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Tractability of linear ill-posed problems in Hilbert space

Peter Mathé, Bernd Hofmann

TL;DR

The paper develops a tractability framework for linear ill-posed inverse problems in Hilbert spaces under noisy data, by importing the Information-based Complexity viewpoint and introducing a discretization level $k_{\ast}(\delta,d)$ that reflects the interaction between noise $\delta$ and dimension $d$. It establishes a one-to-one link between the tractability of inverse problems and their companion direct problems, and proves that entering the asymptotic regularization regime requires discretization levels governed by the decay of singular values via $\Theta(t)=\sqrt{t}\,\varphi(t)$. In the power-type decay case, discretization growth is tied to $d$ and the leading constant $c(1/d)$, yielding either intractability in high dimensions or tractability when the constant decays sufficiently fast; in contrast, the multivariate integration operator exhibits weak tractability in $d$, implying tractability for both direct and inverse problems in that setting. Overall, the work clarifies when discretization costs thwart achieving optimal rates and when high-dimensional ill-posed problems remain computationally feasible.

Abstract

We introduce a notion of tractability for ill-posed operator equations in Hilbert space. For such operator equations the asymptotics of the best possible rate of reconstruction in terms of the underlying noise level is known in many cases. However, the relevant question is, which level of discretization, again driven by the noise level, is required in order to achieve this best possible accuracy. The proposed concept adapts the one from Information-based Complexity. Several examples indicate the relevance of this concept in the light of the curse of dimensionality.

Tractability of linear ill-posed problems in Hilbert space

TL;DR

The paper develops a tractability framework for linear ill-posed inverse problems in Hilbert spaces under noisy data, by importing the Information-based Complexity viewpoint and introducing a discretization level that reflects the interaction between noise and dimension . It establishes a one-to-one link between the tractability of inverse problems and their companion direct problems, and proves that entering the asymptotic regularization regime requires discretization levels governed by the decay of singular values via . In the power-type decay case, discretization growth is tied to and the leading constant , yielding either intractability in high dimensions or tractability when the constant decays sufficiently fast; in contrast, the multivariate integration operator exhibits weak tractability in , implying tractability for both direct and inverse problems in that setting. Overall, the work clarifies when discretization costs thwart achieving optimal rates and when high-dimensional ill-posed problems remain computationally feasible.

Abstract

We introduce a notion of tractability for ill-posed operator equations in Hilbert space. For such operator equations the asymptotics of the best possible rate of reconstruction in terms of the underlying noise level is known in many cases. However, the relevant question is, which level of discretization, again driven by the noise level, is required in order to achieve this best possible accuracy. The proposed concept adapts the one from Information-based Complexity. Several examples indicate the relevance of this concept in the light of the curse of dimensionality.
Paper Structure (9 sections, 7 theorems, 41 equations)

This paper contains 9 sections, 7 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Information complexity, problem formulation
  3. Discretization
  4. Problem formulation
  5. Tractability
  6. Tractability of families of ill-posed equations
  7. Relation to tractability of direct problems
  8. Power-type decay of the singular values
  9. Multivariate integration operator

Key Result

Proposition 1

Suppose that the operator $A$ has an SVD with singular values $\{s_j\}_{j=1}^\infty$, and that smoothness is given as in Definition de:smoothness with an index function $\varphi$. For $j \in\mathbb N$ we have that

Theorems & Definitions (25)

  • Definition 1: source set
  • Proposition 1
  • Remark 1
  • Remark 2
  • Proposition 2
  • proof
  • Example 1: moderately ill-posed operator
  • Example 2: mildly ill-posed operator
  • Example 3: Sobolev embeddings
  • Definition 2: weak tractability
  • ...and 15 more