D problems with low- and high-dimensional random inputs demonstrate the good performance of the proposed methods.

Hyperbolicity-preserving and well-balanced stochastic Galerkin method for two-dimensional shallow water equations

Stochastic Galerkin formulations of the two-dimensional shallow water systems parameterized with random variables may lose hyperbolicity, and hence change the nature of the original model. In this work, we present a hyperbolicity-preserving stochastic Galerkin formulation by carefully selecting the polynomial chaos approximations to the nonlinear terms in the shallow water equations. We derive a sufficient condition to preserve the hyperbolicity of the stochastic Galerkin system which requires only a finite collection of positivity conditions on the stochastic water height at selected quadrature points in parameter space. Based on our theoretical results for the stochastic Galerkin formulation, we develop a corresponding well-balanced hyperbolicity-preserving central-upwind scheme. We demonstrate the accuracy and the robustness of the new scheme on several challenging numerical tests.

Efficient polynomial chaos approximations: Active, local and basis-adapted

Metamodels provide an efficient means for the approximation of response surfaces of systems, particularly for resource-intensive experiment designs. It is oftentimes the case that interest is focused on a specific region of the parameter space. We propose an efficient recipe for the local approximation of response surfaces using Polynomial Chaos techniques. For systems embedded in high-dimensional settings, a basis-adapted spectral representation is exploited locally for dimension reduction. The proposed approach comprises an initial heuristic global solution for parameter space exploration using an approximate global Polynomial Chaos metamodel, followed by a local design being refined through an active learning scheme. The problem of turbulent flow around a symmetric airfoil is considered. Statistical estimators based on the local, active, basis-adapted approach show less bias and faster convergence as compared to the estimators from a global solution.

Robust probabilistic calibration of a stochastic lattice discrete particle model for concrete

Numerical modelling of quasi-brittle materials arising from lattice or particle formulations is based on “a priori” discretisation of a medium according to an idealization of its granularity. This paper concentrates on the so-called Lattice Discrete Particle Model (LDPM), which provides accurate modelling of damage initiation and crack propagation at various length scales. However, its simulations are computationally demanding. We propose an automated identification procedure that would facilitate widespread utilisation without requiring deep expert knowledge about details of the model. Such an automated procedure is complicated, namely due to stochasticity of the LDPM related to the random generation of the particle configuration. The particle size distribution is generated so that it statistically corresponds to prescribed concrete granulometric distributions, but each realisation of particle configuration is created at random. For the proposed identification procedure, probabilistic Bayesian formulation is used to obtain robustness and stability, and time requirements are kept at a reasonable level thanks to the LDPM’s polynomial chaos approximation. The Bayesian formulation solves such an inverse problem as well–posed and provides a quantitative assessment of the underlying uncertainty for the values of material properties identified. The procedure is applied to identify seven material parameters from an unconfined compression cube test and notched three-point bending test. Its efficiency is verified on synthetic data with known parameter values in order to quantify the accuracy of the estimates and it is also validated using experimental data to prove its robustness.

Stochastic linear quadratic control via random parameter‐dependent truncated balanced realization

AbstractThis article deals with stochastic linear quadratic (LQ) control by using random parameter‐dependent truncated balanced realization (RPD‐TBR). The main contribution of the article is theoretical formulation to approximate high‐order stochastic LQ control problems, using the RPD‐TBR instead of the original RPD full state model. The whole approach uses the nonintrusive and intrusive generalized polynomial chaos (GPC) approximations to generate an RPD‐TBR and the corresponding reduced‐order stochastic LQ control formulation. The latter is then rewritten in the GPC space using an intrusive Galerkin‐type projection. Solving the reduced‐order Riccati equation for the resulting deterministic LQ problem enables to suitably approximate the solution of the original full‐order stochastic LQ control problem. The feasibility and effectiveness of the proposed approach are assessed by considering the LQ control of the tip‐tilt mirror for an Extremely Large Telescope‐class adaptive optics system, when the actuator's dynamics depends on an uncertain parameter with known distribution. Through numerical simulations, the proposed approach ability to perform LQ gain selection and performance evaluation for plants with probabilistic uncertainty more efficiently than the standard Monte Carlo approach is illustrated.

Practical uncertainty quantification analysis involving statistically dependent random variables

This article presents a practical refinement of generalized polynomial chaos expansion for uncertainty quantification under dependent input random variables. Unlike the Rodrigues-type formula, which exists for select probability measures, a three-step computational algorithm is put forward to generate a sequence of any approximate measure-consistent multivariate orthonormal polynomials. For uncertainty quantification analysis under dependent random variables, two regression methods, comprising existing standard least-squares and newly developed partitioned diffeomorphic modulation under observable response preserving homotopy (D-MORPH), are proposed to estimate the coefficients of generalized polynomial chaos expansion for the very first time. In contrast to the existing regression devoted so far to the classical polynomial chaos expansion, no tensor-product structure is required or enforced. The partitioned D-MORPH regression is applicable to either an underdetermined or overdetermined system, thus substantially enhancing the ability of the original D-MORPH regression. Numerical results obtained for Gaussian and non-Gaussian probability measures with rectangular or non-rectangular domains point to highly accurate orthonormal polynomials produced by the three-step algorithm. More significantly, the generalized polynomial chaos approximations of mathematical functions and stochastic responses from solid-mechanics problems, in tandem with the partitioned D-MORPH regression, provide excellent estimates of the second-moment properties and reliability from only hundreds of function evaluations or finite element analyses.

Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modeled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection–extraction problem.

Analyticity, Regularity, and Generalized Polynomial Chaos Approximation of Stochastic, Parametric Parabolic Two-Scale Partial Differential Equations

We study two-scale parabolic partial differential equations whose coefficient is stochastic and depends linearly on a sequence of pairwise independent random variables which are uniformly distributed in a compact interval. We cast the problem into a deterministic two-scale parabolic problem which depends on a sequence of real parameters in a compact interval. Passing to the limit when the microscale tends to zero, using two-scale homogenization, we obtain the parametric two-scale homogenized problem. This problem contains the solution to the homogenized equation which describes the solution to the original two-scale parabolic problem macroscopically, and the corrector which encodes the microscopic information. The solution of this two-scale homogenized equation is represented as a generalized polynomial chaos (gpc) expansion according to a polynomial basis of the L2 space of the parameters. We use a semidiscrete Galerkin approximation which projects the solution into a space of parametric functions which contain a finite number of pre-chosen gpc modes. Analyticity of the solution of the two-scale homogenized equation with respect to the parameters is established. Under mild assumptions, we show the summability of the coefficients of the solution’s gpc expansion. From this, an explicit error estimate for the semidiscrete Galerkin approximation in terms of the number of the chosen gpc modes is derived when these gpc modes are chosen as the N best ones according to the norms of the gpc coefficients. Regularity of the gpc coefficients and summability of their norms in the regularity spaces are also established. Using the solution of the best N term semidiscrete Galerkin approximation, we derive an approximation for the solution of the original two-scale problem, with an explicit convergence rate in terms of the microscopic scale and the number of gpc modes in the Galerkin approximation.

Calculating the minimal/maximal eigenvalue of symmetric parameterized matrices using projection

SummaryIn applications of linear algebra including nuclear physics and structural dynamics, there is a need to deal with uncertainty in the matrices. We focus on matrices that depend on a set of parameters ω and we are interested in the minimal eigenvalue of a matrix pencil (A,B) with A,B symmetric and B positive definite. If ω can be interpreted as the realization of random variables, one may be interested in statistical moments of the minimal eigenvalue. In order to obtain statistical moments, we need a fast evaluation of the eigenvalue as a function of ω. Because this is costly for large matrices, we are looking for a small parameterized eigenvalue problem whose minimal eigenvalue makes a small error with the minimal eigenvalue of the large eigenvalue problem. The advantage, in comparison with a global polynomial approximation (on which, e.g., the polynomial chaos approximation relies), is that we do not suffer from the possible nonsmoothness of the minimal eigenvalue. The small‐scale eigenvalue problem is obtained by projection of the large‐scale problem. Our main contribution is that, for constructing the subspace, we use multiple eigenvectors and derivatives of eigenvectors. We provide theoretical results and document numerical experiments regarding the beneficial effect of adding multiple eigenvectors and derivatives.

Some greedy algorithms for sparse polynomial chaos expansions

Compressed sensing algorithms approximate functions using limited evaluations by searching for a sparse representation among a dictionary of basis functions. Orthogonal matching pursuit (OMP) is a greedy algorithm for selecting basis functions whose computational cost scales with the size of the dictionary. For polynomial chaos (PC) approximations, the size of the dictionary grows quickly with the number of model inputs and the maximum polynomial degree, making them often prohibitive to use with greedy methods. We propose two new algorithms for efficiently constructing sparse PC expansions for problems with high-dimensional inputs. The first algorithm is a parallel OMP method coupled with an incremental QR factorization scheme, wherein the model construction step is interleaved with a ν-fold cross-validation procedure. The second approach is a randomized greedy algorithm that leverages a probabilistic argument to only evaluate a subset of basis functions from the dictionary at each iteration of the incremental algorithm. The randomized algorithm is demonstrated to recover model outputs with a similar level of sparsity and accuracy as OMP, but with a cost that is independent of the dictionary size. Both algorithms are validated with a numerical comparison of their performance on a series of algebraic test problems and PDEs with high-dimensional inputs.

Polynomial (chaos) approximation of maximum eigenvalue functions

This paper is concerned with polynomial approximations of the spectral abscissa function (the supremum of the real parts of the eigenvalues) of a parameterized eigenvalue problem, which are closely related to polynomial chaos approximations if the parameters correspond to realizations of random variables. Unlike in existing works, we highlight the major role of the smoothness properties of the spectral abscissa function. Even if the matrices of the eigenvalue problem are analytic functions of the parameters, the spectral abscissa function may not be everywhere differentiable, even not everywhere Lipschitz continuous, which is related to multiple rightmost eigenvalues or rightmost eigenvalues with multiplicity higher than one. The presented analysis demonstrates that the smoothness properties heavily affect the approximation errors of the Galerkin and collocation-based polynomial approximations, and the numerical errors of the evaluation of coefficients with integration methods. A documentation of the experiments, conducted on the benchmark problems through the software Chebfun, is publicly available.

Compressive sensing adaptation for polynomial chaos expansions

Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorithms, resulting in a reduced polynomial chaos approximation with optimal sparsity. The developed adaptation algorithm consists of a two-step optimization procedure that computes the optimal coefficients and the input projection matrix of a low dimensional chaos expansion with respect to an optimally rotated basis. We demonstrate the attractive features of our algorithm through several numerical examples including the application on Large-Eddy Simulation (LES) calculations of turbulent combustion in a HIFiRE scramjet engine.

Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in UQ

We estimate the expressive power of certain deep neural networks (DNNs for short) on a class of countably-parametric, holomorphic maps [Formula: see text] on the parameter domain [Formula: see text]. Dimension-independent rates of best [Formula: see text]-term truncations of generalized polynomial chaos (gpc for short) approximations depend only on the summability exponent of the sequence of their gpc expansion coefficients. So-called [Formula: see text]-holomorphic maps [Formula: see text], with [Formula: see text] for some [Formula: see text], are known to allow gpc expansions with coefficient sequences in [Formula: see text]. Such maps arise for example as response surfaces of parametric PDEs, with applications in PDE uncertainty quantification (UQ) for many mathematical models in engineering and the sciences. Up to logarithmic terms, we establish the dimension independent approximation rate [Formula: see text] for these functions in terms of the total number [Formula: see text] of units and weights in the DNN. It follows that certain DNN architectures can overcome the curse of dimensionality when expressing possibly countably-parametric, real-valued maps with a certain degree of sparsity in the sequences of their gpc expansion coefficients. We also obtain rates of expressive power of DNNs for countably-parametric maps [Formula: see text], where [Formula: see text] is the Hilbert space [Formula: see text].

A Monte Carlo approach to computing stiffness matrices arising in polynomial chaos approximations

We use a Monte Carlo method to assemble finite element matrices for polynomial Chaos approximations of elliptic equations with random coefficients. In this approach, all expectations are approximated by a Monte Carlo method. The resulting methodology requires dealing with sparse block-diagonal matrices instead of block-full matrices. This leads to the solution of a coupled system of elliptic equations where the coupling is given by a Kronecker product matrix involving polynomial evaluation matrices. This generalizes the Classical Monte Carlo approximation and Collocation method for approximating functionals of solutions of these equations.

A preconditioning approach for improved estimation of sparse polynomial chaos expansions

Compressive sampling has been widely used for sparse polynomial chaos (PC) approximation of stochastic functions. The recovery accuracy of compressive sampling highly depends on the incoherence properties of the measurement matrix. In this paper, we consider preconditioning the underdetermined system of equations that is to be solved. Premultiplying a linear equation system by a non-singular matrix results in an equivalent equation system, but it can potentially improve the incoherence properties of the resulting preconditioned measurement matrix and lead to a better recovery accuracy. When measurements are noisy, however, preconditioning can also potentially result in a worse signal-to-noise ratio, thereby deteriorating recovery accuracy. In this work, we propose a preconditioning scheme that improves the incoherence properties of measurement matrix and at the same time prevents undesirable deterioration of signal-to-noise ratio. We provide theoretical motivations and numerical examples that demonstrate the promise of the proposed approach in improving the accuracy of estimated polynomial chaos expansions.

Comments on Truncation Errors for Polynomial Chaos Expansions

Methods based on polynomial chaos expansion allow to approximate the behavior of systems with uncertain parameters by deterministic dynamics. These methods are used in a wide range of applications, spanning from simulation of uncertain systems to estimation and control. For practical purposes the exploited spectral series expansion is typically truncated to allow for efficient computation, which leads to approximation errors. Despite the Hilbert space nature of polynomial chaos, there are only a few results in the literature that explicitly discuss and quantify these approximation errors. This work derives error bounds for polynomial chaos approximations of polynomial and non-polynomial mappings. Sufficient conditions are established, which allow investigating the question whether zero truncation errors can be achieved and which series order is required to achieve this. Furthermore, convex quadratic programs, whose argmin operator is a special case of a piecewise polynomial mapping, are studied due to their relevance in predictive control. Several simulation examples illustrate our findings.

A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty

This paper is concerned with generalized polynomial chaos (gPC) approximation for first-order quasilinear hyperbolic systems with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the Jacobian matrix. Then the gPC stochastic Galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gPC expansion coefficients. The resulting deterministic gPC Galerkin system is discretized by a path-conservative finite volume WENO scheme in space and a third-order total variation diminishing Runge–Kutta method in time. The method is further extended to two-dimensional (2D) quasilinear hyperbolic system with uncertainty, where the symmetric hyperbolicity of the one-dimensional gPC Galerkin system is carried over via an operator splitting technique. Several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gPC stochastic Galerkin method.

Effectively Subsampled Quadratures for Least Squares Polynomial Approximations

This paper proposes a new deterministic sampling strategy for constructing polynomial chaos approximations for expensive physics simulation models. The proposed approach, effectively subsampled quadratures, involves sparsely subsampling an existing tensor grid using QR column pivoting. For polynomial interpolation using hyperbolic or total order sets, we then solve the following square least squares problem. For polynomial approximation, we use a column pruning heuristic that removes columns based on the highest total orders and then solves the tall least squares problem. While we provide bounds on the condition number of such tall submatrices, it is difficult to ascertain how column pruning affects solution accuracy as this is problem specific. We conclude with numerical experiments on an analytical function and a model piston problem that show the efficacy of our approach compared with randomized subsampling. We also show an example where this method fails.

A Christoffel function weighted least squares algorithm for collocation approximations

We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial Chaos approximation in uncertainty quantification where a polynomial approximation is formed from a combination of orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density of orthogonality. Our proposed algorithm samples with respect to the equilibrium measure of the parametric domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.

A stochastic Galerkin method for the Boltzmann equation with uncertainty

We develop a stochastic Galerkin method for the Boltzmann equation with uncertainty. The method is based on the generalized polynomial chaos (gPC) approximation in the stochastic Galerkin framework, and can handle random inputs from collision kernel, initial data or boundary data. We show that a simple singular value decomposition of gPC related coefficients combined with the fast Fourier-spectral method (in velocity space) allows one to compute the high-dimensional collision operator very efficiently. In the spatially homogeneous case, we first prove that the analytical solution preserves the regularity of the initial data in the random space, and then use it to establish the spectral accuracy of the proposed stochastic Galerkin method. Several numerical examples are presented to illustrate the validity of the proposed scheme.