Runge Approximation Research Articles

A learning based numerical method for Helmholtz equations with high frequency

High-frequency issues have been remarkable challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty is using Tikhonov regularization to stably learn the solution operator by utilizing relevant information especially the fundamental solutions. Then applying the solution operator to a new boundary input could quickly update the solution. Based on the method of fundamental solutions and the quantitative Runge approximation, we give the error estimate. This indicates interpretability and generalizability of the present method. Numerical results validate the error analysis and demonstrate the high-precision and high-efficiency features.

The global uniqueness of a dissipative fractional Helmholtz equation

In this paper, we consider the Dirichlet problem for the fractional Helmholtz equation with dissipation, and study the inverse problem of determining the source function, potential function and dissipation of the equation using the Dirichlet-to-Neumann map and Runge approximation. We prove the global uniqueness of these three functions of the equation under low-frequency conditions. As the main result, this implies that one can use the external data to uniquely recover the unknown functions of the equation in the low-frequency case, which will be of great significance in photoacoustic tomography and thermoacoustic tomography.

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.

Local near-field scattering data enables unique reconstruction of rough electric potentials

The focus of this paper is the study of the inverse point-source scattering problem, specifically in relation to a certain class of electric potentials. Our research provides a novel uniqueness result for the inverse problem with local data, obtained from the near field pattern. Our work improves the work of Caro and Garcia, who investigated both the direct problem and the inverse problem with global near field data for critically singular and δ -shell potentials. The primary contribution of our research is the introduction of a Runge approximation result for the near field data on the scattering problem which, in combination with an interior regularity argument, enables us to establish a uniqueness result for the inverse problem with local data. Additionaly, we manage to consider a slightly wider class of potentials.

Stability estimates for an inverse problem for Schrödinger operators at high frequencies from arbitrary partial boundary measurements

In this paper, we study the partial data inverse boundary value problem for the Schrödinger operator at a high frequency in a bounded domain with smooth boundary in , . Assuming that the potential is known in a neighborhood of the boundary, we obtain the logarithmic stability when both Dirichlet data and Neumann data are taken on arbitrary open subsets of the boundary where the two sets can be disjointed. Our results also show that the logarithmic stability can be improved to the one of Hölder type in the high frequency regime. To achieve those goals, we used a method by combining the CGO solution, Runge approximation and Carleman estimate.

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.

An inverse problem for semilinear equations involving the fractional Laplacian

Our work concerns the study of inverse problems of heat and wave equations involving the fractional Laplacian operator with zeroth order nonlinear perturbations. We recover nonlinear terms in the semilinear equations from the knowledge of the fractional Dirichlet-to-Neumann type map combined with the Runge approximation and the unique continuation property of the fractional Laplacian.

Fractional Calderón problems and Poincaré inequalities on unbounded domains

We generalize many recent uniqueness results on the fractional Calderón problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calderón problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincaré inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.

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.

Uniqueness for the Fractional Calderón Problem with Quasilocal Perturbations

We study the fractional Schrödinger equation with quasilocal perturbations and show that the qualitative unique continuation and Runge approximation properties hold in the assumption of sufficient decay. Quantitative versions of both results are also obtained via a propagation of smallness analysis for the Caffarelli--Silvestre extension. The results are then used to show uniqueness in the inverse problem of retrieving a quasilocal perturbation from Dirichlet-to-Neumann (DN) data under suitable geometric assumptions. Our work generalizes recent results regarding the locally perturbed fractional Calderón problem.

Non-zero constraints in elliptic PDE with random boundary values and applications to hybrid inverse problems

Hybrid inverse problems are based on the interplay of two types of waves, in order to allow for imaging with both high resolution and high contrast. The inversion procedure often consists of two steps: first, internal measurements involving the unknown parameters and some related quantities are obtained, and, second, the unknown parameters have to be reconstructed from the internal data. The reconstruction in the second step requires the solutions of certain PDE to satisfy some non-zero constraints, such as the absence of nodal or critical points, or a non-vanishing Jacobian.In this work, we consider a second-order elliptic PDE and show that it is possible to satisfy these constraints with overwhelming probability by choosing the boundary values randomly, following a sub-Gaussian distribution. The proof is based on a new quantitative estimate for the Runge approximation, a result of independent interest.

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

<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>

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>

Holomorphic Legendrian curves in ℂℙ3 and superminimal surfaces in 𝕊4

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into $\mathbb{CP}^3$ is path connected. We also show that holomorphic Legendrian immersions from Riemann surfaces of finite genus and at most countably many ends, none of which are point ends, satisfy the Calabi-Yau property. Coupled with the Runge approximation theorem, we infer that every open Riemann surface embeds into $\mathbb{CP}^3$ as a complete holomorphic Legendrian curve. Under the twistor projection $\pi:\mathbb{CP}^3\to \mathbb S^4$ onto the $4$-sphere, immersed holomorphic Legendrian curves $M\to \mathbb{CP}^3$ are in bijective correspondence with superminimal immersions $M\to\mathbb S^4$ of positive spin according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in $\mathbb S^4$. In particular, superminimal immersions into $\mathbb S^4$ satisfy the Runge approximation theorem and the Calabi-Yau property.

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.

Runge property and approximation by complete systems of solutions for strongly elliptic equations

We give an overview of some concepts related to the approximation of solutions of a strongly elliptic operator. A definition of a complete system of classical solutions is given, and we show how using the Runge property, it is possible to extend the completeness of these systems onto strictly internal domains in the , and norms. This result is applied to Schrödinger equations with potentials possessing some symmetries.

On some partial data Calderón type problems with mixed boundary conditions

In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calderón type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these, we deduce uniqueness for localized bulk and boundary potentials. On the other hand, we construct a family of CGO solutions associated with the corresponding equations. These allow us to deduce uniqueness results for arbitrary bounded, not necessarily localized bulk and boundary potentials. The CGO solutions are constructed by duality to a new Carleman estimate.

A Fractional Parabolic Inverse Problem Involving a Time-dependent Magnetic Potential

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 13 August 2020Accepted: 08 December 2020Published online: 13 January 2021Keywordsfractional partial differential equations, inverse problems, Runge approximation propertyAMS Subject Headings35R11, 35R30Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah