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A multi-level algorithm for the solution of moment problems

Otmar Scherzer, Thomas Strohmer

TL;DR

This work tackles general linear moment problems where the solution lies in a nested family of Hilbert subspaces. It introduces two multi-level iterative schemes, based on CGNE and Landweber–Richardson, that automatically determine an optimal reconstruction level $N$ a posteriori from the computations, without requiring prior knowledge of the true bandwidth; stopping criteria are designed to ensure stability and convergence, with guarantees under exact data and controlled behavior under noise. The framework is cast in terms of frames and reproducing kernel Hilbert spaces, enabling practical application to RKHS-based signal recovery, including band-limited signals from irregular, noisy samples and unknown bandwidth. Numerical experiments demonstrate robust performance, with adaptive level selection often yielding better reconstructions than methods relying on fixed a priori bandwidth and clear regularization effects in the presence of irregular sampling and noise.

Abstract

We study numerical methods for the solution of general linear moment problems, where the solution belongs to a family of nested subspaces of a Hilbert space. Multi-level algorithms, based on the conjugate gradient method and the Landweber--Richardson method are proposed that determine the "optimal" reconstruction level a posteriori from quantities that arise during the numerical calculations. As an important example we discuss the reconstruction of band-limited signals from irregularly spaced noisy samples, when the actual bandwidth of the signal is not available. Numerical examples show the usefulness of the proposed algorithms.

A multi-level algorithm for the solution of moment problems

TL;DR

This work tackles general linear moment problems where the solution lies in a nested family of Hilbert subspaces. It introduces two multi-level iterative schemes, based on CGNE and Landweber–Richardson, that automatically determine an optimal reconstruction level a posteriori from the computations, without requiring prior knowledge of the true bandwidth; stopping criteria are designed to ensure stability and convergence, with guarantees under exact data and controlled behavior under noise. The framework is cast in terms of frames and reproducing kernel Hilbert spaces, enabling practical application to RKHS-based signal recovery, including band-limited signals from irregular, noisy samples and unknown bandwidth. Numerical experiments demonstrate robust performance, with adaptive level selection often yielding better reconstructions than methods relying on fixed a priori bandwidth and clear regularization effects in the presence of irregular sampling and noise.

Abstract

We study numerical methods for the solution of general linear moment problems, where the solution belongs to a family of nested subspaces of a Hilbert space. Multi-level algorithms, based on the conjugate gradient method and the Landweber--Richardson method are proposed that determine the "optimal" reconstruction level a posteriori from quantities that arise during the numerical calculations. As an important example we discuss the reconstruction of band-limited signals from irregularly spaced noisy samples, when the actual bandwidth of the signal is not available. Numerical examples show the usefulness of the proposed algorithms.

Paper Structure

This paper contains 9 sections, 11 theorems, 49 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. A multi--level algorithm based on the conjugate gradient method
  3. Convergence and stability of the multi--level algorithm
  4. A multi--level algorithm based on the Landweber--Richardson method
  5. Convergence and stability of the multi--level algorithm
  6. The multi--level algorithm for the moment problem
  7. Signal recovery in reproducing kernel Hilbert spaces
  8. Reconstructing a signal of unknown bandwidth from irregularly spaced noisy samples
  9. Experimental results

Key Result

Proposition 2.4

Let assumption ass2.1CG hold, and let the iteration at fixed level $N$ be stopped according to the generalized discrepancy principle Then $k_*(N,\delta) < \infty$ and

Figures (8)

  • Figure 6.1: Comparison of $\|x_* - P_N x_*\|^2$ and its estimation $\sum_{j} |x^{\delta}_*(t_j)|^2 w_j - \|x^{\delta}_{k_*,N}\|^2$.
  • Figure 6.2: Approximation using Algorithm \ref{['alg3.1']}, terminated at level 19.
  • Figure 6.3: Approximation using Algorithm \ref{['alg2.3CG']}, terminating at level 13.
  • Figure 6.4: Approximation using Algorithm \ref{['alg2.3CG']}, using stopping criterion \ref{['3.1']}, terminating at level 19.
  • Figure 6.5: Approximation using Algorithm \ref{['alg3.1']}, omitting the factor 2 in the stopping criterion \ref{['3.1']}, terminating at level 23.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Proposition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 3.2
  • Corollary 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Definition 4.1
  • ...and 2 more