arXiv subjects

Lauri Oksanen

Publications and source records attributed to Lauri Oksanen.

95 records · Page 6Linked to original sources

arXiv2012

Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary $(M, g)$ from a restriction $Λ_{\Src, \Rec}$ of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here $\Src$ and $\Rec$ are open sets in $\p M$ and the restriction $Λ_{\Src, \Rec}$ corresponds to the case where the Dirichlet data is supported on $\R_+\times \Src$ and the Neumann data is measured on $\R_+\times \Rec$. In the novel case where $\bar \Src \cap \bar \Rec = \emptyset$, we show that $Λ_{\Src, \Rec}$ determines the manifold $(M,g)$ uniquely, assuming that the wave equation is exactly controllable from the set of sources $\Src$. Moreover, we show that the exact controllability can be replaced by the Hassell-Tao condition for eigenvalues and eigenfunctions, that is, λ_j \le C \norm{\p_νϕ_j}_{L^2(\Src)}^2, \quad j =1, 2, ..., where $λ_j$ are the Dirichlet eigenvalues and $(ϕ_j)_{j=1}^\infty$ is an orthonormal basis of the corresponding eigenfunctions.

math.AP↗

arXiv2011

Inverse obstacle problem for the non-stationary wave equation with an unknown background

We consider boundary measurements for the wave equation on a bounded domain $M \subset \R^2$ or on a compact Riemannian surface, and introduce a method to locate a discontinuity in the wave speed. Assuming that the wave speed consist of an inclusion in a known smooth background, the method can determine the distance from any boundary point to the inclusion. In the case of a known constant background wave speed, the method reconstructs a set contained in the convex hull of the inclusion and containing the inclusion. Even if the background wave speed is unknown, the method can reconstruct the distance from each boundary point to the inclusion assuming that the Riemannian metric tensor determined by the wave speed gives simple geometry in $M$. The method is based on reconstruction of volumes of domains of influence by solving a sequence of linear equations. For $τ\in C(\p M)$ the domain of influence $M(τ)$ is the set of those points on the manifold from which the distance to some boundary point $x$ is less than $τ(x)$.

math.AP↗

arXiv2011

Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements

We consider the inverse problem for the wave equation on a compact Riemannian manifold or on a bounded domain of $\R^n$, and generalize the concept of {\em domain of influence}. We present an efficient minimization algorithm to compute the volume of a domain of influence using boundary measurements and time-reversed boundary measurements. Moreover, we show that if the manifold is simple, then the volumes of the domains of influence determine the manifold. For a continuous real valued function $τ$ on the boundary of the manifold, the domain of influence is the set of those points on the manifold from which the travel time to some boundary point $y$ is less than $τ(y)$.

math.AP↗

arXiv2010

An inverse problem for the wave equation with one measurement and the pseudorandom noise

We consider the wave equation $(\p_t^2-Δ_g)u(t,x)=f(t,x)$, in $\R^n$, $u|_{\R_-\times \R^n}=0$, where the metric $g=(g_{jk}(x))_{j,k=1}^n$ is known outside an open and bounded set $M\subset \R^n$ with smooth boundary $\p M$. We define a deterministic source $f(t,x)$ called the pseudorandom noise as a sum of point sources, $f(t,x)=\sum_{j=1}^\infty a_jδ_{x_j}(x)δ(t)$, where the points $x_j,\ j\in\Z_+$, form a dense set on $\p M$. We show that when the weights $a_j$ are chosen appropriately, $u|_{\R\times \p M}$ determines the scattering relation on $\p M$, that is, it determines for all geodesics which pass through $M$ the travel times together with the entering and exit points and directions. The wave $u(t,x)$ contains the singularities produced by all point sources, but when $a_j=λ^{-λ^{j}}$ for some $λ>1$, we can trace back the point source that produced a given singularity in the data. This gives us the distance in $(\R^n, g)$ between a source point $x_j$ and an arbitrary point $y \in \p M$. In particular, if $(\bar M,g)$ is a simple Riemannian manifold and $g$ is conformally Euclidian in $\bar M$, these distances are known to determine the metric $g$ in $M$. In the case when $(\bar M,g)$ is non-simple we present a more detailed analysis of the wave fronts yielding the scattering relation on $\p M$.

math.AP↗

arXiv2010

Inverse problem for wave equation with sources and observations on disjoint sets

We consider an inverse problem for a hyperbolic partial differential equation on a compact Riemannian manifold. Assuming that $Γ_1$ and $Γ_2$ are two disjoint open subsets of the boundary of the manifold we define the restricted Dirichlet-to-Neumann operator $Λ_{Γ_1,Γ_2}$. This operator corresponds the boundary measurements when we have smooth sources supported on $Γ_1$ and the fields produced by these sources are observed on $Γ_2$. We show that when $Γ_1$ and $Γ_2$ are disjoint but their closures intersect at least at one point, then the restricted Dirichlet-to-Neumann operator $Λ_{Γ_1,Γ_2}$ determines the Riemannian manifold and the metric on it up to an isometry. In the Euclidian space, the result yields that an anisotropic wave speed inside a compact body is determined, up to a natural coordinate transformations, by measurements on the boundary of the body even when wave sources are kept away from receivers. Moreover, we show that if we have three arbitrary non-empty open subsets $Γ_1,Γ_2$, and $Γ_3$ of the boundary, then the restricted Dirichlet-to-Neumann operators $Λ_{Γ_j,Γ_k}$ for $1\leq j<k\leq 3$ determine the Riemannian manifold to an isometry. Similar result is proven also for the finite-time boundary measurements when the hyperbolic equation satisfies an exact controllability condition.

math.AP↗