Approximation of high-dimensional parametric PDEs

Abstract

Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, and uncertainty quantification. In most of these applications, the number of parameters is large or perhaps even infinite. Thus, the development of numerical methods for these parametric problems is faced with the possible curse of dimensionality. This article is directed at (i) identifying and understanding which properties of parametric equations allow one to avoid this curse and (ii) developing and analyzing effective numerical methodd which fully exploit these properties and, in turn, are immune to the growth in dimensionality.The first part of this article studies the smoothness and approximability of the solution map, that is, the map a → u(a) where a is the parameter value and u(a) is the corresponding solution to the PDE. It is shown that for many relevant parametric PDEs, the parametric smoothness of this map is typically holomorphic and also highly anisotropic in that the relevant parameters are of widely varying importance in describing the solution. These two properties are then exploited to establish convergence rates of n-term approximations to the solution map for which each term is separable in the parametric and physical variables. These results reveal that, at least on a theoretical level, the solution map can be well approximated by discretizations of moderate complexity, thereby showing how the curse of dimensionality is broken. This theoretical analysis is carried out through concepts of approximation theory such as best n-term approximation, sparsity, and n-widths. These notions determine a priori the best possible performance of numerical methods and thus serve as a benchmark for concrete algorithms.The second part of this article turns to the development of numerical algorithms based on the theoretically established sparse separable approximations. The numerical methods studied fall into two general categories. The first uses polynomial expansions in terms of the parameters to approximate the solution map. The second one searches for suitable low dimensional spaces for simultaneously approximating all members of the parametric family. The numerical implementation of these approaches is carried out through adaptive and greedy algorithms. An a priori analysis of the performance of these algorithms establishes how well they meet the theoretical benchmarks.

Greedy Algorithms for Optimal Measurements Selection in State Estimation Using Reduced Models

“…. , y d ), using polynomial approximations of the form 8) where Λ n is a conveniently chosen set of multi-indices such that #(Λ n ) = n. See in particular [7,8] where convergence estimates of the form sup y∈U u(y) − u n (y) ≤ Cn −s , (1.9) are established for some s > 0 even when d = ∞. Thus, all solutions M are approximated in the space V n := span{v ν : ν ∈ Λ n }.…”

Fully Discrete Approximation of Parametric and Stochastic Elliptic PDEs

“…We take ρ j = 1, up to rescaling. We notice that when a ∈ W 1,∞ (D), we can write the equation in the strong form −a∆u = ∇a · ∇u + f , where all terms in the equality belong to 12) for all y ∈ U , and we can differentiate at y = 0, which leads to the identities 13) where all terms in the equality belong to H −1 (D) and to L τ (D). So pointwise 14) where κ j := |∇ψ i | j≥1 |∇ψ i | and ω j := |ψ i | j≥1 |ψ i | so that j≥1 κ j = j≥1 ω j = 1, and where C > 1 is a fixed constant.…”

Optimal low-rank Dynamic Mode Decomposition

“…As a response to this computational bottleneck, reducedorder models aim to approximate the trajectories of the system for a range of regimes determined by a set of initial conditions [1]. A common approach is to assume that the trajectories of interest are well approximated in a sub-space of R n .…”