Runge Approximation Property Research Articles

Well-posedness and inverse problems for semilinear nonlocal wave equations

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form f(x,u) under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension n∈N.

Low regularity theory for the inverse fractional conductivity problem

We characterize partial data uniqueness for the inverse fractional conductivity problem with Hs,n/s regularity assumptions in all dimensions. This extends the earlier results for H2s,n2s∩Hs conductivities by Covi and the authors. We construct counterexamples to uniqueness on domains bounded in one direction whenever measurements are performed in disjoint open sets having positive distance to the domain. In particular, we provide counterexamples in the special cases s∈(n/4,1), n=2,3, missing in the literature due to the earlier regularity conditions. We also give a new proof of the uniqueness result which is not based on the Runge approximation property. Our work can be seen as a fractional counterpart of Haberman’s uniqueness theorem for the classical Calderón problem with W1,n conductivities when n=3,4. One motivation of this work is Brown’s conjecture that uniqueness for the classical Calderón problem holds for W1,n conductivities also in dimensions n≥5.

On inverse problems arising in fractional elasticity

We first formulate an inverse problem for a linear fractional Lamé system. We determine the Lamé parameters from exterior partial measurements of the Dirichlet-to-Neumann map. We further study an inverse obstacle problem as well as an inverse problem for a nonlinear fractional Lamé system. Our arguments are based on the unique continuation property for the fractional operator as well as the associated Runge approximation property.

Conductivity reconstruction from power density data in limited view

In acousto-electric tomography, the objective is to extract information about the interior electrical conductivity in a physical body from knowledge of the interior power density data generated from prescribed boundary conditions for the governing elliptic partial differential equation. In this note, we consider the problem when the controlled boundary conditions are applied only on a small subset of the full boundary. We demonstrate using the unique continuation principle that the Runge approximation property is valid also for this special case of limited view data. As a consequence, we guarantee the existence of finitely many boundary conditions such that the corresponding solutions locally satisfy a non-vanishing gradient condition. This condition is essential for conductivity reconstruction from power density data. In addition, we adapt an existing reconstruction method intended for the full data situation to our setting. We implement the method numerically and investigate the opportunities and shortcomings when reconstructing from two fixed boundary conditions.

On inverse problems for uncoupled space-time fractional operators involving time-dependent coefficients

We study the uncoupled space-time fractional operators involving time-dependent coefficients and formulate the corresponding inverse problems. Our goal is to determine the variable coefficients from the exterior partial measurements of the Dirichlet-to-Neumann map. We exploit the integration by parts formula for Riemann-Liouville and Caputo derivatives to derive the Runge approximation property for our space-time fractional operator based on the unique continuation property of the fractional Laplacian. This enables us to extend early unique determination results for space-fractional but time-local operators to the space-time fractional case.

An inverse problem for a fractional diffusion equation with fractional power type nonlinearities

<p style='text-indent:20px;'>We study the well-posedness of a semi-linear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map. Our arguments are based on a first order linearization as well as the parabolic Runge approximation property.</p>

On an inverse problem for a fractional semilinear elliptic equation involving a magnetic potential

We study a class of fractional semilinear elliptic equations and formulate the corresponding Calder\'on problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order linearization and the Runge approximation property.

Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

We prove a unique continuation property for the fractional Laplacian $(-\Delta)^s$ when $s \in (-n/2,\infty)\setminus \mathbb{Z}$. In addition, we study Poincar\'e-type inequalities for the operator $(-\Delta)^s$ when $s\geq 0$. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schr\"odinger equation. We also study the higher order fractional Schr\"odinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the $d$-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.

Combining the Runge Approximation and the Whitney Embedding Theorem in Hybrid Imaging

Abstract This paper addresses enforcing non-vanishing constraints for solutions to a 2nd-order elliptic partial differential equation by appropriate choices of boundary conditions. We show that, in dimension $d\geq 2$, under suitable regularity assumptions, the family of $2d$ solutions such that their Jacobian has maximal rank in the domain is both open and dense. The case of less regular coefficients is also addressed, together with other constraints, which are relevant for applications to recent hybrid imaging modalities. Our approach is based on the combination of the Runge approximation property and the Whitney projection argument [ 44]. The method is very general and can be used in other settings.

The Calderón problem for the fractional magnetic operator

We introduce a fractional magnetic operator involving a magnetic potential and an electric potential. We formulate an inverse problem for the fractional magnetic operator. We determine the electric potential from the exterior partial measurements of the associated Dirichlet-to-Neumann map by using Runge approximation property.

An inverse problem for the fractional Schrödinger equation in a magnetic field

This paper shows global uniqueness in an inverse problem for a fractional magnetic Schrödinger equation (FMSE): an unknown electromagnetic field in a bounded domain is uniquely determined up to a natural gauge by infinitely many measurements of solutions taken in arbitrary open subsets of the exterior. The proof is based on Alessandrini’s identity and the Runge approximation property, thus generalizing some previous works on the fractional Laplacian. Moreover, we show with a simple model that the FMSE relates to a long jump random walk with weights.

Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients

In this paper we derive a propagation of smallness result for a scalar second elliptic equation in divergence form whose leading order coefficients are Lipschitz continuous on two sides of a C2 hypersurface that crosses the domain, but may have jumps across this hypersurface. Our propagation of smallness result is in the most general form regarding the locations of domains, which may intersect the interface of discontinuity. At the end, we also list some consequences of the propagation of smallness result, including stability results for the associated Cauchy problem, a propagation of smallness result from sets of positive measure, and a quantitative Runge approximation property.

Global Uniqueness and Lipschitz-Stability for the Inverse Robin Transmission Problem

In this paper, we consider the inverse problem of detecting a corrosion coefficient between two layers of a conducting medium from the Neumann-to-Dirichlet map. This inverse problem is motivated by the description of the index of corrosion in non-destructive testing. We show a monotonicity estimates between the Robin coefficient and the Neumann-to-Dirichlet operator. We prove a global uniqueness result and Lipschitz stability estimate, and show how to quantify the Lipschitz stability constant for a given setting. Our quantification of the Lipschitz constant does not rely on quantitative unique continuation or analytic estimates of special functions. Instead of deriving an analytic estimate, we show that the Lipschitz constant for a given setting can be explicitly calculated from the a priori data by solving finitely many well-posed PDEs. Our arguments rely on standard (non-quantitative) unique continuation, a Runge approximation property, the monotonicity result and the method of localized potentials. To solve the problem numerically, we reformulate the inverse problem into a minimization problem using a least square functional. The reformulation of the minimization problem as a suitable saddle point problem allows us to obtain the optimality conditions by using differentiability properties of the min-sup formulation. The reconstruction is then performed by means of the BFGS algorithm. Finally, numerical results are presented to illustrate the efficiency of the proposed alogorithm.

Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

In this note we discuss the conditional stability issue for the finite dimensional Calderon problem for the fractional Schrodinger equation with a finite number of measurements. More precisely, we assume that the unknown potential \begin{document}$ q \in L^{\infty}(\Omega) $\end{document} in the equation \begin{document}$ ((- \Delta)^s+ q)u = 0 \mbox{ in } \Omega\subset \mathbb{R}^n $\end{document} satisfies the a priori assumption that it is contained in a finite dimensional subspace of \begin{document}$ L^{\infty}(\Omega) $\end{document} . Under this condition we prove Lipschitz stability estimates for the fractional Calderon problem by means of finitely many Cauchy data depending on \begin{document}$ q $\end{document} . We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schrodinger equation. Our result relies on the strong Runge approximation property of the fractional Schrodinger equation.

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators

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}$\mathscr{L}_A^s: = (-\nabla·(A(x)\nabla))^s$\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.

Quantitative Runge Approximation and Inverse Problems

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.

Reconstruction of penetrable obstacles in the anisotropic acoustic scattering

We develop an enclosure-type reconstruction scheme to identify penetrableobstacles in acoustic waves with anisotropic medium in $\mathbb{R}^{3}$.The main difficulty of treating this problem lies in the fact thatthere are no complex geometrical optics solutions available for theacoustic equation with anisotropic medium in $\mathbb{R}^{3}$. Instead,we will use another type of special solutions called oscillating-decayingsolutions. Even though that oscillating-decaying solutions are definedonly on the half space, we are able to give necessary boundary inputsby the Runge approximation property. Moreover, since we are consideringa Helmholtz-type equation, we turn to Meyers' $L^{p}$ estimate tocompare the integrals coming from oscillating-decaying solutions andthose from reflected solutions.

Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields

This paper concerns the reconstruction of a complex-valued anisotropic tensor $\gamma = \sigma + \iota\omega\varepsilon$ from knowledge of several internal magnetic fields $H$, where $H$ satisfies the anisotropic Maxwell system on a bounded domain with prescribed boundary conditions. We show that $\gamma$ can be uniquely reconstructed with a loss of two derivatives from errors in the acquisition of $H$. A minimum number of $6$ such functionals is sufficient to obtain a local reconstruction of $\gamma$ in dimension three provided that the electric field satisfies appropriate boundary conditions. When $\gamma$ is close to a scalar tensor, such boundary conditions are shown to exist using the notion of complex geometric optics (CGO) solutions. For arbitrary symmetric tensors $\gamma$, a Runge approximation property is used instead to obtain partial results. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography.

Runge approximation on convex sets implies the Oka property

We prove that the classical Oka property of a complex manifold Y, concerning the existence and homotopy classification of holomorphic mappings from Stein manifolds to Y, is equivalent to a Runge approximation property for holomorphic maps from compact convex sets in Euclidean spaces to Y .

Oscillating–decaying solutions, Runge approximation property for the anisotropic elasticity system and their applications to inverse problems

We construct oscillating–decaying solutions for the general inhomogeneous anisotropic elasticity system. We also prove the Runge approximation property for the inhomogeneous transversely isotropic elasticity system. We apply the oscillating–decaying solutions and the Runge approximation property to the inverse problem of identifying an inclusion or a cavity embedded in a transversely isotropic elastic medium.