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Matti Lassas

Publications and source records attributed to Matti Lassas.

At least 145 records · Page 8Linked to original sources
arXiv2013

Determination of structures in the space-time from local measurements: a detailed exposition

We consider inverse problems for the Einstein equation with a time-depending metric on a 4-dimensional globally hyperbolic Lorentzian manifold $(M,g)$. We formulate the concept of active measurements for relativistic models. We do this by coupling the Einstein equation with equations for scalar fields and study the system $Ein(g)=T$, $T=T(g,ϕ)+F_1$, and $\square_g ϕ=F_2+S(g,ϕ,F_1,F_2)$. Here $F=(F_1,F_2)$ correspond to the perturbations of the physical fields which we control and $S$ is a secondary source corresponding to the adaptation of the system to the perturbation so that the conservation law $div_g(T)=0$ will be satisfied. The inverse problem we study is the question, do the observation of the solutions $(g,ϕ)$ in an open subset $U\subset M$ of the space-time corresponding to sources $F$ supported in $U$ determine the properties of the metric in a larger domain $W\subset M$ containing $U$. To study this problem we define the concept of light observation sets and show that these sets determine the conformal class of the metric. This corresponds to passive observations from a distant area of the space which is filled by light sources (e.g. we see light from stars varying in time). One can apply the obtained result to solve inverse problems encountered in general relativity and in various practical imaging problems.

math.AP↗
arXiv2012

Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

We analyze the inverse problem, originally formulated by Dix in geophysics, of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of seismic waves. We consider a domain $\tilde M$ with a varying and possibly anisotropic wave speed which we model as a Riemannian metric $g$. For our data, we assume that $\tilde M$ contains a dense set of point scatterers and that in a subset $U\subset \tilde M$, modeling a region containing measurement devices, we can measure the wave fronts of the single scattered waves diffracted from the point scatterers. The inverse problem we study is to recover the metric $g$ in local coordinates anywhere on a set $M \subset \tilde M$ up to an isometry (i.e. we recover the isometry type of $M$). To do this we show that the shape operators related to wave fronts produced by the point scatterers within $\tilde M$ satisfy a certain system of differential equations which may be solved along geodesics of the metric. In this way, assuming we know $g$ as well as the shape operator of the wave fronts in the region $U$, we may recover $g$ in certain coordinate systems (e.g. Riemannian normal coordinates centered at point scatterers). This generalizes the well-known geophysical method of Dix to metrics which may depend on all spatial variables and be anisotropic. In particular, the novelty of this solution lies in the fact that it can be used to reconstruct the metric also in the presence of the caustics.

math.AP↗
arXiv2012

Reconstruction of a conformally Euclidean metric from local boundary diffraction travel times

We consider a region $M$ in $\mathbb{R}^n$ with boundary $\partial M$ and a metric $g$ on $M$ conformal to the Euclidean metric. We analyze the inverse problem, originally formulated by Dix, of reconstructing $g$ from boundary measurements associated with the single scattering of seismic waves in this region. In our formulation the measurements determine the shape operator of wavefronts outside of $M$ originating at diffraction points within $M$. We develop an explicit reconstruction procedure which consists of two steps. In the first step we reconstruct the directional curvatures and the metric in what are essentially Riemmanian normal coordinates; in the second step we develop a conversion to Cartesian coordinates. We admit the presence of conjugate points. In dimension $n \geq 3$ both steps involve the solution of a system of ordinary differential equations. In dimension $n=2$ the same is true for the first step, but the second step requires the solution of a Cauchy problem for an elliptic operator which is unstable in general. The first step of the procedure applies for general metrics.

math.AP↗
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

The borderlines of the invisibility and visibility for Calderon's inverse problem

We consider the determination of a conductivity function in a two-dimensional domain from the Cauchy data of the solutions of the conductivity equation on the boundary. We prove uniqueness results for this inverse problem, posed by Calderon, for conductivities that are degenerate, that is, they may not be bounded from above or below. In particular, for scalar conductivities we solve the inverse problem in a class which is larger than $L^\infty$. Also, we give new counterexamples for the uniqueness of the inverse conductivity problem. We say that a conductivity is visible if the inverse problem is solvable so that the inside of the domain can be uniquely determined, up to a change of coordinates, using the boundary measurements. The present counterexamples for the inverse problem have been related to the invisibility cloaking. This means that there are conductivities for which a part of the domain is shielded from detection via boundary measurements. Such conductivities are called invisibility cloaks. In the present paper we identify the borderline of the visible conductivities and the borderline of invisibility cloaking conductivities. Surprisingly, these borderlines are not the same. We show that between the visible and the cloaking conductivities there are the electric holograms, conductivities which create an illusion of a non-existing body. The electric holograms give counterexamples for the uniqueness of the inverse problem which are less degenerate than the previously known ones.

math.AP↗
arXiv2011

Conic singularities, generalized scattering matrix, and inverse scattering on asymptotically hyperbolic surfaces

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = Γ\backslash{\bf H}^2$ associated with a Fuchsian group of the 1st kind $Γ$ containing parabolic elements. $\mathcal M$ is then non-compact, and has a finite number of cusps and elliptic singular points, which is regarded as a hyperbolic orbifold. We introduce a class of Riemannian surfaces with conical singularities on its finite part, having cusps and regular ends at infinity, whose metric is asymptotically hyperbolic. By observing solutions of the Helmholtz equation at the cusp, we define a generalized S-matrix. We then show that this generalized S-matrix determines the Riemannian metric and the structure of conical singularities.

math.AP↗
arXiv2011

Schrodinger's Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics

The advent of transformation optics and metamaterials has made possible devices producing extreme effects on wave propagation. Here we give theoretical designs for devices, Schrödinger hats, acting as invisible concentrators of waves. These exist for any wave phenomenon modeled by either the Helmholtz or Schrödinger equations, e.g., polarized waves in EM, pressure waves in acoustics and matter waves in QM, and occupy one part of a parameter space continuum of wave-manipulating structures which also contains standard transformation optics based cloaks, resonant cloaks and cloaked sensors. For EM and acoustic Schrödinger hats, the resulting centralized wave is a localized excitation. In QM, the result is a new charged quasiparticle, a \emph{quasmon}, which causes conditional probabilistic illusions. We discuss possible solid state implementations.

math-ph↗
arXiv2011

The Novikov-Veselov Equation and the Inverse Scattering Method, Part I: Analysis

The Novikov-Veselov (NV) equation is a (2+1)-dimensional nonlinear evolution equation that generalizes the (1+1)-dimensional Korteweg-deVries (KdV) equation. Solution of the NV equation using the inverse scattering method has been discussed in the literature, but only formally (or with smallness assumptions in case of nonzero energy) because of the possibility of exceptional points, or singularities in the scattering data. In this work, absence of exceptional points is proved at zero energy for evolutions with compactly supported, smooth and rotationally symmetric initial data of the conductivity type: $q_0=γ^{-1/2}Δγ^{1/2}$ with a strictly positive function $γ$. The inverse scattering evolution is shown to be well-defined, real-valued, and preserving conductivity-type. There is no smallness assumption on the initial data.

math.AP↗
arXiv2011

Inverse problems with partial data for a magnetic Schrödinger operator in an infinite slab and on a bounded domain

In this paper we study inverse boundary value problems with partial data for the magnetic Schrödinger operator. In the case of an infinite slab in $R^n$, $n\ge 3$, we establish that the magnetic field and the electric potential can be determined uniquely, when the Dirichlet and Neumann data are given either on the different boundary hyperplanes of the slab or on the same hyperplane. This is a generalization of the results of [41], obtained for the Schrödinger operator without magnetic potentials. In the case of a bounded domain in $R^n$, $n\ge 3$, extending the results of [2], we show the unique determination of the magnetic field and electric potential from the Dirichlet and Neumann data, given on two arbitrary open subsets of the boundary, provided that the magnetic and electric potentials are known in a neighborhood of the boundary. Generalizing the results of [31], we also obtain uniqueness results for the magnetic Schrödinger operator, when the Dirichlet and Neumann data are known on the same part of the boundary, assuming that the inaccessible part of the boundary is a part of a hyperplane.

math.AP↗
arXiv2011

Determining a first order perturbation of the biharmonic operator by partial boundary measurements

We consider an operator $Δ^2 + A(x)\cdot D+q(x)$ with the Navier boundary conditions on a bounded domain in $R^n$, $n\ge 3$. We show that a first order perturbation $A(x)\cdot D+q$ can be determined uniquely by measuring the Dirichlet--to--Neumann map on possibly very small subsets of the boundary of the domain. Notice that the corresponding result does not hold in general for a first order perturbation of the Laplacian.

math.AP↗
arXiv2011

Inverse boundary value problems for the perturbed polyharmonic operator

We show that a first order perturbation $A(x)\cdot D+q(x)$ of the polyharmonic operator $(-Δ)^m$, $m\ge 2$, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in $R^n$, $n\ge 3$. Notice that the corresponding result does not hold in general when $m=1$.

math.AP↗
arXiv2011

Two dimensional invisibility cloaking for Helmholtz equation and non-local boundary conditions

Transformation optics constructions have allowed the design of cloaking devices that steer electromagnetic, acoustic and quantum waves around a region without penetrating it, so that this region is hidden from external observations. The material pa- rameters used to describe these devices are anisotropic, and singular at the interface between the cloaked and uncloaked regions, making physical realization a challenge. These singular material parameters correspond to singular coefficient functions in the partial differential equations modeling these constructions and the presence of these singularities causes various mathematical problems and physical effects on the interface surface. In this paper, we analyze the two dimensional cloaking for Helmholtz equation when there are sources or sinks present inside the cloaked region. In particular, we consider nonsingular approximate invisibility cloaks based on the truncation of the singular transformations. Using such truncation we analyze the limit when the approximate cloaking approaches the ideal cloaking. We show that, surprisingly, a non-local boundary condition appears on the inner cloak interface. This effect in the two dimensional (or cylindrical) invisibility cloaks, which seems to be caused by the infinite phase velocity near the interface between the cloaked and uncloaked regions, is very different from the earlier studied behavior of the solutions in the three dimensional cloaks.

math.AP↗
arXiv2010

Determining electrical and heat transfer parameters using coupled boundary measurements

Let $Ω\subset\R^n$, $n\ge 3$, be a smooth bounded domain and consider a coupled system in $Ω$ consisting of a conductivity equation $\nabla \cdot γ(x) \nabla u(t,x)=0$ and an anisotropic heat equation $κ^{-1}(x)\partial_tψ(t,x)=\nabla\cdot (A(x)\nabla ψ(t,x))+(γ\nabla u(t,x))\cdot \nabla u(t,x), \quad t\ge 0$. It is shown that the coefficients $γ$, $κ$ and $A=(a_{jk})$ are uniquely determined from the knowledge of the boundary map $u|_{\partialΩ}\mapsto ν\cdot A\nabla ψ|_{\partialΩ}$, where $ν$ is the unit outer normal to $\partialΩ$. The coupled system models the following physical phenomenon. Given a fixed voltage distribution, maintained on the boundary $\partialΩ$, an electric current distribution appears inside $Ω$. The current in turn acts as a source of heat inside $Ω$, and the heat flows out of the body through the boundary. The boundary measurements above then correspond to the map taking a voltage distribution on the boundary to the resulting heat flow through the boundary. The presented mathematical results suggest a new hybrid diffuse imaging modality combining electrical prospecting and heat transfer-based probing.

math.AP↗
arXiv2010

Cloaking a sensor via transformation optics

It is generally believed that transformation optics based cloaking, besides rendering the cloaked region invisible to detection by scattering of incident waves, also shields the region from those same waves. We demonstrate a coupling between the cloaked and uncloaked regions, exposing a difference between cloaking for rays and waves. Interior resonances allow this coupling to be amplified, and careful choice of parameters leads to effective cloaks with degraded shielding. As one application, we describe how to use transformation optics to hide sensors in the cloaked region and yet enable the sensors to efficiently measure waves incident on the exterior of the cloak, an effect similar to the plasmon based approach of Alu' and Engheta.

physics.optics↗
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 problems for differential forms on Riemannian manifolds with boundary

Consider a real-analytic orientable connected complete Riemannian manifold $M$ with boundary of dimension $n\ge 2$ and let $k$ be an integer $1\le k\le n$. In the case when $M$ is compact of dimension $n\ge 3$, we show that the manifold and the metric on it can be reconstructed, up to an isometry, from the set of the Cauchy data for harmonic $k$-forms, given on an open subset of the boundary. This extends a result of [13] when $k=0$. In the two-dimensional case, the same conclusion is obtained when considering the set of the Cauchy data for harmonic $1$-forms. Under additional assumptions on the curvature of the manifold, we carry out the same program when $M$ is complete non-compact. In the case $n\ge 3$, this generalizes the results of [12] when $k=0$. In the two-dimensional case, we are able to reconstruct the manifold from the set of the Cauchy data for harmonic $1$-forms.

math.AP↗
arXiv2010

Reconstruction of Betti numbers of manifolds for anisotropic Maxwell and Dirac systems

We consider an invariant formulation of the system of Maxwell's equations for an anisotropic medium on a compact orientable Riemannian 3-manifold $(M,g)$ with nonempty boundary. The system can be completed to a Dirac type first order system on the manifold. We show that the Betti numbers of the manifold can be recovered from the dynamical response operator for the Dirac system given on a part of the boundary. In the case of the original physical Maxwell system, assuming that the entire boundary is known, all Betti numbers of the manifold can also be determined from the dynamical response operator given on a part of the boundary. Physically, this operator maps the tangential component of the electric field into the tangential component of the magnetic field on the boundary.

math.AP↗