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arxiv: 2410.11467 · v2 · submitted 2024-10-15 · 🧮 math.AP · cs.NA· math.NA

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

Rima Alaifari , Giovanni S. Alberti , Tandri Gauksson This is my paper

Pith reviewed 2026-05-23 19:01 UTC · model grok-4.3

classification 🧮 math.AP cs.NAmath.NA
keywords L^∞ stabilitywave equationFourier multipliersregularizationinverse problemscompact operatorshyperbolic PDEs
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The pith

The linear wave equation is unstable in L^∞, but regularizing its Fourier multipliers produces a stable solution method that extends to inverse problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the standard solution operator for the linear wave equation fails to be stable when measured in the L^∞ norm. It introduces an alternative method that regularizes the Fourier multipliers appearing in the solution formula, thereby restoring stability in this norm. The same regularization strategy is then applied to the inversion of compact linear operators, yielding an L^∞-stable reconstruction procedure. These constructions matter because the L^∞ norm is better suited than energy norms for capturing localized phenomena. The paper also notes a link between the resulting stability and the behavior of deep neural networks whose layers are governed by hyperbolic PDEs.

Core claim

The linear wave equation is not stable in L^∞, and we design an alternative solution method based on the regularization of Fourier multipliers, which is stable in L^∞. 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^∞.

What carries the argument

Regularization of Fourier multipliers, which damps high-frequency components to control the L^∞ norm of wave solutions and operator inverses.

If this is right

  • Wave solutions can be computed with explicit L^∞ stability bounds via the regularized multipliers.
  • Inversion formulas for compact operators become stable in L^∞ after the same regularization step.
  • Stability analysis for hyperbolic PDEs can shift focus from L² energy estimates to pointwise L^∞ control.
  • Neural networks modeled by hyperbolic PDEs can inherit L^∞ stability from the regularized forward map.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The regularization parameter could be chosen adaptively from data to balance stability and accuracy in applications with noisy observations.
  • The method might extend to other linear hyperbolic systems whose solutions involve Fourier multipliers, such as the Maxwell equations.
  • Numerical experiments on concrete inverse problems like limited-angle tomography could quantify the practical gain in local feature recovery.

Load-bearing premise

Regularizing the Fourier multipliers achieves L^∞ stability without unacceptable loss of the original wave propagation properties or introduction of artifacts that invalidate the solution.

What would settle it

A numerical test in which the regularized wave solution differs from the exact solution by more than a fixed tolerance in the L^∞ norm for a smooth, compactly supported initial condition.

Figures

Figures reproduced from arXiv: 2410.11467 by Giovanni S. Alberti, Rima Alaifari, Tandri Gauksson.

Figure 1
Figure 1. Figure 1: Radial component of spherical waves at initial time [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Radial component of spherical waves at initial time [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Radial component of spherical waves at times [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The regularization filters, kα and hβ, used to define the preprocessing operator Tα,β : f 7→ kα ∗ (hβf) used in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The kernel k for the forward operator Ax = k ∗ x, defined by (12), and the three signals of Figures 6-8. Example 4.1. We see that Tikhonov regularization struggles to suppress the resulting spike, while the effects are milder for T c α. In addition, we show a truncated SVD approximation of the solution, T TSVD α , which can approximate smooth signals uniformly but is ill-suited for rough signals. In partic… view at source ↗
Figure 6
Figure 6. Figure 6: Regularization of adversarially perturbed measurements [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Analogous to Figure 6 but with [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Analogous to Figure 6 but with [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript asserts that the standard linear wave equation is unstable in the L^∞ norm. It introduces a regularization of Fourier multipliers as an alternative evolution that achieves L^∞ stability for wave propagation. The approach is extended to construct a regularization method for the inversion of compact operators that is stable in L^∞. The paper also discusses connections between these ideas and the stability of deep neural networks modeled by hyperbolic PDEs.

Significance. If the regularized evolution is shown to converge to the original wave solution with explicit error control while preserving key properties such as finite propagation speed, the results could provide a useful tool for applications that require L^∞ control on localized phenomena. The extension to stable L^∞ inversion of compact operators would similarly be of interest if the approximation quality is quantified. The link to DNN stability is potentially novel but its significance depends on the depth of the connection established.

major comments (1)
  1. [Main results on wave propagation (likely §3–4)] The central claim that the regularized Fourier-multiplier evolution serves as a stable proxy for the original wave equation requires explicit quantification of the approximation error (e.g., an L^∞ bound on the difference between the two solutions that vanishes as the regularization parameter tends to zero) together with verification that the regularized symbol retains the hyperbolicity and domain-of-dependence properties of the wave operator. No such estimates appear in the abstract or are referenced in the provided description of the main results; without them the L^∞ stability result applies only to a modified evolution whose relation to the original PDE remains unquantified.
minor comments (2)
  1. [Introduction] The abstract states that the linear wave equation 'is not stable in L^∞' but does not cite the precise sense (e.g., failure of continuous dependence on initial data in L^∞) or the counter-example used; a brief reference or short derivation in the introduction would clarify the starting point.
  2. [Final section on DNN connection] The discussion of deep neural networks modeled by hyperbolic PDEs is mentioned only at the end of the abstract; the manuscript should indicate whether the regularization is applied directly to the network dynamics or only used as an analogy.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The major comment identifies a genuine gap in the current manuscript regarding error quantification and preservation of structural properties. We address this point below and will incorporate the requested material in the revision.

read point-by-point responses

  1. Referee: [Main results on wave propagation (likely §3–4)] The central claim that the regularized Fourier-multiplier evolution serves as a stable proxy for the original wave equation requires explicit quantification of the approximation error (e.g., an L^∞ bound on the difference between the two solutions that vanishes as the regularization parameter tends to zero) together with verification that the regularized symbol retains the hyperbolicity and domain-of-dependence properties of the wave operator. No such estimates appear in the abstract or are referenced in the provided description of the main results; without them the L^∞ stability result applies only to a modified evolution whose relation to the original PDE remains unquantified.

    Authors: We agree that the manuscript currently lacks explicit L^∞ error bounds between the regularized and original solutions, as well as a direct verification that the regularized multiplier preserves hyperbolicity and finite propagation speed. In the revised version we will add these estimates in a new subsection of §3 (or §4). Specifically, we will derive an L^∞ convergence rate as the regularization parameter tends to zero by combining the Fourier representation with standard multiplier estimates and Sobolev embedding. We will also show that the regularized symbol remains strictly hyperbolic and that the associated fundamental solution has support contained in the light cone of the original wave operator, thereby retaining the domain-of-dependence property. These additions will make the approximation relation quantitative and will be referenced already in the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: no derivation chain or equations visible for inspection

full rationale

The abstract states the linear wave equation lacks L^∞ stability and proposes regularization of Fourier multipliers as an alternative, plus extensions to inverse problems. However, the provided text contains zero equations, no derivation steps, no self-citations, and no fitted parameters or ansatzes. Without any load-bearing mathematical steps to examine, no reduction to inputs by construction can be exhibited. This is the default honest outcome when the paper's claimed chain is not supplied for review.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information on free parameters, axioms, or invented entities is available.

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Reference graph

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