[Paper Review] On instability mechanisms for inverse problems

Herbert Koch, Angkana Rüland+1 more|arXiv (Cornell University)|Dec 3, 2020

Numerical methods in inverse problems90 references10 citations

TL;DR

This paper introduces three robust instability mechanisms for linear and nonlinear inverse problems, based on strong compression properties derived from global or microlocal smoothing of forward operators. It establishes that logarithmic instability is inherent in Calderón-type problems, unique continuation, and backward heat equations—even in rough geometries or with low-regularity coefficients—offering a general framework to quantify ill-posedness without relying on symmetry or explicit computations.

ABSTRACT

In this article we present three robust instability mechanisms for linear and nonlinear inverse problems. All of these are based on strong compression properties (in the sense of singular value or entropy number bounds) which we deduce through either strong global smoothing, only weak global smoothing or microlocal smoothing for the corresponding forward operators, respectively. As applications we for instance present new instability arguments for unique continuation, for the backward heat equation and for linear and nonlinear Calderón type problems in general geometries, possibly in the presence of rough coefficients. Our instability mechanisms could also be of interest in the context of control theory, providing estimates on the cost of (approximate) controllability in rather general settings. This is a revised version of the article ``On instability mechanisms for inverse problems'' Ars Inveniendi Analytica (2021), Paper No. 7, 93 pp by the same authors.

Motivation & Objective

• To identify general, robust instability mechanisms for inverse problems that are independent of symmetry or explicit operator structure.
• To extend instability results to nonlinear and variable-coefficient inverse problems in general domains, including those with low-regularity coefficients.
• To provide a systematic framework to quantify the degree of ill-posedness (e.g., logarithmic or double-logarithmic stability) from forward operator properties.
• To unify and generalize existing instability arguments—such as those by Mandache—beyond settings requiring constant coefficients or symmetric geometries.
• To demonstrate applicability to key problems: Calderón-type inverse problems, unique continuation, backward heat equation, and control-theoretic controllability costs.

Proposed method

• Derive instability mechanisms via bounds on singular values or entropy numbers of forward operators, linking them to global or microlocal smoothing properties.
• Use pseudodifferential and Fourier integral operator theory in Gevrey or analytic regularity classes (Gσ) to analyze the smoothing behavior of forward operators.
• Apply oscillatory integral estimates with analytic or Gevrey wave front sets to control the decay of solutions and infer instability via phase space localization.
• Construct nearly analytic cut-off functions (via Lemma D.4) to localize in phase space while preserving derivative bounds, enabling stationary phase arguments in non-smooth settings.
• Employ the method of stationary phase with large parameters to derive decay estimates for oscillatory integrals, showing that singular values decay slower than any polynomial.
• Establish that the inverse problem’s instability modulus ω is bounded below by logarithmic or double-logarithmic rates, derived from the forward operator’s compression strength.

Experimental results

Research questions

• RQ1 Can instability in inverse problems be established without relying on explicit singular value computation or symmetry assumptions?
• RQ2 To what extent do strong global smoothing or microlocal smoothing properties of forward operators imply inherent instability in inverse problems?
• RQ3 Can the instability mechanisms be generalized to nonlinear and variable-coefficient problems in nonsmooth domains with low-regularity coefficients?
• RQ4 What is the precise quantitative link between the compression strength (via entropy numbers or singular values) of a forward operator and the stability modulus of its inverse?
• RQ5 How do these mechanisms apply to control theory, particularly in estimating the cost of approximate controllability for elliptic, parabolic, and hyperbolic PDEs?

Key findings

• The paper establishes that logarithmic stability is optimal for the Calderón problem in general domains with rough coefficients, even without symmetry or constant principal parts.
• For the backward heat equation, the instability mechanism shows that the inverse problem cannot be better than logarithmic in stability, regardless of domain regularity.
• The unique continuation principle is shown to be inherently unstable in general geometries, with instability rates quantified via entropy number bounds on the forward operator.
• The three instability mechanisms—based on strong global smoothing, weak global smoothing, and microlocal smoothing—apply uniformly across linear and nonlinear inverse problems.
• The cost of approximate controllability in control theory is bounded below by logarithmic or double-logarithmic rates, derived from the same compression mechanisms.
• The revised version corrects assumptions in Theorems 4.1 and 4.2 to exclude singleton closed sets, improving the generality and robustness of the instability results.