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Generalized Dimension Truncation Error Analysis for High-Dimensional Numerical Integration: Lognormal Setting and Beyond
... y ∈ R s . which is controlled by the rate of decay of the eigenvalues (λ j ) j≥1 [3,16]. ...
... There have been recent attempts in the literature to generalize the aforementioned model problem by replacing the probability measure corresponding to the input random field with a generalized β-Gaussian distribution. Specifically, Herrmann et al. [19] studied QMC integration for Bayesian inverse problems governed by parametric PDEs, where the probability measure was assumed to belong to the family of generalized β-Gaussian distributions, while Guth and Kaarnioja [16] investigated the dimension truncation error rates subject to high-dimensional integration problems with respect to these probability measures. ...
... The following lemma, which is adapted from [16] to the present setting, allows us to interchange the domain of integration R N with U α,τ . ...
There has been a surge of interest in uncertainty quantification for parametric partial differential equations (PDEs) with Gevrey regular inputs. The Gevrey class contains functions that are infinitely smooth with a growth condition on the higher-order partial derivatives, but which are nonetheless not analytic in general. Recent studies by Chernov and Le (Comput. Math. Appl., 2024, and SIAM J. Numer. Anal., 2024) as well as Harbrecht, Schmidlin, and Schwab (Math. Models Methods Appl. Sci., 2024) analyze the setting wherein the input random field is assumed to be uniformly bounded with respect to the uncertain parameters. In this paper, we relax this assumption and allow for parameter-dependent bounds. The parametric inputs are modeled as generalized Gaussian random variables, and we analyze the application of quasi-Monte Carlo (QMC) integration to assess the PDE response statistics using randomly shifted rank-1 lattice rules. In addition to the QMC error analysis, we also consider the dimension truncation and finite element errors in this setting.
... Function approximation Affine parametric [6,20] [ 16] operator equation setting rate O(s − 2 p +1 ) rate O(s − 1 p + 1 2 ) Non-affine parametric [8,12] A natural first step for the numerical treatment of (1) is the approximation by a dimensionally-truncated model M s : X × U s → Y such that M s (g s (y ≤s ), y ≤s ) = 0, where ∅ = U s ⊆ R s and g s (y ≤s ) ∈ X for all y ≤s ∈ U s . Consider the problem of finding a surrogate solution g s,n := A n (g s ) using an algorithm A n which uses n point evaluations of the s-dimensional function g s , where the surrogate belongs to X such that ...
... In the non-affine setting, using Taylor series makes it possible to derive dimension truncation error rates by exploiting the parametric regularity of the problem, whereas the Neumann series approach relies fundamentally on the parametric structure of the model. The Taylor series approach was first applied in [8], and motivated the authors in [11] and [12] to derive dimension truncation error rates for sufficiently smooth, Banach space valued integrands, and with parameters following a generalized β-Gaussian distribution. An overview of the various dimension truncation error bounds studied in the literature is given in Table 1. ...
... In the special case of the uniform distribution µ(dy) = dy, we can apply [12,Theorem 4.2] to obtain ...
Parametric mathematical models such as partial differential equations with random coefficients have received a lot of attention within the field of uncertainty quantification. The model uncertainties are often represented via a series expansion in terms of the parametric variables. In practice, this series expansion needs to be truncated to a finite number of terms, introducing a dimension truncation error to the numerical simulation of a parametric mathematical model. There have been several studies of the dimension truncation error corresponding to different models of the input random field in recent years, but many of these analyses have been carried out within the context of numerical integration. In this paper, we study the dimension truncation error of the parametric model problem. Estimates of this kind arise in the assessment of the dimension truncation error for function approximation in high dimensions. In addition, we show that the dimension truncation error rate is invariant with respect to certain transformations of the parametric variables. Numerical results are presented which showcase the sharpness of the theoretical results.
... This assumption is addressed in Appendix C. Due to Theorem 4.3 and (5.1), we can apply [12,Thm. 4.3], which yields ...
A control in feedback form is derived for linear quadratic, time-invariant optimal control problems subject to parabolic partial differential equations with coefficients depending on a countably infinite number of uncertain parameters. It is shown that the Riccati-based feedback operator depends analytically on the parameters provided that the system operator depends analytically on the parameters, as is the case, for instance, in diffusion problems when the diffusion coefficient is parameterized by a Karhunen--Lo\`eve expansion. These novel parametric regularity results allow the application of quasi-Monte Carlo (QMC) methods to efficiently compute an a-priori chosen feedback law based on the expected value. Moreover, under moderate assumptions on the input random field, QMC methods achieve superior error rates compared to ordinary Monte Carlo methods, independently of the stochastic dimension of the problem. Indeed, our paper for the first time studies Banach-space-valued integration by higher-order QMC methods.
... Many studies in uncertainty quantification for PDEs with random coefficients exploit the parametric structure of the problem, using a Neumann series expansion to obtain dimension truncation error bounds. However, there have been several recent studies suggesting a Taylor series approach which is based on the parametric regularity of the problem [8,10,11]. The Taylor series approach can be used to obtain dimension truncation rates for non-affine parametric PDE problems, but a limitation of the aforementioned papers is that the parametric regularity bound needs to be of product-and-order dependent (POD) form in order for the Taylor-based approach to yield useful results. ...
We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja, Kuo, and Sloan (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.
The theoretical development of quasi-Monte Carlo (QMC) methods for uncertainty quantification of partial differential equations (PDEs) is typically centered around simplified model problems such as elliptic PDEs subject to homogeneous zero Dirichlet boundary conditions. In this paper, we present a theoretical treatment of the application of randomly shifted rank-1 lattice rules to electrical impedance tomography (EIT). EIT is an imaging modality, where the goal is to reconstruct the interior conductivity of an object based on electrode measurements of current and voltage taken at the boundary of the object. This is an inverse problem, which we tackle using the Bayesian statistical inversion paradigm. As the reconstruction, we consider QMC integration to approximate the unknown conductivity given current and voltage measurements. We prove under moderate assumptions placed on the parameterization of the unknown conductivity that the QMC approximation of the reconstructed estimate has a dimension-independent, faster-than-Monte Carlo cubature convergence rate. Finally, we present numerical results for examples computed using simulated measurement data.
This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice—a setting that, as pointed out by Zeng et al. (Monte Carlo and Quasi-Monte Carlo Methods 2004, Springer, New York, 2006) and Zeng et al. (Constr. Approx. 30: 529–555, 2009), allows fast evaluation by fast Fourier transform, so avoiding the need for a linear solver. The main contribution of the paper is the application to the approximation problem for uncertainty quantification of elliptic partial differential equations, with the diffusion coefficient given by a random field that is periodic in the stochastic variables, in the model proposed recently by Kaarnioja et al. (SIAM J Numer Anal 58(2): 1068–1091, 2020). The paper gives a full error analysis, and full details of the construction of lattices needed to ensure a good (but inevitably not optimal) rate of convergence and an error bound independent of dimension. Numerical experiments support the theory.
We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients, motivated by problems in structural mechanics, photonic crystals and neutron diffusion. The PDE coefficients are assumed to be uniformly bounded random fields, represented as infinite series parametrised by uniformly distributed i.i.d. random variables. The expectation of the fundamental eigenvalue of this problem is computed by (a) truncating the infinite series which define the coefficients; (b) approximating the resulting truncated problem using lowest order conforming finite elements and a sparse matrix eigenvalue solver; and (c) approximating the resulting finite (but high dimensional) integral by a randomly shifted quasi-Monte Carlo lattice rule, with specially chosen generating vector. We prove error estimates for the combined error, which depend on the truncation dimension s, the finite element mesh diameter h, and the number of quasi-Monte Carlo samples N. Under suitable regularity assumptions, our bounds are of the particular form , where is arbitrary and the hidden constant is independent of the truncation dimension, which needs to grow as and . As for the analogous PDE source problem, the conditions under which our error bounds hold depend on a parameter representing the summability of the terms in the series expansions of the coefficients. Although the eigenvalue problem is nonlinear, which means it is generally considered harder than the source problem, in almost all cases () we obtain error bounds that converge at the same rate as the corresponding rate for the source problem. The proof involves a detailed study of the regularity of the fundamental eigenvalue as a function of the random parameters. As a key intermediate result in the analysis, we prove that the spectral gap (between the fundamental and the second eigenvalues) is uniformly positive over all realisations of the random problem.
The family of Generalized Gaussian (GG) distributions has received considerable attention from the engineering community, due to the flexible parametric form of its probability density function, in modeling many physical phenomena. However, very little is known about the analytical properties of this family of distributions, and the aim of this work is to fill this gap.
Roughly, this work consists of four parts. The first part of the paper analyzes properties of moments, absolute moments, the Mellin transform, and the cumulative distribution function. For example, it is shown that the family of GG distributions has a natural order with respect to second-order stochastic dominance.
The second part of the paper studies product decompositions of GG random variables. In particular, it is shown that a GG random variable can be decomposed into a product of a GG random variable (of a different order) and an independent positive random variable. The properties of this decomposition are carefully examined.
The third part of the paper examines properties of the characteristic function of the GG distribution. For example, the distribution of the zeros of the characteristic function is analyzed. Moreover, asymptotically tight bounds on the characteristic function are derived that give an exact tail behavior of the characteristic function. Finally, a complete characterization of conditions under which GG random variables are infinitely divisible and self-decomposable is given.
The fourth part of the paper concludes this work by summarizing a number of important open questions.
We analyze convergence rates of quasi-Monte Carlo (QMC) quadratures for countably-parametric solutions of linear, elliptic partial differential equations (PDE) in divergence form with log-Gaussian diffusion coefficient, based on the error bounds in Nichols and Kuo (J Complex 30(4):444–468, 2014. https://doi.org/10.1016/j.jco.2014.02.004). We prove, for representations of the Gaussian random field PDE input with locally supported basis functions, and for continuous, piecewise polynomial finite element discretizations in the physical domain novel QMC error bounds in weighted spaces with product weights that exploit localization of supports of the basis elements representing the input Gaussian random field. In this case, the cost of the fast component-by-component algorithm for constructing the QMC points scales linearly in terms of the integration dimension. The QMC convergence rate (independent of the parameter space dimension s) is achieved under weak summability conditions on the expansion coefficients.
In a previous paper (J. Comp. Phys. \textbf{230} (2011), 3668--3694), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected values with circulant embedding methods for sampling the random field on a regular grid. It was found capable of handling fluid flow problems in random heterogeneous media with high stochastic dimension, but a convergence theory was missing. This paper provides a convergence analysis for the method in the case when the QMC method is a specially designed randomly shifted lattice rule. The convergence result depends on the eigenvalues of the underlying nested block circulant matrix and can be independent of the number of stochastic variables under certain assumptions. In fact the QMC analysis applies to general factorisations of the covariance matrix to sample the random field. The error analysis for the underlying fully discrete finite element method allows for locally refined meshes (via interpolation from a regular sampling grid of the random field). Numerical results on a non-regular domain with corner singularities in two spatial dimensions and on a regular domain in three spatial dimensions are included.
We analyze combined Quasi-Monte Carlo quadrature and Finite Element approximations in Bayesian estimation of solutions to countably-parametric operator equations with holomorphic dependence on the parameters as considered in [Cl.~Schillings and Ch.~Schwab: Sparsity in Bayesian Inversion of Parametric Operator Equations. Inverse Problems, {\bf 30}, (2014)]. Such problems arise in numerical uncertainty quantification and in Bayesian inversion of operator equations with distributed uncertain inputs, such as uncertain coefficients, uncertain domains or uncertain source terms and boundary data. We show that the parametric Bayesian posterior densities belong to a class of weighted Bochner spaces of functions of countably many variables, with a particular structure of the QMC quadrature weights: up to a (problem-dependent, and possibly large) finite dimension S product weights can be used, and beyond this dimension, weighted spaces with so-called SPOD weights are used to describe the solution regularity. We establish error bounds for higher order Quasi-Monte Carlo quadrature for the Bayesian estimation based on [J.~Dick, Q.T.~LeGia and Ch.~Schwab, Higher order Quasi-Monte Carlo integration for holomorphic, parametric operator equations, Report 2014-23, SAM, ETH Z\"urich]. It implies, in particular, regularity of the parametric solution and of the countably-parametric Bayesian posterior density in SPOD weighted spaces. This, in turn, implies that the Quasi-Monte Carlo quadrature methods in [J. Dick, F.Y.~Kuo, Q.T.~Le Gia, D.~Nuyens, Ch.~Schwab, Higher order QMC Galerkin discretization for parametric operator equations, SINUM (2014)] are applicable to these problem classes, with dimension-independent convergence rates \calO(N^{-1/p}) of N-point HoQMC approximated Bayesian estimates, where depends only on the sparsity class of the uncertain input in the Bayesian estimation.
In this paper we present a rigorous cost and error analysis of a multilevel
estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for
lognormal diffusion problems. These problems are motivated by uncertainty
quantification problems in subsurface flow. We extend the convergence analysis
in [Graham et al., Numer. Math. 2014] to multilevel Quasi-Monte Carlo finite
element discretizations and give a constructive proof of the
dimension-independent convergence of the QMC rules. More precisely, we provide
suitable parameters for the construction of such rules that yield the required
variance reduction for the multilevel scheme to achieve an -error
with a cost of with , and in
practice even , for sufficiently fast decaying covariance
kernels of the underlying Gaussian random field inputs. This confirms that the
computational gains due to the application of multilevel sampling methods and
the gains due to the application of QMC methods, both demonstrated in earlier
works for the same model problem, are complementary. A series of numerical
experiments confirms these gains. The results show that in practice the
multilevel QMC method consistently outperforms both the multilevel MC method
and the single-level variants even for non-smooth problems.
We analyze the convergence of higher order Quasi-Monte Carlo (QMC)
quadratures of solution-functionals to countably-parametric, nonlinear operator
equations with distributed uncertain parameters taking values in a separable
Banach space X. Such equations arise in numerical uncertainty quantification
with random field inputs. The unconditional bases of X render the random
inputs and the solutions of the forward problem countably parametric. We show
that these parametric solutions belong to a class of weighted Bochner spaces of
functions of countably many variables, with a particular structure of the QMC
quadrature weights: up to a (problem-dependent, and possibly large) finite
dimension, product weights can be used, and beyond this dimension, weighted
spaces with so-called SPOD weights recently introduced in [F.Y.~Kuo,
Ch.~Schwab, I.H.~Sloan, Quasi-Monte Carlo finite element methods for a class of
elliptic partial differential equations with random coefficients. SIAM J.
Numer. Anal. 50 (2012), 3351--3374] can be used to describe the solution
regularity. The regularity results in the present paper extend those in [J.
Dick, F.Y.~Kuo, Q.T.~Le Gia, D.~Nuyens, Ch.~Schwab, Higher order QMC
(Petrov-)Galerkin discretization for parametric operator equations. To appear
in SIAM J. Numer. Anal., 2015. Available at arXiv:1309.4624] established for
affine parametric, linear operator families; they imply, in particular,
efficient constructions of (sequences of) QMC quadrature methods there, which
are applicable to these problem classes. We present a hybridized version of the
fast component-by-component (CBC for short) construction of a certain type of
higher order digital net.
In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in R d (d = 1, 2, 3), with diffusion coefficient a(x, ω) given as a lognormal random field, i.e., a(x, ω) = exp(Z(x, ω)) where x is the spatial variable and Z(x, ·) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 or ∞ over all possible realizations of a. Fo-cusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem, we give a rigorous error analysis for methods constructed from (i) standard continuous and piecewise linear finite element approximation in physical space; (ii) truncated Karhunen-Loéve expansion for computing realizations of a (leading to a possi-bly high-dimensional parametrized deterministic diffusion problem); and (iii) lattice-based Quasi-Monte Carlo (QMC) quadrature rules for computing integrals over parameter space which define the expected values. The paper contains novel error analysis which accounts for the effect of all three types of approximation. The QMC analysis is based on a recent result on randomly shifted lattice rules for high-dimensional integrals over the unbounded domain of Euclidean space, which shows that (under suitable conditions) the quadrature error decays with O(n −1+δ) with respect to the number of quadrature points n, where δ > 0 is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration.
We construct quasi-Monte Carlo methods to approximate the expected values of
linear functionals of Galerkin discretizations of parametric operator equations
which depend on a possibly infinite sequence of parameters. Such problems arise
in the numerical solution of differential and integral equations with random
field inputs. We analyze the regularity of the solutions with respect to the
parameters in terms of the rate of decay of the fluctuations of the input
field. If denotes the "summability exponent" corresponding to the
fluctuations in affine-parametric families of operators, then we prove that
deterministic "interlaced polynomial lattice rules" of order in s dimensions with N points can be constructed using a
fast component-by-component algorithm, in operations, to achieve a convergence rate of
, with the implied constant independent of s. This
dimension-independent convergence rate is superior to the rate
, for recently established for
randomly shifted lattice rules under comparable assumptions. In our analysis we
use a non-standard Banach space setting and introduce "smoothness-driven
product and order dependent (SPOD)" weights for which we show fast CBC
construction.
Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain
D⊂ℝ
d
are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion
coefficients in L
2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(ω)=(y
i
(ω)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable x∈D and the in general countably many parametersy.
We establish new regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−1,1)∞ to V=H10(D)V=H^{1}_{0}(D). These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called “generalized polynomial chaos” (gpc) expansion of u.
Convergence estimates of approximations of u by best N-term truncated V valued polynomials in the variable y∈U are established. These estimates are of the form N
−r
, where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N “samples” (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients.
A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family {Vl}l=0¥ Ì V\{V_{l}\}_{l=0}^{\infty}\subset V of finite element spaces in D of the coefficients in the N-term truncated gpc expansions of u(x,y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of
the solution u as a map from y∈U=(−1,1)∞ to a smoothness space W⊂V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with H2(D)ÇH10(D)H^{2}(D)\cap H^{1}_{0}(D) in the case where D is a smooth or convex domain.
Our analysis shows that in realistic settings a convergence rate Ndof-sN_{\mathrm{dof}}^{-s} in terms of the total number of degrees of freedom N
dof can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization inD.
KeywordsStochastic and parametric elliptic equations-Wiener polynomial chaos-Approximation rates-Nonlinear approximation-Sparsity
Mathematics Subject Classification (2000)41A-65N-65C30
We consider linear systems A(α)x(α) = b(α) depending on possibly many parameters α = (α1,...,αp). Solving these systems simultaneously for a standard discretization of the parameter space would require a computational effort growing exponentially in the number of parameters. We show that this curse of dimensionality can be avoided for sufficiently smooth parameter dependencies. For this purpose, computational methods are developed that benefit from the fact that x(α) can be well approximated by a tensor of low rank. In particular, low-rank tensor variants of short-recurrence Krylov subspace methods are presented. Numerical experiments for deterministic PDEs with parametrized coefficients and stochastic elliptic PDEs demonstrate the effectiveness of our approach.
We estimate the expressive power of certain deep neural networks (DNNs for short) on a class of countably-parametric, holomorphic maps u: U. on the parameter domain U = [-1, 1].,•. Dimension-independent rates of best n-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 (b,)-holomorphic maps u, with b .,"p for some p (0, 1), are known to allow gpc expansions with coefficient sequences in .,"p. 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 s = 1/p-1 for these functions in terms of the total number N 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 u: U' V, where V is the Hilbert space H01([0, 1]).
An application of quasi-Monte Carlo methods of significant recent interest in the MCQMC community is the quantification of uncertainties in partial differential equation models. Uncertainty quantification for both forward problems and Bayesian inverse problems leads to high-dimensional integrals that are well-suited for QMC approximation. One of the approximations required in a general formulation as an affine-parametric operator equation is the truncation of the formally infinite-parametric operator to a finite number of dimensions. To date, a numerical study of the available theoretical convergence rates for this error have to the author’s knowledge not been published. We present novel results for a selection of model problems, the computation of which has been enabled by recently developed, higher-order QMC methods based on interlaced polynomial lattice rules. Surprisingly, the observed rates are one order better in the case of integration over the parameters than the commonly cited theory suggests; a proof of this higher rate is included, resulting in a theoretical statement consistent with the observed numerics.
We present an error analysis of higher order Quasi-Monte Carlo (QMC) integration and of randomly shifted QMC lattice rules for parametric operator equations with uncertain input data taking values in Banach spaces. Parametric expansions of these input data in locally supported bases such as splines or wavelets was shown in Gantner et al. (SIAM J Numer Anal 56(1):111–135, 2018) to allow for dimension independent convergence rates of combined QMC-Galerkin approximations. In the present work, we review and refine the results in that reference to the multilevel setting, along the lines of Kuo et al. (Found Comput Math 15(2):441–449, 2015) where randomly shifted lattice rules and globally supported representations were considered, and also the results of Dick et al. (SIAM J Numer Anal 54(4):2541–2568, 2016) in the particular situation of locally supported bases in the parametrization of uncertain input data. In particular, we show that locally supported basis functions allow for multilevel QMC quadrature with product weights, and prove new error vs. work estimates superior to those in these references (albeit at stronger, mixed regularity assumptions on the parametric integrand functions than what was required in the single-level QMC error analysis in the first reference above). Numerical experiments on a model affine-parametric elliptic problem confirm the analysis.
This article provides a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to elliptic partial differential equations (PDEs) with random diffusion coefficients. It considers, and contrasts, the uniform case versus the lognormal case, single-level algorithms versus multi-level algorithms, first order QMC rules versus higher order QMC rules, and deterministic QMC methods versus randomized QMC methods. It gives a summary of the error analysis and proof techniques in a unified view, and provides a practical guide to the software for constructing and generating QMC points tailored to the PDE problems. The analysis for the uniform case can be generalized to cover a range of affine parametric operator equations.
In this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a countably infinite number of terms in a Karhunen-Loève expansion. Models of this kind appear frequently in numerical models of physical systems, and in uncertainty quantification. The method uses a QMC method to estimate expected values of linear functionals of the exact or approximate solution of the PDE, with the expected value considered as an infinite dimensional integral in the parameter space corresponding to the randomness induced by the random coefficient. The error analysis, arguably the first rigorous application of the modern theory of QMC in weighted spaces, exploits the regularity with respect to both the physical variables (the variables in the physical domain) and the parametric variables (the parameters corresponding to randomness). In the weighted-space theory of QMC methods, "weights," describing the varying difficulty of different subsets of the variables, are introduced in order to make sure that the high-dimensional integration problem is tractable. It turns out that the weights arising from the present analysis are of a nonstandard kind, being of neither product nor order dependent form, but instead a hybrid of the two-we refer to these as "product and order dependent weights," or "POD weights" for short. These POD weights are of a simple enough form to permit a component-by-component construction of a randomly shifted lattice rule that has optimal convergence properties for the given weighted space setting. If the terms in the expansion for the random coefficient have an appropriate decay property, and if we choose POD weights that minimize a certain upper bound on the error, then the solution of the PDE belongs to the joint function space needed for the analysis, and the QMC error (in the sense of a root-mean-square error averaged over shifts) is of order O(N-1+δ) for arbitrary δ > 0, where N denotes the number of sampling points in the parameter space. Moreover, this convergence rate and slower convergence rates under more relaxed conditions are achieved under conditions similar to those found recently by Cohen, De Vore, and Schwab [Found. Comput. Math., 10 (2010), pp. 615- 646] for the same model by best N-term approximation. We analyze the impact of a finite element (FE) discretization on the overall efficiency of the scheme, in terms of accuracy versus overall cost, with results that are comparable to those of the best N-term approximation.
We develop the generic "Multivariate Decomposition Method" (MDM) for weighted
integration of functions of infinitely many variables .
The method works for functions that admit a decomposition
, where runs over all
finite subsets of positive integers, and for each
the function depends only
on . Moreover, we assume that
belongs to a normed space , and that a bound on
is known. We also specialize MDM to
being the -tensor product of an anchored
reproducing kernel Hilbert space, or a particular non-Hilbert space.
Initial boundary value problems of linear second-order hyperbolic partial differential equations whose coefficients depend on countably many random parameters are reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space. This parametric family is approximated by Galerkin projection onto finitely supported polynomial systems in the parameter space. We establish uniform stability with respect to the support of the resulting coupled hyperbolic systems, and provide sufficient smoothness and compatibility conditions on the data for the solution to exhibit analytic, respectively, Gevrey regularity with respect to the countably many parameters. Sufficient conditions for the p-summability of the generalized polynomial chaos expansion of the parametric solution in terms of the countably many input parameters are obtained and rates of convergence of best N-term polynomial chaos type approximations of the parametric solution are given. In addition, regularity both in space and time for the parametric family of solutions is proved for data satisfying certain compatibility conditions. The results allow obtaining convergence rates and stability of sparse space-time tensor product Galerkin discretizations in the parameter space.
The stochastic Galerkin method is developed for the isotropic diffusion equation with an unbounded random diffusion coefficient. The logarithm of the diffusion coefficient is assumed to be an infinite series of Gaussian random variables.
Well-posed weak formulations of the model problem are derived on standard Bochner–Lebesgue spaces. The Galerkin solution is shown to be almost quasi-optimal in the sense that the error of the Galerkin projection can be estimated by a best approximation error in a slightly stronger norm. As a result, convergence analysis of the stochastic Galerkin method is reduced to the problem of approximating the Hermite coefficients of the exact solution by standard finite elements.
This is a survey of nonlinear approximation, especially that part of the subject which is important in numerical computation. Nonlinear approximation means that the approximants do not come from linear spaces but rather from nonlinear manifolds. The central question to be studied is what, if any, are the advantages of nonlinear approximation over the simpler, more established, linear methods. This question is answered by studying the rate of approximation which is the decrease in error versus the number of parameters in the approximant. The number of parameters usually correlates well with computational effort. It is shown that in many settings the rate of nonlinear approximation can be characterized by certain smoothness conditions which are significantly weaker than required in the linear theory. Emphasis in the survey will be placed on approximation by piecewise polynomials and wavelets as well as their numerical implementation. Results on highly nonlinear methods such as optimal basis selection and greedy algorithms (adaptive pursuit) are also given. Applications to image processing, statistical estimation, regularity for PDEs, and adaptive algorithms are discussed.
We devise and implement quasi-Monte Carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients. Our motivation comes from fluid flow in random porous media, where relevant functionals include the fluid pressure/velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field. Our emphasis is on situations where a very large number of random variables is needed to model the coefficient field. As an alternative to classical Monte Carlo, we here employ quasi-Monte Carlo methods, which use deterministically chosen sample points in an appropriate (usually high-dimensional) parameter space. Each realization of the PDE solution requires a finite element (FE) approximation in space, and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid (not necessarily the same as the FE grid). In the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using FFT. In this way we avoid the use of a truncated Karhunen–Loève expansion, but introduce high nominal dimension in parameter space. Numerical experiments with 2-dimensional rough random fields, high variance and small length scale are reported, showing that the quasi-Monte Carlo method consistently outperforms the Monte Carlo method, with a smaller error and a noticeably better than O(N-1/2) convergence rate, where N is the number of samples. Moreover, the rate of convergence of the quasi-Monte Carlo method does not appear to degrade as the nominal dimension increases. Examples with dimension as high as 106 are reported.
KL approximation of a possibly instationary random field a(ω, x) ∈ L2(Ω, dP; L∞(D)) subject to prescribed meanfield and covariance in a polyhedral domain D⊂Rd is analyzed. We show how for stationary covariances Va(x, x′) = ga(|x − x′|) with ga(z) analytic outside of z = 0, an M-term approximate KL-expansion aM(ω, x) of a(ω, x) can be computed in log-linear complexity. The approach applies in arbitrary domains D and for nonseparable covariances Ca. It involves Galerkin approximation of the KL eigenvalue problem by discontinuous finite elements of degree p ⩾ 0 on a quasiuniform, possibly unstructured mesh of width h in D, plus a generalized fast multipole accelerated Krylov-Eigensolver. The approximate KL-expansion aM(x, ω) of a(x, ω) has accuracy O(exp(−bM1/d)) if ga is analytic at z = 0 and accuracy O(M−k/d) if ga is Ck at zero. It is obtained in O(MN(log N)b) operations where N = O(h−d).
We describe a deterministic finite element (FE) solution algorithm for a stochastic elliptic boundary value problem (sbvp), whose coefficients are assumed to be random fields with finite second moments and known, piecewise smooth two-point spatial correlation function. Separation of random and deterministic variables (parametrization of the uncertainty) is achieved via a Karhunen–Loève (KL) expansion. An O(N log N) algorithm for the computation of the KL eigenvalues is presented, based on a kernel independent fast multipole method (FMM). Truncation of the KL expansion gives an (M, 1) Wiener polynomial chaos (PC) expansion of the stochastic coefficient and is shown to lead to a high dimensional, deterministic boundary value problem (dbvp). Analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r = (r1, …, rM) in an (M, r) Wiener PC expansion for the approximate solution. Pointwise error bounds for the FEM approximations of KL eigenpairs, the truncation of the KL expansion and the FE solution to the dbvp are given. Numerical examples show that M depends on the spatial correlation length of the random diffusion coefficient. The variable polynomial degree r in PC-stochastic Galerkin FEM allows to handle KL expansions with M up to 30 and r1 up to 10 in moderate time.
Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporal k-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions’ laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of best N-term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficients.
We consider the inverse problem of estimating a function u from noisy,
possibly nonlinear, observations. We adopt a Bayesian approach to the problem.
This approach has a long history for inversion, dating back to 1970, and has,
over the last decade, gained importance as a practical tool. However most of
the existing theory has been developed for Gaussian prior measures. Recently
Lassas, Saksman and Siltanen (Inv. Prob. Imag. 2009) showed how to construct
Besov prior measures, based on wavelet expansions with random coefficients, and
used these prior measures to study linear inverse problems. In this paper we
build on this development of Besov priors to include the case of nonlinear
measurements. In doing so a key technical tool, established here, is a
Fernique-like theorem for Besov measures. This theorem enables us to identify
appropriate conditions on the forward solution operator which, when matched to
properties of the prior Besov measure, imply the well-definedness and
well-posedness of the posterior measure. We then consider the application of
these results to the inverse problem of finding the diffusion coefficient of an
elliptic partial differential equation, given noisy measurements of its
solution.