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Lauri Oksanen

Publications and source records attributed to Lauri Oksanen.

At least 91 records · Page 5Linked to original sources
arXiv2017

Recovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction

In this paper, we study the inverse boundary value problem for the wave equation with a view towards an explicit reconstruction procedure. We consider both the anisotropic problem where the unknown is a general Riemannian metric smoothly varying in a domain, and the isotropic problem where the metric is conformal to the Euclidean metric. Our objective in both cases is to construct the metric, using either the Neumann-to-Dirichlet (N-to-D) map or Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we construct the metric in the boundary normal (or semi-geodesic) coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. Both cases utilize a variant of the Boundary Control method, and work by probing the interior using special boundary sources. We provide a computational experiment to demonstrate our procedure in the isotropic case with N-to-D data.

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arXiv2017

Fully discrete finite element data assimilation method for the heat equation

We consider a finite element discretization for the reconstruction of the final state of the heat equation, when the initial data is unknown, but additional data is given in a sub domain in the space time. For the discretization in space we consider standard continuous affine finite element approximation, and the time derivative is discretized using a backward differentiation. We regularize the discrete system by adding a penalty of the $H^1$-semi-norm of the initial data, scaled with the mesh-parameter. The analysis of the method uses techniques developed in E. Burman and L. Oksanen, Data assimilation for the heat equation using stabilized finite element methods, arXiv, 2016, combining discrete stability of the numerical method with sharp Carleman estimates for the physical problem, to derive optimal error estimates for the approximate solution. For the natural space time energy norm, away from $t=0$, the convergence is the same as for the classical problem with known initial data, but contrary to the classical case, we do not obtain faster convergence for the $L^2$-norm at the final time.

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arXiv2017

Inverse problems for the connection Laplacian

We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a Calderon problem for connections on a cylinder.

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arXiv2017

Application of the boundary control method to partial data Borg-Levinson inverse spectral problem

We consider the multidimensional Borg-Levinson problem of determining a potential $q$, appearing in the Dirichlet realization of the Schrödinger operator $A_q=-Δ+q$ on a bounded domain $Ω\subset \mathbb{R}^n$, $n\geq2$, from the boundary spectral data of $A_q$ on an arbitrary portion of $\partialΩ$. More precisely, for $γ$ an open and non-empty subset of $\partialΩ$, we consider the boundary spectral data on $γ$ given by $\mathrm{BSD}(q,γ):=\{(λ_{k},{\partial_νϕ_{k}}_{|\overlineγ}):\ k \geq1\}$, where $\{ λ_k:\ k \geq1\}$ is the non-decreasing sequence of eigenvalues of $A_q$, $\{ ϕ_k:\ k \geq1 \}$ an associated Hilbertian basis of eigenfunctions, and $ν$ is the unit outward normal vector to $\partialΩ$. We prove that the data $\mathrm{BSD}(q,γ)$ uniquely determine a bounded potential $q\in L^\infty(Ω)$. Previous uniqueness results, with arbitrarily small $γ$, assume that $q$ is smooth. Our approach is based on the Boundary Control method, and we give a self-contained presentation of the method, focusing on the analytic rather than geometric aspects of the method.

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arXiv2016

Correlation based passive imaging with a white noise source

Passive imaging refers to problems where waves generated by unknown sources are recorded and used to image the medium through which they travel. The sources are typically modelled as a random variable and it is assumed that some statistical information is available. In this paper we study the stochastic wave equation $\partial_t^2 u - Δ_g u = χW$, where $W$ is a random variable with the white noise statistics on ${\mathbb R}^{1+n}$, $n \ge 3$, $χ$ is a smooth function vanishing for negative times and outside a compact set in space, and $Δ_g$ is the Laplace-Beltrami operator associated to a smooth non-trapping Riemannian metric tensor $g$ on ${\mathbb R}^n$. The metric tensor $g$ models the medium to be imaged, and we assume that it coincides with the Euclidean metric outside a compact set. We consider the empirical correlations on an open set $\mathcal X \subset {\mathbb R}^n$, $$ C_T(t_1, x_1, t_2, x_2) = \frac 1 T \int_0^T u(t_1+s,x_1) u(t_2+s,x_2) ds, \quad t_1,t_2>0,\ x_1,x_2\in \mathcal X, $$ for $T>0$. Supposing that $χ$ is non-zero on $\mathcal X$ and constant in time after $t > 1$, we show that in the limit $T \to \infty$, the data $C_T$ becomes statistically stable, that is, independent of the realization of $W$. Our main result is that, with probability one, this limit determines the Riemannian manifold $({\mathbb R}^n,g)$ up to an isometry. To our knowledge, this is the first result showing that a medium can be determined in a passive imaging setting, without assuming a separation of scales.

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arXiv2016

Time reversal method with stabilizing boundary conditions for Photoacoustic tomography

We study an inverse initial source problem that models Photoacoustic tomography measurements with array detectors, and introduce a method that can be viewed as a modification of the so called back and forth nudging method. We show that the method converges at an exponential rate under a natural visibility condition, with data given only on a part of the boundary of the domain of wave propagation. In this paper we consider the case of noiseless measurements.

math.AP↗
arXiv2016

Data assimilation for the heat equation using stabilized finite element methods

We consider data assimilation for the heat equation using a finite element space semi-discretization. The approach is optimization based, but the design of regularization operators and parameters rely on techniques from the theory of stabilized finite elements. The space semi-discretized system is shown to admit a unique solution. Combining sharp estimates of the numerical stability of the discrete scheme and conditional stability estimates of the ill-posed continuous pde-model we then derive error estimates that reflect the approximation order of the finite element space and the stability of the continuous model. Two different data assimilation situations with different stability properties are considered to illustrate the framework. Full detail on how to adapt known stability estimates for the continuous model to work with the numerical analysis framework is given in appendix.

math.NA↗
arXiv2016

Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d \geq 2$, with boundary $\partial M$, we study the inverse boundary value problem of determining a time-dependent potential $q$, appearing in the wave equation $\partial_t^2u-Δ_g u+q(t,x)u=0$ in $\bar M=(0,T)\times M$ with $T>0$. Under suitable geometric assumptions we prove global unique determination of $q\in L^\infty(\bar M)$ given the Cauchy data set on the whole boundary $\partial \bar M$, or on certain subsets of $\partial \bar M$. Our problem can be seen as an analogue of the Calderón problem on the Lorentzian manifold $(\bar M, dt^2 - g)$.

math.AP↗
arXiv2016

Global uniqueness in an inverse problem for time fractional diffusion equations

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d \geq 2$, with boundary $\partial M$, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T) \times M$, $T>0$, with time-fractional Caputo derivative of order $α\in (0,1) \cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solution on a subset of $\partial M$ at fixed time. In the "flat" case where $M$ is a compact subset of $\mathbb R^d$, two out the three coefficients $ρ$ (weight), $a$ (conductivity) and $q$ (potential) appearing in the equation $ρ\partial_t^αu-\textrm{div}(a \nabla u)+ q u=0$ on $(0,T)\times Ω$ are recovered simultaneously.

math.AP↗
arXiv2015

Uniqueness for a seismic inverse source problem modeling a subsonic rupture

We consider an inverse problem for an inhomogeneous wave equation with discrete-in-time sources, modeling a seismic rupture. We assume that the sources occur along a path with subsonic velocity, and that data are collected over time on some detection surface. We explore the question of uniqueness for these problems, show how to recover the times and locations of sources microlocally, and then reconstruct the smooth part of the source assuming that it is the same at each source location.

math.AP↗
arXiv2015

Regularization strategy for inverse problem for 1+1 dimensional wave equation

An inverse boundary value problem for a 1+1 dimensional wave equation with wave speed $c(x)$ is considered. We give a regularisation strategy for inverting the map $\mathcal A:c\mapsto Λ,$ where $Λ$ is the hyperbolic Neumann-to-Dirichlet map corresponding to the wave speed $c$. More precisely, we consider the case when we are given a perturbation of the Neumann-to-Dirichlet map $\tilde Λ=Λ+\mathcal E $, where $\mathcal E$ corresponds to the measurement errors, and reconstruct an approximate wave speed $\tilde c$. We emphasize that $\tilde Λ$ may not not be in the range of the map $\mathcal A$. We show that the reconstructed wave speed $\tilde c$ satisfies $\| \tilde c-c\|_{L^\infty}<C \|E\|^{1/18}$. Our regularization strategy is based on a new formula to compute $c$ from $Λ$.

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arXiv2015

On the Construction of Virtual Interior Point Source Travel Time Distances from the Hyperbolic Neumann-to-Dirichlet Map

We introduce a new algorithm to construct travel time distances between a point in the interior of a Riemannian manifold and points on the boundary of the manifold, and describe a numerical implementation of the algorithm. It is known that the travel time distances for all interior points determine the Riemannian manifold in a stable manner. We do not assume that there are sources or receivers in the interior, and use the hyperbolic Neumann-to-Dirichlet map, or its restriction, as our data. Our algorithm is a variant of the Boundary Control method, and to our knowledge, this is the first numerical implementation of the method in a geometric setting.

math.AP↗
arXiv2015

Determination of the Spacetime from Local Time Measurements

We consider an inverse problem for a Lorentzian spacetime $(M,g)$, and show that time measurements, that is, the knowledge of the Lorentzian time separation function on a submanifold $Σ\subset M$ determine the $C^\infty$-jet of the metric in the Fermi coordinates associated to $Σ$. We use this result to study the global determination of the spacetime $(M,g)$ when it has a real-analytic structure or is stationary and satisfies the Einstein-scalar field equations. In addition to this, we require that $(M,g)$ is geodesically complete modulo scalar curvature singularities. The results are Lorentzian counterparts of extensively studied inverse problems in Riemannian geometry - the determination of the jet of the metric and the boundary rigidity problem. We give also counterexamples in cases when the assumptions are not valid, and discuss inverse problems in general relativity.

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arXiv2013

Inverse problem for the wave equation with a white noise source

We consider a smooth Riemannian metric tensor $g$ on $\R^n$ and study the stochastic wave equation for the Laplace-Beltrami operator $\p_t^2 u - Δ_g u = F$. Here, $F=F(t,x,ω)$ is a random source that has white noise distribution supported on the boundary of some smooth compact domain $M \subset \R^n$. We study the following formally posed inverse problem with only one measurement. Suppose that $g$ is known only outside of a compact subset of $M^{int}$ and that a solution $u(t,x,ω_0)$ is produced by a single realization of the source $F(t,x,ω_0)$. We ask what information regarding $g$ can be recovered by measuring $u(t,x,ω_0)$ on $\R_+ \times \p M$? We prove that such measurement together with the realization of the source determine the scattering relation of the Riemannian manifold $(M, g)$ with probability one. That is, for all geodesics passing through $M$, the travel times together with the entering and exit points and directions are determined. In particular, if $(M,g)$ is a simple Riemannian manifold and $g$ is conformally Euclidian in $M$, the measurement determines the metric $g$ in $M$.

math.AP↗
arXiv2013

Photoacoustic and thermoacoustic tomography with an uncertain wave speed

We consider the mathematical model of photoacoustic and thermoacoustic tomography in media with a variable sound speed. When the sound speed is known, the explicit reconstruction formula by P. Stefanov and G. Uhlmann (Inverse Problems, 25(7):075011, 16, 2009) can be used. We study how a modelling error in the sound speed affects the reconstruction formula and quantify the effect in terms of a stability estimate.

math.AP↗
arXiv2012

A Lipschitz stable reconstruction formula for the inverse problem for the wave equation

We consider the problem to reconstruct a wave speed $c \in C^\infty(M)$ in a domain $M \subset \R^n$ from acoustic boundary measurements modelled by the hyperbolic Dirichlet-to-Neumann map $Λ$. We introduce a reconstruction formula for $c$ that is based on the Boundary Control method and incorporates features also from the complex geometric optics solutions approach. Moreover, we show that the reconstruction formula is locally Lipschitz stable for a low frequency component of $c^{-2}$ under the assumption that the Riemannian manifold $(M, c^{-2} dx^2)$ has a strictly convex function with no critical points. That is, we show that for all bounded $C^2$ neighborhoods $U$ of $c$, there is a $C^1$ neighborhood $V$ of $c$ and constants $C, R > 0$ such that |\F\ll(\tilde c^{-2} - c^{-2}\rr)(ξ)| \le C e^{2R |ξ|} \norm{\tilde Λ- Λ}_*, \quad ξ\in \R^n, for all $\tilde c \in U \cap V$, where $\tilde Λ$ is the Dirichlet-to-Neumann map corresponding to the wave speed $\tilde c$ and $\norm{\cdot}_*$ is a norm capturing certain regularity properties of the Dirichlet-to-Neumann maps.

math.AP↗
arXiv2012

Solving an inverse obstacle problem for the wave equation by using the boundary control method

We introduced in [arXiv:1106.3204] a method to locate discontinuities of a wave speed in dimension two from acoustic boundary measuments modelled by the hyperbolic Neumann-to-Dirichlet operator. Here we extend the method for sound hard obstacles in arbitrary dimension. We present numerical experiments with simulated noisy data suggesting that the method is robust against measurement noise.

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