Dependence on the Dimension for Complexity of Approximation of Random Fields

We consider an \eps-approximation by n-term partial sums of the Karhunen-Lo\`eve expansion to d-parametric random fields of tensor product-type in the average case setting. We investigate the behavior, as d tends to infinity, of the information complexity n(\eps,d) of approximation with error not exceeding a given level \eps. It was recently shown by M.A. Lifshits and E.V. Tulyakova that for this problem one observes the curse of dimensionality (intractability) phenomenon. The aim of this paper is to give the exact asymptotic expression for the information complexity n(\eps,d).Comment: 18 pages. The published in Theory Probab. Appl. (2010) extended English translation of the original paper "Zavisimost slozhnosti approximacii sluchajnyh polej ot rasmernosti", submitted on 15.01.2007 and published in Theor. Veroyatnost. i Primenen. 54:2, 256-27

Average Case Tractability of Non-homogeneous Tensor Product Problems

We study d-variate approximation problems in the average case setting with respect to a zero-mean Gaussian measure. Our interest is focused on measures having a structure of non-homogeneous linear tensor product, where covariance kernel is a product of univariate kernels. We consider the normalized average error of algorithms that use finitely many evaluations of arbitrary linear functionals. The information complexity is defined as the minimal number n(h,d) of such evaluations for error in the d-variate case to be at most h. The growth of n(h,d) as a function of h^{-1} and d depends on the eigenvalues of the covariance operator and determines whether a problem is tractable or not. Four types of tractability are studied and for each of them we find the necessary and sufficient conditions in terms of the eigenvalues of univariate kernels. We illustrate our results by considering approximation problems related to the product of Korobov kernels characterized by a weights g_k and smoothnesses r_k. We assume that weights are non-increasing and smoothness parameters are non-decreasing. Furthermore they may be related, for instance g_k=g(r_k) for some non-increasing function g. In particular, we show that approximation problem is strongly polynomially tractable, i.e., n(h,d)\le C h^{-p} for all d and 0<h<1, where C and p are independent of h and d, iff liminf |ln g_k|/ln k >1. For other types of tractability we also show necessary and sufficient conditions in terms of the sequences g_k and r_k

The Complexity of the Poisson Problem for Spaces of Bounded Mixed Derivatives

We are interested in the complexity of the Poisson problem with homogeneous Dirichlet boundary conditions on the d-dimensional unit cube Ω. Error is measured in the energy norm, and only standard information (consisting of function evaluations) is available. In previous work on this problem, the standard assumption has been that the class F of problem elements has been the unit ball of a Sobolev space of fixed smoothness r, in which case the ϵ-complexity is proportional to ϵ to -d/r. Given this exponential dependence on d, the problem is intractable for such classes F. In this paper, we seek to overcome this intractability by allowing F to be the unit ball of a space Hp(Ω) of bounded mixed derivatives, with p a fixed multi-index with positive entries. We find that the complexity is proportional to c(d)(1/ϵ) 1/pmin[ln(1/ϵ)] and we give bounds on b = bp,d. Hence, the problem is tractable in 1/ϵ with exponent at most 1/ϵmin. The upper bound on the complexity (which is close to the lower bound) is attained by a modified finite element method (MFEM) using discrete blending spline spaces; we obtain an explicit bound (with no hidden constants) on the cost of using this MFEM to compute ϵ-approximations. Finally, we show that for any positive multi-index p, the Poisson problem is strongly tractable, and that the MFEM using discrete blended piecewise polynomial splines of degree p is a strongly polynomial time algorithm. In particular, for the case p=1, the MFEM using discrete blended piecewise linear splines produces an ϵ-approximation with cost at most 0.839262(c(d)+2)(1/ϵ)to 5.07911

On tractability of path integration

Do we really need to use randomized algorithms for path integrals? Perhaps we can find a deterministic algorithm that is more effective even in the worst case setting. To answer this question we study the worst case complexity of path integration which roughly speaking is defined as the minimal number of the integrand evaluations needed to compute an approximation with error at most e. We consider path integration with respect to a Gaussian measure and for various classes of integrands

On the quasi-Monte Carlo quadrature with Halton points for elliptic PDEs with log-normal diffusion

This article is dedicated to the computation of the moments of the solution to elliptic partial differential equations with random, log-normally distributed diffusion coefficients by the quasi-Monte Carlo method. Our main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters. Especially, we attain this rather mild dependence on the stochastic dimensionality without any randomization of the quasi-Monte Carlo method under consideration. For the proof of the main result, we require related regularity estimates for the solution and its powers. These estimates are also provided here. Numerical experiments are given to validate the theoretical findings. This article is dedicated to the computation of the moments of the solution to elliptic partial differential equations with random, log-normally distributed diffusion coefficients by the quasi-Monte Carlo method. Our main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters. Especially, we attain this rather mild dependence on the stochastic dimensionality without any randomization of the quasi-Monte Carlo method under consideration. For the proof of the main result, we require related regularity estimates for the solution and its powers. These estimates are also provided here. Numerical experiments are given to validate the theoretical findings

A unified treatment of tractability for approximation problems defined on Hilbert spaces

A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions $1, 2, \ldots$ grows at a moderate rate, i.e., the sequence of problems is tractable. Here, we focus on the situation where the space of available information consists of all linear functionals and the problems are defined as linear operator mappings between Hilbert spaces. We unify the proofs of known tractability results and generalize a number of existing results. These generalizations are expressed as five theorems that provide equivalent conditions for (strong) tractability in terms of sums of functions of the singular values of the solution operators

Tractability of the Quasi-Monte Carlo quadrature with Halton points for elliptic PDEs with random diffusion

This article is dedicated to the computation of the moments of the solution to stochastic partial differential equations with log-normal distributed diffusion coefficient by the Quasi-Monte Carlo method. Our main result is the polynomial tractability for the Quasi-Monte Carlo method based on the Halton sequence. As a by-product, we obtain also the strong tractability of stochastic partial differential equations with uniformly elliptic diffusion coefficient by the Quasi-Monte Carlo method. Numerical experiments are given to validate the theoretical findings