Approximation of high-dimensional parametric PDEs (1502.06797v2)

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 $a\mapsto u(a)$ 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.

• The paper demonstrates that holomorphic extensions and anisotropic smoothness enable sparse n-term polynomial approximations to combat the curse of dimensionality.
• The paper introduces non-intrusive polynomial interpolation and recursive Taylor expansion techniques to achieve stable and efficient numerical computations.
• The paper outlines reduced basis algorithms and greedy selection strategies to construct low-dimensional spaces for uniform error minimization in PDE solutions.

Overview of the Approximation of High-Dimensional Parametric PDEs

In the paper of high-dimensional parametric partial differential equations (PDEs), researchers Albert Cohen and Ronald DeVore address both the theoretical underpinnings and practical algorithms for overcoming the curse of dimensionality. Their work examines conditions under which parametric PDEs can be effectively represented and approximated using manageable computational resources, despite the potential complexity arising from a large or infinite number of parameters.

The authors' research is bifurcated into two principal themes: the theoretical exploration of conditions that mitigate the curse of dimensionality and the development of numerical methods that fully exploit these conditions. They focus particularly on the anisotropic smoothness and holomorphic properties of solution maps and leverage these to construct sparse approximations using polynomial expansions and separable representations.

Theoretical Insights and Techniques

1. Parametric Smoothness and Anisotropy: The authors examine solution maps $a \mapsto u(a)$, where $a$ represents parameter values, highlighting the holomorphic nature and high anisotropy due to varying importance of parameters. This is crucial for explaining why high-dimensional parametric PDEs remain tractable.
2. Holomorphic Extensions: By establishing holomorphic extensions of the solution maps to complex domains, the authors reveal that solution maps are not just smooth but possess higher regularity, which can be systematically exploited for approximation.
3. Sparse Approximation and $n$-Term Approximations: The research demonstrates that holomorphic and anisotropic properties enable $n$-term polynomial approximations that avoid the curse of dimensionality. The authors use notions from approximation theory, such as $n$-widths and best $n$-term approximations, to theoretically benchmark these approximations.

Development of Numerical Methods

1. Polynomial Interpolation: Non-intrusive polynomial interpolation algorithms are used, where discretized instances of the solution map are interpolated over a grid of points. These algorithms are structured to be progressive and stable, facilitated by the selection of points such as Leja sequences, ensuring the bounded growth of Lebesgue constants.
2. Taylor Series Expansion: For linear PDEs, the authors propose an intrusive method that relies on the recursive computation of Taylor coefficients for solution maps. This method efficiently evaluates the coefficients using a recursive procedure tailored for downward closed index sets.
3. Reduced Basis Algorithms: The research discusses reduced basis methods for constructing low-dimensional spaces to approximate the solution manifold, targeting uniform error minimization. Greedy algorithms are emphasized for offline stage instance selection, with convergence rates paralleling the $n$-width of solution manifolds.

Practical Implications and Future Directions

The results provide a concrete pathway to design numerical methods that remain efficient as parameter dimensionality grows, offering a framework that combines theoretical insights with practical algorithmic implementations. By breaking the curse of dimensionality for certain classes of parametric PDEs, this research presents significant implications for inverse problems, optimization, and uncertainty quantification. The continued evolution of these techniques promises to refine computational strategies in scientific computing, with potential expansions into more complex nonlinear and domain-dependent PDEs.

In summary, Cohen and DeVore's work represents a significant step in parametrically high-dimensional PDE approximation, balancing rigorous theoretical advancement with computational algorithm development, aimed at solving practical problems in scientific and engineering applications. The precise integration of approximation theory with computational practices provides a robust toolkit for researchers aiming to transcend traditional limitations in parametric modeling.