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Mikko Salo

Publications and source records attributed to Mikko Salo.

At least 37 records · Page 2Linked to original sources
arXiv2024

Coefficient Determination for Non-Linear Schrödinger Equations on manifolds

We consider an inverse problem of recovering the unknown coefficients $β(t,x)$ and $V(t,x)$ appearing in a time-dependent nonlinear Schrödinger equation $ (\mathrm{i} \partial_t +Δ+V)u + βu^2=0$ in $(0,T) \times M$, on Euclidean geometry as well as on Riemannian geometry. We consider measurements in $Ω\subset M$ that is a neighborhood of the boundary of $M$ and the source-to-solution map $ L_{β, V}$ that maps a source $f$ supported in $ Ω\times (0,T) $ to the restriction of the solution $u$ in $ Ω\times (0,T) $. We show that the map $L_{β, V}$ uniquely determines the time-dependent potential and the coefficient of the non-linearity, for the above non-linear Schrödinger equation and for the Gross-Pitaevskii equation, with a cubic non-linear term $β|u|^2 \, u$, that is encountered in quantum physics.

math.AP↗
arXiv2024

Increasing resolution and instability for linear inverse scattering problems

In this work we study the increasing resolution of linear inverse scattering problems at a large fixed frequency. We consider the problem of recovering the density of a Herglotz wave function, and the linearized inverse scattering problem for a potential. It is shown that the number of features that can be stably recovered (stable region) becomes larger as the frequency increases, whereas one has strong instability for the rest of the features (unstable region). To show this rigorously, we prove that the singular values of the forward operator stay roughly constant in the stable region and decay exponentially in the unstable region. The arguments are based on structural properties of the problems and they involve the Courant min-max principle for singular values, quantitative Agmon-Hörmander estimates, and a Schwartz kernel computation based on the coarea formula.

math.AP↗
arXiv2024

Applications of the Stone-Weierstrass theorem in the Calderón problem

We give examples on the use of the Stone-Weierstrass theorem in inverse problems. We show uniqueness in the linearized Calderón problem on holomorphically separable Kähler manifolds, and in the Calderón problem for nonlinear equations on conformally transversally anisotropic manifolds. We also study the holomorphic separability condition in terms of plurisubharmonic functions. The Stone-Weierstrass theorem allows us to generalize and simplify earlier results. It also makes it possible to circumvent the use of complex geometrical optics solutions and inversion of explicit transforms in certain cases.

math.CV↗
arXiv2024

On scattering behavior of corner domains with anisotropic inhomogeneities

This paper investigates the possible scattering and non-scattering behavior of an anisotropic and inhomogeneous Lipschitz medium at a fixed wave number and with a single incident field. We connect the anisotropic non-scattering problem to a Bernoulli type free boundary problem. By invoking methods from the theory of free boundaries, we show that an anisotropic medium with Lipschitz but not $C^{1,α}$ boundary scatters every incident wave that satisfies a non-degeneracy condition.

math.AP↗
arXiv2024

Inverse problems for semilinear Schrödinger equations at large frequency via polynomial resolvent estimates on manifolds

We study inverse boundary problems for semilinear Schrödinger equations on smooth compact Riemannian manifolds of dimensions $\ge 2$ with smooth boundary, at a large fixed frequency. We show that certain classes of cubic nonlinearities are determined uniquely from the knowledge of the nonlinear Dirichlet--to--Neumann map at a large fixed frequency on quite general Riemannian manifolds. In particular, in contrast to the previous results available, here the manifolds need not satisfy any product structure, may have trapped geodesics, and the geodesic ray transform need not be injective. Only a mild assumption about the geometry of intersecting geodesics is required. We also establish a polynomial resolvent estimate for the Laplacian on an arbitrary smooth compact Riemannian manifold without boundary, valid for most frequencies. This estimate, along with the invariant construction of Gaussian beam quasimodes with uniform bounds for underlying constants and a stationary phase lemma with explicit control over all involved constants, constitutes the key elements in proving the uniqueness results for the considered inverse problems.

math.AP↗
arXiv2023

On geometric inverse problems and microlocal analysis

This work gives an expository account of certain applications of microlocal analysis in three geometric inverse problems. We will discuss the geodesic X-ray transform inverse problem, the Gelfand problem for the wave equation on a Riemannian manifold, and the Calderón problem for the Laplace equation on a Riemannian manifold.

math.AP↗
arXiv2023

Jacobian of solutions to the conductivity equation in limited view

The aim of hybrid inverse problems such as Acousto-Electric Tomography or Current Density Imaging is the reconstruction of the electrical conductivity in a domain that can only be accessed from its exterior. In the inversion procedure, the solutions to the conductivity equation play a central role. In particular, it is important that the Jacobian of the solutions is non-vanishing. In the present paper we address a two-dimensional limited view setting, where only a part of the boundary of the domain can be controlled by a non-zero Dirichlet condition, while on the remaining boundary there is a zero Dirichlet condition. For this setting, we propose sufficient conditions on the boundary functions so that the Jacobian of the corresponding solutions is non-vanishing. In that regard we allow for discontinuous boundary functions, which requires the use of solutions in weighted Sobolev spaces. We implement the procedure of reconstructing a conductivity from power density data numerically and investigate how this limited view setting affects the Jacobian and the quality of the reconstructions.

math.FA↗
arXiv2023

The anisotropic Calderón problem at large fixed frequency on manifolds with invertible ray transform

We consider the inverse problem of recovering a potential from the Dirichlet to Neumann map at a large fixed frequency on certain Riemannian manifolds. We extend the earlier result of [G. Uhlmann and Y. Wang, arXiv:2104.03477] to the case of simple manifolds, and more generally to manifolds where the geodesic ray transform is stably invertible. The argument involves an invariantly formulated construction of Gaussian beam quasimodes with uniform bounds for the underlying constants.

math.AP↗
arXiv2023

A general support theorem for analytic double fibration transforms

We develop a systematic approach for resolving the analytic wave front set for a class of integral geometry transforms appearing in various tomography problems. Combined with microlocal analytic continuation, this leads to uniqueness and support theorems for analytic integral transforms which are in the microlocal double fibration framework introduced by Guillemin. For the case of ray transforms, we show that the double fibration setup has a concrete interpretation in terms of curve families obtained by projecting integral curves of a fixed vector field on some fiber bundle down to the base. This setup includes geodesic X-ray type transforms, null bicharacteristic ray transforms and transforms related to real principal type systems. We also study transforms integrating over submanifolds of any codimension, and give geometric characterizations for the Bolker condition required for recovering singularities. Our approach is based on a general result related to recovering the analytic wave front set of a function from its transform given by a suitable analytic elliptic Fourier integral operator. This approach extends and unifies a number of previous works. We use wave packet transforms to extrapolate the geometric features of wave front set propagation for such operators when their canonical relation satisfies the Bolker condition.

math.AP↗
arXiv2023

Fixed angle inverse scattering for sound speeds close to constant

We study the fixed angle inverse scattering problem of determining a sound speed from scattering measurements corresponding to a single incident wave. The main result shows that a sound speed close to constant can be stably determined by just one measurement. Our method is based on studying the linearized problem, which turns out to be related to the acoustic problem in photoacoustic imaging. We adapt the modified time-reversal method from [P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems 25 (2009), 075011] to solve the linearized problem in a stable way, and we use this to give a local uniqueness result for the nonlinear inverse problem.

math.AP↗
arXiv2023

On positivity sets for Helmholtz solutions

We address the question of finding global solutions of the Helmholtz equation that are positive in a given set. This question arises in inverse scattering for penetrable obstacles. In particular, we show that there are solutions that are positive on the boundary of a bounded Lipschitz domain.

math.AP↗
arXiv2022

Uniqueness in an inverse problem of fractional elasticity

We study an inverse problem for fractional elasticity. In analogy to the classical problem of linear elasticity, we consider the unique recovery of the Lamé parameters associated to a linear, isotropic fractional elasticity operator from fractional Dirichlet-to-Neumann data. In our analysis we make use of a fractional matrix Schrödinger equation via a generalization of the so-called Liouville reduction, a technique classically used in the study of the scalar conductivity equation. We conclude that unique recovery is possible if the Lamé parameters agree and are constant in the exterior, and their Poisson ratios agree everywhere. Our study is motivated by the significant recent activity in the field of nonlocal elasticity.

math.AP↗
arXiv2022

The linearized Calderón problem for polyharmonic operators

In this article we consider a linearized Calderón problem for polyharmonic operators of order $2m\ (m\ge 2)$ in the spirit of Calderón's original work [Cal80]. We give a uniqueness result for determining coefficients of order $\leq 2m-1$ up to gauge, based on inverting momentum ray transforms.

math.AP↗
arXiv2022

Increasing stability in the linearized inverse Schrödinger potential problem with power type nonlinearities

We consider increasing stability in the inverse Schrödinger potential problem with power type nonlinearities at a large wavenumber. Two linearization approaches, with respect to small boundary data and small potential function, are proposed and their performance on the inverse Schrödinger potential problem is investigated. It can be observed that higher order linearization for small boundary data can provide an increasing stability for an arbitrary power type nonlinearity term if the wavenumber is chosen large. Meanwhile, linearization with respect to the potential function leads to increasing stability for a quadratic nonlinearity term, which highlights the advantage of nonlinearity in solving the inverse Schrödinger potential problem. Noticing that both linearization approaches can be numerically approximated, we provide several reconstruction algorithms for the quadratic and general power type nonlinearity terms, where one of these algorithms is designed based on boundary measurements of multiple wavenumbers. Several numerical examples shed light on the efficiency of our proposed algorithms.

math.AP↗
arXiv2022

Inverse problems for semilinear elliptic PDE with measurements at a single point

We consider the inverse problem of determining a potential in a semilinear elliptic equation from the knowledge of the Dirichlet-to-Neumann map. For bounded Euclidean domains we prove that the potential is uniquely determined by the Dirichlet-to-Neumann map measured at a single boundary point, or integrated against a fixed measure. This result is valid even when the Dirichlet data is only given on a small subset of the boundary. We also give related uniqueness results on Riemannian manifolds.

math.AP↗
arXiv2022

The non-Abelian X-ray transform on surfaces

This paper settles the question of injectivity of the non-Abelian X-ray transform on simple surfaces for the general linear group of invertible complex matrices. The main idea is to use a factorization theorem for Loop Groups to reduce to the setting of the unitary group, where energy methods and scalar holomorphic integrating factors can be used. We also show that our main theorem extends to cover the case of an arbitrary Lie group.

math.DG↗