Enforcing Local Non-Zero Constraints in PDEs and Applications to Hybrid Imaging Problems

Arch Rational Mech Anal

Bal

Cristo

2017

Self Cite

This paper concerns the existence of critical points for solutions to second order elliptic equations of the form ∇ · σ (x)∇u = 0 posed on a bounded domain X with prescribed boundary conditions. In spatial dimension n = 2, it is known that the number of critical points (where ∇u = 0) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient σ . We show that the situation is different in dimension n ≥ 3. More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for u on ∂ X , there exists an open set of smooth coefficients σ (x) such that ∇u vanishes at least at one point in X . By using estimates related to the Laplacian with mixed boundary conditions, the result is first obtained for a piecewise constant conductivity with infinite contrast, a problem of independent interest. A second step shows that the topology of the vector field ∇u on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients σ (x). These results find applications in the class of hybrid inverse problems, where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points. Our results show that for any (finite number of) prescribed boundary conditions, there are coefficients σ (x) for which the stability of the reconstructions will inevitably degrade.

Section: Dirichlet Boundary Conditionsmentioning

confidence: 99%

“…In other words, the tube T l connects the two points x l (1) and x l (2) . We now construct suitable inclusions for each l = 1, .…”

Section: Dirichlet Boundary Conditionsmentioning

confidence: 99%

Section: Dirichlet Boundary Conditionsmentioning

confidence: 99%

Section: Dirichlet Boundary Conditionsmentioning

confidence: 99%

See 2 more Smart Citations

Critical Points for Elliptic Equations with Prescribed Boundary Conditions

Arch Rational Mech Anal

Bal

Cristo

2017

Self Cite

“…The quantitative step is normally solved with PDE-based methods, by combining the internal data with the PDE modeling the problem. Such approach is sometimes very powerful in the reconstruction [21,13,16,2,3]. However, there may be difficulties in using these methods.…”

Section: Introductionmentioning

confidence: 99%

Disjoint sparsity for signal separation and applications to hybrid inverse problems in medical imaging

Applied and Computational Harmonic Analysis

Ammari

2017

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Abstract. The main focus of this work is the reconstruction of the signals f and g i , i = 1, . . . , N , from the knowledge of their sums h i = f + g i , under the assumption that f and the g i 's can be sparsely represented with respect to two different dictionaries A f and Ag. This generalizes the well-known "morphological component analysis" to a multi-measurement setting. The main result of the paper states that f and the g i 's can be uniquely and stably reconstructed by finding sparse representations of h i for every i with respect to the concatenated dictionary [A f , Ag], provided that enough incoherent measurements g i are available. The incoherence is measured in terms of their mutual disjoint sparsity.This method finds applications in the reconstruction procedures of several hybrid imaging inverse problems, where internal data are measured. These measurements usually consist of the main unknown multiplied by other unknown quantities, and so the disjoint sparsity approach can be directly applied. As an example, we show how to apply the method to the reconstruction in quantitative photoacoustic tomography, also in the case when the Grüneisen parameter, the optical absorption and the diffusion coefficient are all unknown.

Absence of Critical Points of Solutions to the Helmholtz Equation in 3D

Arch Rational Mech Anal

2016

Self Cite

Abstract. The focus of this paper is to show the absence of critical points for the solutions to the Helmholtz equation in a bounded domain Ω ⊂ R 3 , given by −div(a ∇uWe prove that for an admissible g there exists a finite set of frequencies K in a given interval and an open cover Ω = ∪ ω∈K Ωω such that |∇u g ω (x)| > 0 for every ω ∈ K and x ∈ Ωω. The set K is explicitly constructed. If the spectrum of this problem is simple, which is true for a generic domain Ω, the admissibility condition on g is a generic property.