scholarly journals Quantitative Runge Approximation and Inverse Problems

2018 ◽  
Vol 2019 (20) ◽  
pp. 6216-6234 ◽  
Author(s):  
Angkana Rüland ◽  
Mikko Salo
Keyword(s):  
Inverse Problems ◽  
Approximation Property ◽  
Short Note ◽  
Unique Continuation ◽  
Local Data ◽  
Model Application ◽  
Quantitative Version ◽  
Calderón Problem ◽  
Quantitative Unique Continuation ◽  
Second Order Elliptic Operators

AbstractIn this short note, we provide a quantitative version of the classical Runge approximation property for second-order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We show that these estimates are essentially optimal. As a model application, we provide a new proof of the result from [8], [2] on stability for the Calderón problem with local data.

scholarly journals Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
María Ángeles García-Ferrero ◽  
Angkana Rüland ◽  
Wiktoria Zatoń
Keyword(s):  
Helmholtz Equation ◽  
Frequency Dependence ◽  
High Frequency ◽  
Convex Sets ◽  
Unique Continuation ◽  
Approximation Properties ◽  
Frequency Limit ◽  
Partial Data ◽  
Calderón Problem ◽  
Quantitative Unique Continuation

<p style='text-indent:20px;'>In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on [<xref ref-type="bibr" rid="b3">3</xref>,<xref ref-type="bibr" rid="b35">35</xref>]. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.</p>

scholarly journals ON NONUNIQUENESS FOR THE ANISOTROPIC CALDERÓN PROBLEM WITH PARTIAL DATA

2020 ◽  
Vol 8 ◽  
Author(s):  
THIERRY DAUDÉ ◽  
NIKY KAMRAN ◽  
FRANÇOIS NICOLEAU
Keyword(s):  
Infinite Number ◽  
Riemannian Manifolds ◽  
Riemannian Metrics ◽  
Unique Continuation ◽  
Manifolds With Boundary ◽  
Conformal Class ◽  
Hölder Continuous ◽  
Continuation Principle ◽  
Partial Data ◽  
Calderón Problem

We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order $\unicode[STIX]{x1D70C}<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M,g)$ and we show that there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}=c^{4}g$ such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric $g$ and do not satisfy the unique continuation principle.

scholarly journals Band Edge Localization Beyond Regular Floquet Eigenvalues

2020 ◽  
Vol 21 (7) ◽  
pp. 2151-2166
Author(s):  
Albrecht Seelmann ◽  
Matthias Täufer
Keyword(s):  
Floquet Theory ◽  
Large Deviation ◽  
Random Variables ◽  
Unique Continuation ◽  
Band Edge ◽  
Random Schrödinger Operators ◽  
Initial Scale ◽  
Quantitative Unique Continuation ◽  
Scale Estimate ◽  
Floquet Eigenvalues

Abstract We prove that localization near band edges of multi-dimensional ergodic random Schrödinger operators with periodic background potential in $$L^2({\mathbb {R}}^d)$$ L 2 ( R d ) is universal. By this, we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furthermore, our reasoning is sufficiently flexible to prove this initial scale estimate in a non-ergodic setting, which promises to be an ingredient for understanding band edge localization also in these situations.

On the topography of the cost functional in linear and nonlinear inverse problems

Geophysics ◽  
2012 ◽  
Vol 77 (1) ◽  
pp. W1-W15 ◽  
Author(s):  
Juan L. Fernández Martínez ◽  
M. Zulima Fernández Muñiz ◽  
Michael J. Tompkins
Keyword(s):  
Inverse Problems ◽  
Null Space ◽  
Order Approximation ◽  
Singular Values ◽  
Physical Models ◽  
Local Data ◽  
Cost Functional ◽  
Nonlinear Inverse Problems ◽  
Linear And Nonlinear ◽  
The Cost

We analyze, through linear algebra, the topography of the cost functional in linear and nonlinear inverse problems with the aim of illuminating general characteristics. To a first-order approximation, the local data misfit function in any inverse problem is valley-shaped and elongated in the directions of the null space of the Jacobian and/or in the directions of the smallest singular values. In nonlinear inverse problems, valleys persist; however, local minima might also coexist in the misfit space and might be related to nonlinear effects ignored by the Gauss-Newton approximation to the Hessian, the regularization term designed to provide convexity to the misfit function, or to noise in the data. Furthermore, noise perturbs the size of the equivalence region making location of solutions easier but finding a global minimum harder (in the case of existence). Understanding the behavior of the cost functional is an important step in the developing techniques to appraise inverse solutions and estimate uncertainties caused by noise, incomplete sampling, regularization, and more fundamentally, simplified physical models.

Sign in / Sign up

Export Citation Format

Share Document