Abstract. This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of 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 treats easily a wide 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 of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables. Introduction. Thanks to the rapidly growing power of computers, numerical simulations are increasingly used to produce predictions of the behavior of complex physical or engineering systems. Some sources of errors arising in computer simulations now can be controlled and reduced by using sophisticated techniques such as a posteriori error estimation [1,3,57], mesh adaptivity, and the more recent modeling error analysis [49,50,13]. All this has increased the accuracy of numerical predictions as well as our confidence in them.Yet many engineering applications are affected by a relatively large amount of uncertainty in the input data, such as model coefficients, forcing terms, boundary conditions, and geometry. In this case, to obtain a reliable numerical prediction, one has to include uncertainty quantification due to the uncertainty in the input data.Uncertainty can be described in several ways, depending on the amount of information available; among others, we mention the worst-case scenario analysis, fuzzy set theory, evidence theory, and probabilistic setting. See [6,35] and the references therein. In this paper we focus on elliptic partial differential equations (PDEs) with a probabilistic description of the uncertainty in the input data. The model problem has the formwhere L is an elliptic operator in a domain D ⊂ R d , which depends on some coefficients a (x, ω), with x ∈ D, ω ∈ Ω, and Ω indicating the set of possible outcomes. Similarly, the forcing term f = f (x, ω) can be assumed to be random as well.The first step in developing a stochastic collocation method consists in parametrizing the input randomness by means of a finite number of random variables. This can be the case i...

This work proposes and analyzes a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems as in the Monte Carlo method. If the number of random variables needed to describe the input data is moderately large, full tensor product spaces are computationally expensive to use due to the curse of dimensionality. In this case the sparse grid approach is still expected to be competitive with the classical Monte Carlo method. Therefore, it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than Monte Carlo. This work provides error estimates for the fully discrete solution using L q norms and analyzes the computational efficiency of the proposed method. In particular, it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem (number of input random variables) in the final estimates. The derived estimates are then used to compare the method with Monte Carlo, indicating for which problems the former is more efficient than the latter. Computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and Monte Carlo approaches.

We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the computations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method generates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The second method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h-or p-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.

Abstract. This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of 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 treats easily a wide 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 of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables. Introduction. Thanks to the rapidly growing power of computers, numerical simulations are increasingly used to produce predictions of the behavior of complex physical or engineering systems. Some sources of errors arising in computer simulations now can be controlled and reduced by using sophisticated techniques such as a posteriori error estimation [1,3,57], mesh adaptivity, and the more recent modeling error analysis [49,50,13]. All this has increased the accuracy of numerical predictions as well as our confidence in them.Yet many engineering applications are affected by a relatively large amount of uncertainty in the input data, such as model coefficients, forcing terms, boundary conditions, and geometry. In this case, to obtain a reliable numerical prediction, one has to include uncertainty quantification due to the uncertainty in the input data.Uncertainty can be described in several ways, depending on the amount of information available; among others, we mention the worst-case scenario analysis, fuzzy set theory, evidence theory, and probabilistic setting. See [6,35] and the references therein. In this paper we focus on elliptic partial differential equations (PDEs) with a probabilistic description of the uncertainty in the input data. The model problem has the formwhere L is an elliptic operator in a domain D ⊂ R d , which depends on some coefficients a (x, ω), with x ∈ D, ω ∈ Ω, and Ω indicating the set of possible outcomes. Similarly, the forcing term f = f (x, ω) can be assumed to be random as well.The first step in developing a stochastic collocation method consists in parametrizing the input randomness by means of a finite number of random variables. This can be the case i...

This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms (input data of the model). The method consists of a Galerkin approximation in the space variables and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem. These procedures seem to be very effective for the problems under study. The proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh approximately equally in the solution, while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables, as in the case where input random variables are Karhunen-Loève truncations of "smooth" random fields. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates (sub)exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime, with respect to the total number of collocation points. It also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo. In particular, for moderately large-dimensional problems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods.

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Abstract. We propose and analyze a novel Multi-Index Monte Carlo (MIMC) method for weak approximation of stochastic models that are described in terms of differential equations either driven by random measures or with random coefficients. The MIMC method is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Inspired by Giles's seminal work, we use in MIMC high-order mixed differences instead of using first-order differences as in MLMC to reduce the variance of the hierarchical differences dramatically. This in turn yields new and improved complexity results, which are natural generalizations of Giles's MLMC analysis and which increase the domain of the problem parameters for which we achieve the optimal convergence, O(TOL −2 ). Moreover, in MIMC, the rate of increase of required memory with respect to TOL is independent of the number of directions up to a logarithmic term which allows far more accurate solutions to be calculated for higher dimensions than what is possible when using MLMC.We motivate the setting of MIMC by first focusing on a simple full tensor index set. We then propose a systematic construction of optimal sets of indices for MIMC based on properly defined profits that in turn depend on the average cost per sample and the corresponding weak error and variance. Under standard assumptions on the convergence rates of the weak error, variance and work per sample, the optimal index set turns out to be the total degree (TD) type. In some cases, using optimal index sets, MIMC achieves a better rate for the computational complexity than the corresponding rate when using full tensor index sets. We also show the asymptotic normality of the statistical error in the resulting MIMC estimator and justify in this way our error estimate, which allows both the required accuracy and the confidence level in our computational results to be prescribed. Finally, we include numerical experiments involving a partial differential equation posed in three spatial dimensions and with random coefficients to substantiate the analysis and illustrate the corresponding computational savings of MIMC.

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