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Valter Pohjola

Publications and source records attributed to Valter Pohjola.

16 recordsLinked to original sources
arXiv2026

Reconstruction of shear modulus inclusions in elastostatics via divergence-free localization

We formulate an improved version of the monotonicity method for shape reconstruction in the inverse problem of linear elastostatics. We show that this method can reconstruct inclusions in the second Lamé parameter, i.e., the shear modulus, independently of inhomogeneities in the first Lamé parameter. This improves on earlier methods that only recover inclusions in the bulk modulus under certain assumptions on the local interplay between the Lamé parameters. Our proofs are based on linearization and divergence-free localization of energy using harmonic vector fields.

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arXiv2026

Reconstruction for an inverse scattering problem with a Kerr type nonlinearity

We study the inverse scattering problem for the Kerr-nonlinear Helmholtz equation \[ Δu + k^2(1+q(x)|u|^2)u = 0 \quad \text{in }\mathbb{R}^n,\; n\geq 2, \] where the aim is to recover the unknown potential $q$ from the scattering amplitude. We obtain uniqueness for full data and partial data cases of backscattering, fixed angle scattering, and fixed energy scattering. For the linear Helmholtz equation, uniqueness in backscattering and fixed angle cases are classical and largely open problems. We are able to explicitly reconstruct individual Fourier modes of the potential, and if the measured directions and energies cover an open subset, we recover $q$. The simplicity of the approach leads to an efficient numerical method, and numerical experiments show accurate reconstructions, even in the presence of noise.

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arXiv2025

On the growth properties of interior transmission eigenfunctions near corners

We investigate the localization and vanishing of $L^2$ interior transmission eigenfunctions at corners. Past numerical computations suggest that these eigenfunctions localize at non-convex corners. This phenomenon has, however, not been proven theoretically. We show that localization does indeed occur for some eigenfunctions at a non-convex corner. We also investigate the vanishing of interior transmission eigenfunctions at a convex corner. We prove that these eigenfunctions vanish at convex corners with reduced smoothness assumptions compared to earlier results.

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arXiv2025

Gaussian beam interactions and inverse source problems for nonlinear wave equations

We study the inverse source problem for the semilinear wave equation \[ (\Box_g + q_1)u + q_2 u^2 = F, \] on a globally hyperbolic Lorentzian manifold. We demonstrate that the coefficients $q_1$ and $q_2$, as well as the source term $F$, can be recovered up to a natural gauge symmetry inherent in the problem from local measurements. Furthermore, if $q_1$ is known, we establish the unique recovery of the source $F$, which is in a striking contrast to inverse source problems for linear equations where unique recovery is not possible. Our results also generalize previous works by eliminating the assumption that $u= 0$ is a solution, and by accommodating quadratic nonlinearities. A key contribution is the development of a calculus for nonlinear interactions of Gaussian beams. This framework provides an explicit representation for waves that correspond to sources involving products of two or more Gaussian beams. We anticipate this calculus will serve as a versatile tool in related problems, offering a concrete alternative to Fourier integral operator methods.

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arXiv2024

The linearized monotonicity method for elastic waves and the separation of material parameters

We derive a linearized version of the monotonicity method for shape reconstruction using time harmonic elastic waves. The linearized method provides an efficient version of the method, drastically reducing computation time. Here we show that the linearized method has some additional advantages. The linearized method can in particular be used to obtain additional information on the material parameters, and is able to partially separate and identify the supports of the Lamé parameters.

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arXiv2022

On quantitative Runge approximation for the time harmonic Maxwell equations

Here we derive some results on so called quantitative Runge approximation in the case of the time-harmonic Maxwell equations. This provides a Runge approximation having more explicit quantitative information. We additionally derive some results on the conditional stability of the Cauchy problem for the time-harmonic Maxwell equations.

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arXiv2021

Cones with convoluted geometry that always scatter or radiate

We investigate fixed energy scattering from conical potentials having an irregular cross-section. The incident wave can be any arbitrary non-trivial Herglotz wave. We show that a large number of such local conical scatterers scatter all incident waves, meaning that the far-field will always be non-zero. In essence there are no incident waves for which these potentials would seem transparent at any given energy. We show more specifically that there is a large collection of star-shaped cones whose local geometries always produce a scattered wave. In fact, except for a countable set, all cones from a family of deformations between a circular and a star-shaped cone will always scatter any non-trivial incident Herglotz wave. Our methods are based on the use of spherical harmonics and a deformation argument. We also investigate the related problem for sources. In particular if the support of the source is locally a thin cone, with an arbitrary cross-section, then it will produce a non-zero far-field.

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arXiv2019

Dimension bounds in monotonicity methods for the Helmholtz equation

The article [HPS] established a monotonicity inequality for the Helmholtz equation and presented applications to shape detection and local uniqueness in inverse boundary problems. The monotonicity inequality states that if two scattering coefficients satisfy $q_1 \leq q_2$, then the corresponding Neumann-to-Dirichlet operators satisfy $Λ(q_1) \leq Λ(q_2)$ up to a finite dimensional subspace. Here we improve the bounds for the dimension of this space. In particular, if $q_1$ and $q_2$ have the same number of positive Neumann eigenvalues, then the finite dimensional space is trivial.

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arXiv2019

Monotonicity and local uniqueness for the Helmholtz equation

This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schrödinger) equation $(Δ+ k^2 q) u = 0$ in a bounded domain for fixed non-resonance frequency $k>0$ and real-valued scattering coefficient function $q$. We show a monotonicity relation between the scattering coefficient $q$ and the local Neumann-Dirichlet operator that holds up to finitely many eigenvalues. Combining this with the method of localized potentials, or Runge approximation, adapted to the case where finitely many constraints are present, we derive a constructive monotonicity-based characterization of scatterers from partial boundary data. We also obtain the local uniqueness result that two coefficient functions $q_1$ and $q_2$ can be distinguished by partial boundary data if there is a neighborhood of the boundary where $q_1\geq q_2$ and $q_1\not\equiv q_2$.

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arXiv2016

Multidimensional Borg--Levinson theorems for unbounded potentials

We prove that the Dirichlet eigenvalues and Neumann boundary data of the corresponding eigenfunctions of the operator $-Δ+ q$, determine the potential $q$, when $q \in L^{n/2}(Ω,\mathbb{R})$ and $n \geq 3$. We also consider the case of incomplete spectral data, in the sense that the above spectral data is unknown for some finite number of eigenvalues. In this case we prove that the potential $q$ is uniquely determined for $q \in L^p(Ω,\mathbb{R})$ with $p=n/2$, for $n\geq4$ and $p>n/2$, for $n=3$.

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arXiv2013

An Inverse problem for the Magnetic Schrödinger Operator on a Half Space with partial data

In this paper we prove uniqueness for an inverse boundary value problem for the magnetic Schrödinger equation in a half space, with partial data. We prove that the curl of the magnetic potential $A$, when $A\in W_{comp}^{1,\infty}(\ov{\R^3_{-}},\R^3)$, and the electric pontetial $q \in L_{comp}^{\infty}(\ov{\R^3_{-}},\C)$ are uniquely determined by the knowledge of the Dirichlet-to-Neumann map on parts of the boundary of the half space.

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arXiv2012

An Inverse Boundary Value Problem for the Magnetic Schrödinger Operator on a Half Space

This licentiate thesis is concerned with an inverse boundary value problem for the magnetic Schrödinger equation in a half space, for compactly supported potentials $A\in W^{1,\infty}(\bar{\mathbb{R}^3_{-}},\R^3)$ and $q \in L^{\infty}(\bar{\mathbb{R}^3_{-}},\C)$. We prove that $q$ and the curl of $A$ are uniquely determined by the knowledge of the Dirichlet-to-Neumann map on parts of the boundary of the half space. The existence and uniqueness of the corresponding direct problem are also considered.

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