arXiv subjects

Lauri Oksanen

Publications and source records attributed to Lauri Oksanen.

At least 73 records · Page 4Linked to original sources

arXiv2020

A fully discrete numerical control method for the wave equation

We present a fully discrete finite element method for the interior null controllability problem subject to the wave equation. For the numerical scheme, piece-wise affine continuous elements in space and finite differences in time are considered. We show that if the sharp geometric control condition holds, our numerical scheme yields the optimal rate of convergence with respect to the space-time mesh parameter $h$. The approach is based on the design of stabilization terms for the discrete scheme with the goal of minimizing the computational error.

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arXiv2020

Inverse problems for real principal type operators

We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determination methods for general operators, and prove global uniqueness results for determining coefficients in nonlinear real principal type equations. The article presents a unified approach for treating inverse boundary problems for transport and wave equations, and highlights the role of propagation of singularities in the solution of related inverse problems.

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arXiv2020

An inverse problem for a semi-linear elliptic equation in Riemannian geometries

We study the inverse problem of unique recovery of a complex-valued scalar function $V:\mathcal M \times \mathbb C\to \mathbb C$, defined over a smooth compact Riemannian manifold $(\mathcal M,g)$ with smooth boundary, given the Dirichlet to Neumann map, in a suitable sense, for the elliptic semi-linear equation $-Δ_{g}u+V(x,u)=0$. We show that under some geometrical assumptions uniqueness can be proved for a large class of non-linearities. The proof is constructive and is based on a multiple-fold linearization of the semi-linear equation near complex geometric optic solutions for the linearized operator and the resulting non-linear interactions. These non-linear interactions result in the study of a weighted transform along geodesics, that we call the Jacobi weighted ray transform.

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arXiv2019

Space time stabilized finite element methods for a unique continuation problem subject to the wave equation

We consider a stabilized finite element method based on a spacetime formulation, where the equations are solved on a global (unstructured) spacetime mesh. A unique continuation problem for the wave equation is considered, where data is known in an interior subset of spacetime. For this problem, we consider a primal-dual discrete formulation of the continuum problem with the addition of stabilization terms that are designed with the goal of minimizing the numerical errors. We prove error estimates using the stability properties of the numerical scheme and a continuum observability estimate, based on the sharp geometric control condition by Bardos, Lebeau and Rauch. The order of convergence for our numerical scheme is optimal with respect to stability properties of the continuum problem and the interpolation errors of approximating with polynomial spaces. Numerical examples are provided that illustrate the methodology.

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arXiv2019

The Light Ray transform in Stationary and Static Lorentzian geometries

Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity holds if either a convex foliation condition is satisfied on a Cauchy surface on the manifold or the manifold is real analytic and null geodesics do not have cut points. Next, we consider the light ray transform on tensor fields of arbitrary rank in the more restrictive class of static Lorentzian manifolds and show that if the geodesic ray transform on tensors defined on the spatial part of the manifold is injective up to the natural gauge, then the light ray transform on tensors is also injective up to its natural gauge. Finally, we provide applications of our results to some inverse problems about recovery of coefficients for hyperbolic partial differential equations from boundary data.

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arXiv2019

A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime

The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit dependence on the Péclet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in local $H^1$- or $L^2$-norms that are optimal with respect to the approximation order, the problem's stability and perturbations in data. The convergence order is the same for both norms, but the $H^1$-estimate requires an additional divergence assumption for the convective field. The theory is illustrated in some computational examples.

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arXiv2019

The Light Ray transform on Lorentzian manifolds

We study the weighted light ray transform $L$ of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze $L$ as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the spacelike singularities of a function $f$ from its the weighted light ray transform $Lf$ by a suitable filtered back-projection.

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arXiv2019

Recovery of zeroth order coefficients in non-linear wave equations

This paper is concerned with the resolution of an inverse problem related to the recovery of a scalar (potential) function $V$ from the source to solution map, of the semi-linear equation $(\Box_{g}+V)u+u^3=0$ on a globally hyperbolic Lorentzian manifold $(M,g)$. We first study the simpler model problem where the geometry is the Minkowski space and prove the uniqueness of $V$ through the use of geometric optics and a three-fold wave interaction arising from the cubic non-linearity. Subsequently, the result is generalized to globally hyperbolic Lorentzian manifolds by using Gaussian beams.

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arXiv2019

Uniqueness for the inverse boundary value problem of piecewise homogeneous anisotropic elasticity in the time domain

We consider the inverse boundary value problem of recovering piecewise homogeneous elastic tensor and piecewise homogeneous mass density from a localized lateral Dirichlet-to-Neumann or Neumann-to-Dirichlet map for the elasticity equation in the space-time domain. We derive uniqueness for identifying these tensor and density on all domains of homogeneity that may be reached from the part of the boundary where the measurements are taken by a chain of subdomains whose successive interfaces contain a curved portion.

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arXiv2019

Detection of Hermitian connections in wave equations with cubic non-linearity

We consider the geometric non-linear inverse problem of recovering a Hermitian connection $A$ from the source-to-solution map of the cubic wave equation $\Box_{A}ϕ+κ|ϕ|^{2}ϕ=f$, where $κ\neq 0$ and $\Box_{A}$ is the connection wave operator in the Minkowski space $\mathbb{R}^{1+3}$. The equation arises naturally when considering the Yang-Mills-Higgs equations with Mexican hat type potentials. Our proof exploits the microlocal analysis of nonlinear wave interactions, but instead of employing information contained in the geometry of the wave front sets as in previous literature, we study the principal symbols of waves generated by suitable interactions. Moreover, our approach relies on inversion of a novel non-abelian broken light ray transform.

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arXiv2019

Recovery of time dependent coefficients from boundary data for hyperbolic equations

We study uniqueness of the recovery of a time-dependent magnetic vector-valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet to Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold.

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arXiv2018

A finite element data assimilation method for the wave equation

We design a primal-dual stabilized finite element method for the numerical approximation of a data assimilation problem subject to the acoustic wave equation. For the forward problem, piecewise affine, continuous, finite element functions are used for the approximation in space and backward differentiation is used in time. Stabilizing terms are added on the discrete level. The design of these terms is driven by numerical stability and the stability of the continuous problem, with the objective of minimizing the computational error. Error estimates are then derived that are optimal with respect to the approximation properties of the numerical scheme and the stability properties of the continuous problem. The effects of discretizing the (smooth) domain boundary and other perturbations in data are included in the analysis.

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arXiv2018

Unique continuation for the Helmholtz equation using stabilized finite element methods

In this work we consider the computational approximation of a unique continuation problem for the Helmholtz equation using a stabilized finite element method. First conditional stability estimates are derived for which, under a convexity assumption on the geometry, the constants grow at most linearly in the wave number. Then these estimates are used to obtain error bounds for the finite element method that are explicit with respect to the wave number. Some numerical illustrations are given.

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arXiv2018

Hyperbolic inverse problem with data on disjoint sets

We consider a restricted Dirichlet-to-Neumann map associated to a wave type operator on a Riemannian manifold with boundary. The restriction corresponds to the case where the Dirichlet traces are supported on one subset of the boundary and the Neumann traces are restricted on another subset. We show that the restricted Dirichlet-to-Neumann map determines the lower order terms in the wave equation, up the natural gauge invariances, along a convex foliation of the manifold. We allow the lower order terms to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy.

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arXiv2018

Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation

An inverse boundary value problem for the 1+1 dimensional wave equation $(\partial_t^2 - c(x)^2 \partial_x^2)u(x,t)=0,\quad x\in\mathbb{R}_+$ is considered. We give a discrete regularization strategy to recover wave speed $c(x)$ when we are given the boundary value of the wave, $u(0,t)$, that is produced by a single pulse-like source. The regularization strategy gives an approximative wave speed $\widetilde c$, satisfying a Hölder type estimate $\| \widetilde c-c\|\leq C ε^γ$, where $ε$ is the noise level.

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arXiv2018

Inverse problem for Einstein-scalar field equations

The paper introduces a method to solve inverse problems for hyperbolic systems where the leading order terms are non-linear. We apply the method to the coupled Einstein-scalar field equations and study the question whether the structure of spacetime can be determined by making active measurements near the world line of an observer. We show that such measurements determine the topological, differential and conformal structure of the spacetime in the optimal chronological diamond type set containing the world line. In the case when the unknown part of the spacetime is vacuum, we can also determine the metric itself. We exploit the non-linearity of the equation to obtain a rich set of propagating singularities, produced by a non-linear interaction of singularities that propagate initially as for linear wave equations. This non-linear effect is then used as a tool to solve the inverse problem for the non-linear system. The method works even in cases where the corresponding inverse problems for linear equations remain open, and it can potentially be applied to a large class of inverse problems for non-linear hyperbolic equations encountered in practical imaging problems.

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arXiv2017

Primal dual mixed finite element methods for the elliptic Cauchy problem

We consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known conditional stability estimates and determine the minimal amount of weakly consistent stabilization and Tikhonov regularization that yields optimal convergence for smooth exact solutions. The effect of perturbations in data is also accounted for. A reduced version of the method, obtained by choosing a special stabilization of the dual variable, can be viewed as a variant of the least squares mixed finite element method introduced by Dardé, Hannukainen and Hyvönen in \emph{An {$H\sb {\sf{div}}$}-based mixed quasi-reversibility method for solving elliptic {C}auchy problems}, SIAM J. Numer. Anal., 51(4) 2013. The main difference is that our choice of regularization does not depend on auxiliary parameters, the mesh size being the only asymptotic parameter. Finally, we show that the reduced method can be used for defect correction iteration to determine the solution of the full method. The theory is illustrated by some numerical examples.

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arXiv2017

An Exact Redatuming Procedure for the Inverse Boundary Value Problem for the Wave Equation

Redatuming is a data processing technique to transform measurements recorded in one acquisition geometry to an analogous data set corresponding to another acquisition geometry, for which there are no recorded measurements. We consider a redatuming problem for a wave equation on a bounded domain, or on a manifold with boundary, and model data acquisition by a restriction of the associated Neumann-to-Dirichlet map. This map models measurements with sources and receivers on an open subset $Γ$ contained in the boundary of the manifold. We model the wavespeed by a Riemannian metric, and suppose that the metric is known in some coordinates in a neighborhood of $Γ$. Our goal is to move sources and receivers into this known near boundary region. We formulate redatuming as a collection of unique continuation problems, and provide a two step procedure to solve the redatuming problem. We investigate the stability of the first step in this procedure, showing that it enjoys conditional Hölder stability under suitable geometric hypotheses. In addition, we provide computational experiments that demonstrate our redatuming procedure.

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