An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800–825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

In this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh 𝒯 H ⁢ ( D ) {\mathcal{T}_{H}(D)} with small scale of sparse collocation points η ⁢ ( L , N ) {\eta(L,N)} and solve a linearized equations on a fine mesh 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} using large scale of sparse collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} (where η ⁢ ( L , N ) , η ⁢ ( ℓ , N ) {\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels L , ℓ {L,\ell} in N dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when h ≈ H 3 , η ⁢ ( ℓ , N ) ≈ ( η ⁢ ( L , N ) ) 3 {{h\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}} , the novel two-level discretization method achieves the same convergence accuracy in norm ∥ ⋅ ∥ ℒ ρ 2 ⁢ ( Γ ) ⊗ ℒ 2 ⁢ ( D ) {\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} ( ℒ ρ 2 ⁢ ( Γ ) {\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted ℒ 2 {\mathcal{L}^{2}} space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} and large scale collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} in random spaces.

Summary A sparse grid approach to orbit uncertainty propagation is presented. Efficient and accurate uncertainty propagation methods for nonlinear dynamic systems have been of enormous interest to space object tracking. Recent methods include those based on the time evolution of the probability density function, the statistical moments, the random samples, or a sum of Gaussian components. The idea of the sparse grid method for orbit uncertainty propagation is to represent the initial uncertainty by a sparse grid, propagate the sparse grid points individually through the nonlinear orbit dynamics, and compute the statistical moments from the propagated sparse grid points. The Smolyak rule is used to generate the sparse grid through linear combinations of low-level tensor products; the number of the resultant sparse grid points is polynomial in the dimension of the state. The sparse grid based uncertainty propagation may be considered a generalization of the Unscented Transformation (UT) of the well-known Unscented Kalman Filter but can achieve much higher accuracy levels than UT. Two examples of orbit propagation in different dynamic environments are used to show the efficacy of the sparse grid approach.

Option pricing with a direct adaptive sparse grid approach

These days sparse grids are of increasing interest in numerical simulations. Based upon hierarchical tensor product bases, the sparse grid approach is a very efficient one improving the ratio of invested storage and computing time to the achieved accuracy for many problems in the area of numerical solution of differential equations, for instance in numerical fluid mechanics. The particle tracing algorithms that are available so far cannot cope with sparse grids. Now we present an approach that directly works on sparse grids. As a second aspect in this paper, we suggest to use sparse girds as a data compression method in order to visualize huge data sets even on small workstations. Because the size of data sets used in numerical simulations is still growing, this feature makes it possible that workstations can continue to handle these data sets.

The sparse grid approach uses basis functions which are tensor products of univariate basis functions. This gives a connection between sparse grids and the tensor space setting. On the other hand, in the tensor calculus one is interested in suitable representations of tensors (cf. Hackbusch, Tensor spaces and numerical tensor calculus, Monograph in preparation). Then the storage size is in particular described by certain rank parameters. Limited rank requires an approximation of the tensors (functions). In particular, function approximations like polynomial interpolation may lead to such representations. Here, we exploit the sparse grid approximation. We show that the involved rank is not the sparse grid complexity \(O(N{\log }^{d-1}N),\) but \(O({N{}^{(d-1)/d}\log }^{d-3}N).\) KeywordsSparse GridTensor SpaceNumerical Tensor CalculusUnivariate Basis FunctionsTucker FormatThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Risk-averse structural topology optimization under random fields using stochastic expansion methods

Simulation evaluation of emerging estimation techniques for multinomial probit models

The sparse grid combination technique provides a framework to solve high-dimensional numerical problems with standard solvers by assembling a sparse grid from many coarse and anisotropic full grids called component grids. Hierarchization is one of the most fundamental tasks for sparse grids. It describes the transformation from the nodal basis to the hierarchical basis. In settings where the component grids have to be frequently combined and distributed in a massively parallel compute environment, hierarchization on component grids is relevant to minimize communication overhead.

High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm to efficiently solve the transport equation in moderately complex geometrical domains using a Galerkin method stabilized by streamline diffusion. The ansatz spaces are a tensor product of a sparse grid in space and discontinuous piecewise polynomials in time. Here, the sparse grid is constructed upon nested multilevel finite element spaces to provide geometric flexibility. This results in an implicit time-stepping scheme which we prove to be stable and convergent. If the solution has additional mixed regularity, the convergence of a 2d-dimensional problem equals that of a d-dimensional one up to logarithmic factors. For the implementation, we rely on the representation of sparse grids as a sum of anisotropic full grid spaces. This enables us to store the functions and to carry out the computations on a sequence regular full grids exploiting the tensor product structure of the ansatz spaces. In this way existing finite element libraries and GPU acceleration can be used. The combination technique is used as a preconditioner for an iterative scheme to solve the transport equation on the sequence of time strips. Numerical tests show that the method works well for problems in up to six dimensions. Finally, the method is also used as a building block to solve nonlinear Vlasov-Poisson equations.

Relying on the classical connection between backward stochastic differential equations and nonlinear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional semilinear parabolic PDEs. This scheme is inspired by the approach coming from machine learning and developed using deep neural networks in Han et al. (2018, Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci., 115, 8505–8510. Our algorithm is based on a Picard iteration scheme in which a sequence of linear-quadratic optimization problem is solved by means of stochastic gradient descent algorithm. In the framework of a linear specification of the approximation space, we manage to prove a convergence result for our scheme, under some smallness condition. In practice, in order to be able to treat high-dimensional examples, we employ sparse-grid approximation spaces. In the case of periodic coefficients and using pre-wavelet basis functions, we obtain an upper bound on the global complexity of our method. It shows, in particular, that the curse of dimensionality is tamed in the sense that in order to achieve a root mean squared error of order $\varepsilon $, for a prescribed precision $\varepsilon $, the complexity of the Picard algorithm grows polynomially in $\varepsilon ^{-1}$ up to some logarithmic factor $|\!\log (\varepsilon )|$, whose exponent grows linearly with respect to the PDE dimension. Various numerical results are presented to validate the performance of our method, and to compare them with some recent machine learning schemes proposed in E et al. (2017, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat., 5, 349–380) and Huré et al. (2020, Deep backward schemes for high-dimensional nonlinear PDEs. Math. Comput., 89, 1547–1579).

In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems.

This paper presents a fully automated procedure to estimate the uncertainty of compressor stage performance, due to impeller manufacturing variability. The methodology was originally developed for 2D stages, i.e., stages for which the impeller blade angle and thickness distribution are only defined at the hub end-wall. Here, we extend the procedure to general 3D stages, for which blade angle and thickness distributions can be prescribed independently at the shroud and hub endwalls. Starting from the probability distribution of the impeller geometrical parameters, 3D sample geometries are generated and 1D/2D aerodynamic models are created, which are used to predict the performance of each sample geometry. The original procedure used the Monte Carlo method to propagate uncertainty. However, this requires a large number of samples to compute accurate performance statistics. Here we compare the results from Monte Carlo, with those obtained using Sparse Grid Polynomial Chaos Expansion (PCE) and a Multidimensional Cubature Rule for uncertainty propagation. PCE has exponential convergence in the stochastic space for smooth functions, and the use of sparse grids mitigates the increase of sample points due to the increase in the number of uncertain parameters. The cubature rule has accuracy limitations, but sample points increase only linearly with the number of parameters. For a 3D stage, the probability distributions of the performance characteristics are computed, as well as the sensitivity to the design parameters. The results show that PCE and Multidimensional Cubature give similar results to MC computations, with a much lower computational effort.

An adaptive sparse grid method for elliptic PDEs with stochastic coefficients

Non-intrusive polynomial chaos expansion (PCE) and stochastic collocation (SC) methods are attractive techniques for uncertainty quantification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic variability. PCE estimates coefficients for known orthogonal polynomial basis functions based on a set of response function evaluations, using sampling, linear regression, tensor-product quadrature, or Smolyak sparse grid approaches. SC, on the other hand, forms interpolation functions for known coefficients, and requires the use of structured collocation point sets derived from tensor product or sparse grids. When tailoring the basis functions or interpolation grids to match the forms of the input uncertainties, exponential convergence rates can be achieved with both techniques for a range of probabilistic analysis problems. In addition, analytic features of the expansions can be exploited for moment estimation and stochastic sensitivity analysis. In this paper, the latest ideas for tailoring these expansion methods to numerical integration approaches will be explored, in which expansion formulations are modified to best synchronize with tensor-product quadrature and Smolyak sparse grids using linear and nonlinear growth rules. The most promising stochastic expansion approaches are then carried forward for use in new approaches for mixed aleatory-epistemic UQ, employing second-order probability approaches, and design under uncertainty, employing bilevel, sequential, and multifidelity approaches.