Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On $L^\infty$ stability for wave propagation and for linear inverse problems

Rima Alaifari, Giovanni S. Alberti, Tandri Gauksson

Abstract

Stability is a key property of both forward models and inverse problems, and depends on the norms considered in the relevant function spaces. For instance, stability estimates for hyperbolic partial differential equations are often based on energy conservation principles, and are therefore expressed in terms of $L^2$ norms. The focus of this paper is on stability with respect to the $L^\infty$ norm, which is more relevant to detect localized phenomena. The linear wave equation is not stable in $L^\infty$, and we design an alternative solution method based on the regularization of Fourier multipliers, which is stable in $L^\infty$. Furthermore, we show how these ideas can be extended to inverse problems, and design a regularization method for the inversion of compact operators that is stable in $L^\infty$. We also discuss the connection with the stability of deep neural networks modeled by hyperbolic PDEs.

On $L^\infty$ stability for wave propagation and for linear inverse problems

Abstract

Stability is a key property of both forward models and inverse problems, and depends on the norms considered in the relevant function spaces. For instance, stability estimates for hyperbolic partial differential equations are often based on energy conservation principles, and are therefore expressed in terms of norms. The focus of this paper is on stability with respect to the norm, which is more relevant to detect localized phenomena. The linear wave equation is not stable in , and we design an alternative solution method based on the regularization of Fourier multipliers, which is stable in . Furthermore, we show how these ideas can be extended to inverse problems, and design a regularization method for the inversion of compact operators that is stable in . We also discuss the connection with the stability of deep neural networks modeled by hyperbolic PDEs.

Paper Structure

This paper contains 15 sections, 14 theorems, 120 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Adversarial sequences for linear operators
  3. L infty stability of the wave equation via Fourier multipliers
  4. Instability in wave propagation
  5. Filtering in space and frequency
  6. The wave equation: a numerical example
  7. Uniform approximation with spectral filtering and applications to inverse problems
  8. Instability in Tikhonov regularization
  9. Spectral filtering
  10. Convergence
  11. Convergence rates on source sets
  12. Boundedness of the spectral filter
  13. A numerical example
  14. Proofs
  15. Connections to the stability of deep neural networks

Key Result

Theorem 2.3

Let $(X,X')$ and $(Y,Y')$ be two Banach couples and let $A\colon X\to Y$ be a bounded linear operator. If there exists an adversarial sequence for $A$ (relative to $X'$ and $Y'$), then there exists an adversarial perturbation for $A$ (relative to $X'$ and $Y'$).

Figures (8)

  • Figure 1: Radial component of spherical waves at initial time $t=0$ (left) and end time $t=1$ (right). The initial states $f_n$, $n=1,2,3,4$, are continuously differentiable functions that transition from the value $-1$ to the value 0 in a window of width $w=2^{-n}$, centered at $r=1$. The increasingly steep transition causes the end states $Bf_n$ to grow at $r=0$ as $n$ increases.
  • Figure 2: Radial component of spherical waves at initial time $t=0$ (left) and end time $t=1$ (right). The "clean" initial state is perturbed by $0.01\cdot f_n$, $n=1,\ldots,5$, where $f_n$ are the functions from Figure \ref{['fig:weq-pert']}.
  • Figure 3: Radial component of spherical waves at times $t=0$ and $t=1$. Left: the clean initial state $f$, a perturbed initial state $f+r$, and regularized initial states $T_{\alpha,\beta}(f+r)$. The perturbation satisfies $\left\lVert r\right\rVert_{L^\infty(\mathbb{R}^3)}=0.01\cdot\left\lVert f\right\rVert_{L^\infty(\mathbb{R}^3)}$. The initial state and the perturbation are the same as in Figure \ref{['fig:weq-signalpert']}. Right: the end states $Bf$, $B(f+r)$, and $B_{\alpha,\beta}(f+r)=BT_{\alpha,\beta}(f+r)$.
  • Figure 4: The regularization filters, $k_\alpha$ and $h_\beta$, used to define the preprocessing operator $T_{\alpha,\beta}\colon f \mapsto k_\alpha*(h_\beta f)$ used in Figure \ref{['fig:weq-regularized']}.
  • Figure 5: The kernel $k$ for the forward operator $Ax = k*x$, defined by \ref{['eq:singularkernel']}, and the three signals of Figures \ref{['fig:perconv-test0']}-\ref{['fig:perconv-test2']}.
  • ...and 3 more figures

Theorems & Definitions (38)

  • Definition 2.1: Adversarial perturbations and sequences
  • Example 2.2
  • Theorem 2.3
  • proof
  • Example 2.4
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 28 more