Teemu Tyni

University of Oulu

We show that an inverse scattering problem for a semilinear wave equation can be solved on a manifold having an asymptotically Minkowskian infinity, that is, scattering functionals determine the topology, differentiable structure and the conformal type of the manifold. Moreover, the metric and the coefficient of the non-linearity are determined up...

The main purpose of this article is to establish the Runge-type approximation in $L^2(0,T;\widetilde{H}^s(\Omega))$ for solutions of linear nonlocal wave equations. To achieve this, we extend the theory of very weak solutions for classical wave equations to our nonlocal framework. This strengthened Runge approximation property allows us to extend t...

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 inver...

Abstract. We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inve...

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 inver...

We develop a boundary integral equation-based numerical method to solve for the electrostatic potential in two dimensions, inside a medium with piecewise constant conductivity, where the boundary condition is given by the complete electrode model (CEM). The CEM is seen as the most accurate model of the physical setting where electrodes are placed o...

We consider the recovery of a potential associated with a semi-linear wave equation on Rn+1, n≥1. We show that an unknown potential a(x,t) of the wave equation □u+aum=0 can be recovered in a Hölder stable way from the map u|∂Ω×[0,T]↦〈ψ,∂νu|∂Ω×[0,T]〉L2(∂Ω×[0,T]). This data is equivalent to the inner product of the Dirichlet-to-Neumann map with a mea...

We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse proble...

We consider an inverse problem of recovering a potential associated to a semi-linear wave equation with a quadratic nonlinearity in $1 + 1$ dimensions. We develop a numerical scheme to determine the potential from a noisy Dirichlet-to-Neumann map on the lateral boundary. The scheme is based on the recent higher order linearization method [20]. We a...

We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order ad...

This paper concerns an inverse boundary value problem of recovering a zeroth order time-dependent term of a semi-linear wave equation on a globally hyperbolic Lorentzian manifold. We show that an unknown potential $q$ in the non-linear wave equation $\square_g u +q u^m=0$, $m\geq 4$, can be recovered in a H\"older stable way from the Dirichlet-to-N...

The subject of this work concerns the classical direct and inverse scattering problems for quasi-linear perturbations of the two-dimensional biharmonic operator. The quasi-linear perturbations of the first and zero order might be complex-valued and singular. We show the existence of the scattering solutions to the direct scattering problem in the S...

We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order ad...

We consider the recovery of a potential associated with a semi-linear wave equation on $\mathbb{R}^{n+1}$, $n\geq 1$. We show a H\"older stability estimate for the recovery of an unknown potential $a$ of the wave equation $\square u +a u^m=0$ from its Dirichlet-to-Neumann map. We show that an unknown potential $a(x,t)$, supported in $\Omega\times[t...

We consider an inverse medium problem in two- and three-dimensional cases. Namely, we investigate the problem of reconstruction of unknown compactly supported refractive index (contrast) from L² with a fixed positive wave number. The proof is based on the new estimates for the Green-Faddeev function in L∞ space. The main goal of this work is to pro...

We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely de...

We consider an inverse scattering problem of recovering the unknown coefficients of quasi-linearly perturbed biharmonic operator on the line. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove that the inverse...

We consider a backscattering Born approximation for a perturbed biharmonic operator in three space dimensions. Previous results on this approach for biharmonic operator use the fact that the coefficients are real-valued to obtain the reconstruction of singularities in the coefficients. In this text we drop the assumption about real-valued coefficie...

Some scattering problems for the multidimensional biharmonic operator are studied. The operator is perturbed by first and zero order perturbations, which maybe complex-valued and singular. We show that the solutions to direct scattering problem satisfy a Lippmann-Schwinger equation, and that this integral equation has a unique solution in the weigh...

We consider the inverse backscattering problem for a biharmonic operator with two lower order perturbations in two and three dimensions. The inverse Born approximation is used to recover jumps and singularities of an unknown combination of potentials. Numerical examples are given to illustrate the practical usefulness of the method.

Some inverse scattering problems for operator of order 4 which is the perturbation (in smaller terms) of the biharmonic operator in one and three dimensions are considered. The coefficients of this perturbation are assumed to be from some Sobolev spaces (they might be singular). The classical (as for the Schrödinger operator) scattering theory is d...

We consider a differential operator of order four with three coefficients which may be complex valued. We prove that the Born approximation can be used efficiently to recover essential information about a combination of the coefficients from the knowledge of the reflection coefficient. Numerical examples illustrate the feasibility of this method.