Showing papers by "Yi-Hsuan Lin published in 2019"

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional schrödinger operators

Abstract: Let \begin{document}$A∈{\rm{Sym}}(n× n)$\end{document} be an elliptic 2-tensor. Consider the anisotropic fractional Schrodinger operator \begin{document}$\mathscr{L}_A^s+q$\end{document} , where \begin{document}$\mathscr{L}_A^s: = (- abla·(A(x) abla))^s$\end{document} , \begin{document}$s∈ (0, 1)$\end{document} and \begin{document}$q∈ L^∞$\end{document} . We are concerned with the simultaneous recovery of \begin{document}$q$\end{document} and possibly embedded soft or hard obstacles inside \begin{document}$q$\end{document} by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain \begin{document}$Ω$\end{document} associated with \begin{document}$\mathscr{L}_A^s+q$\end{document} . It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential \begin{document}$q$\end{document} . If multiple measurements are allowed, then the surrounding potential \begin{document}$q$\end{document} can also be uniquely recovered. These are surprising findings since in the local case, namely \begin{document}$s = 1$\end{document} , both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are long-standing problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrodinger operators.

Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

Abstract: We study various partial data inverse boundary value problems for the semilinear elliptic equation $\Delta u+ a(x,u)=0$ in a domain in $\mathbb R^n$ by using the higher order linearization technique introduced in [LLS 19, FO19]. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of $a(x,z)$ at $z=0$ under general assumptions on $a(x,z)$. The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calderon problem [FKSU09], and implies the solution of partial data problems for certain semilinear equations $\Delta u+ a(x,u) = 0$ also proved in [KU19]. The results that we prove are in contrast to the analogous inverse problems for the linear Schrodinger equation. There recovering an unknown cavity (or part of the boundary) and the potential simultaneously are long-standing open problems, and the solution to the Calderon problem with partial data is known only in special cases when $n \geq 3$.

Monotonicity-based Inversion of the Fractional Schrödinger Equation I. Positive Potentials

Abstract: We consider an inverse problem for the fractional Schrodinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their...

Radiating and non-radiating sources in elasticity

Abstract: In this work, we study the inverse source problem of a fixed frequency for the Navier equation. We investigate non-radiating external forces. If the support of such a force has a convex or non-convex corner or edge on its boundary, the force must be vanishing there. The vanishing property at corners and edges holds also for sufficiently smooth transmission eigenfunctions in elasticity. The idea originates from the enclosure method: an energy identity and a new type of exponential solution for the Navier equation.

Inverse problems for elliptic equations with power type nonlinearities

Abstract: We introduce a method for solving Calderon type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension $2$, and a potential on transversally anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show that one can solve the Calderon problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions.

The Calder\'on problem for a space-time fractional parabolic equation

Abstract: In this article we study an inverse problem for the space-time fractional parabolic operator $(\partial_t-\Delta)^s+Q$ with $0

Monotonicity-based inversion of the fractional Schr\"odinger equation II. General potentials and stability

Abstract: In this work, we use monotonicity-based methods for the fractional Schrodinger equation with general potentials $q\in L^\infty(\Omega)$ in a Lipschitz bounded open set $\Omega\subset \mathbb R^n$ in any dimension $n\in \mathbb N$. We demonstrate that if-and-only-if monotonicity relations between potentials and the Dirichlet-to-Neumann map hold up to a finite dimensional subspace. Based on these if-and-only-if monotonicity relations, we derive a constructive global uniqueness results for the fractional Calderon problem and its linearized version. We also derive a reconstruction method for unknown obstacles in a given domain that only requires the background solution of the fractional Schrodinger equation, and we prove uniqueness and Lipschitz stability from finitely many measurements for potentials lying in an a-priori known bounded set in a finite dimensional subset of $L^\infty(\Omega)$.

Boundary determination of electromagnetic and Lam\'e parameters with corrupted data

Abstract: We study boundary determination for an inverse problem associated to the time-harmonic Maxwell equations and another associated to the isotropic elasticity system. We identify the electromagnetic parameters and the Lam\'e moduli for these two systems from the corresponding boundary measurements. In a first step we reconstruct Lipschitz magnetic permeability, electric permittivity and conductivity on the surface from the ideal boundary measurements. Then, we study inverse problems for Maxwell equations and the isotropic elasticity system assuming that the data contains measurement errors. For both systems, we provide explicit formulas to reconstruct the parameters on the boundary as well as its rate of convergence formula.

Leading and Second Order Homogenization of an Elastic Scattering Problem for Highly Oscillating Anisotropic Medium

Abstract: We consider the scattering of elastic waves by highly oscillating anisotropic periodic media with bounded support. Applying the two-scale homogenization, we first obtain a constant coefficient second-order partial differential elliptic equation that describes the wave propagation of the effective or overall wave field. We further pursue a higher-order homogenization with the help of complimentary boundary correctors and provide a detailed analysis on the rate of higher-order convergence. Finally we provide preliminary numerical examples to demonstrate the higher-order homogenization.

Reconstruction of unknown cavity by single measurement

Abstract: In this paper we propose a domain sampling type reconstruction scheme for an inverse boundary value problem to identify an unknown cavity by single measurement on the accessible boundary of a known electric or heat conductive medium. Here the single measurement is to give single current or heat flux which can have a small support over the boundary, and we measure the corresponding voltage or temperature over the whole boundary. For this inverse boundary value problem, we adapted the single NRT (no response test) introduced by Luke and Potthast for inverse scattering problem and show that it can provide such a domain sampling type reconstruction scheme.