Victor Isakov

Uniqueness and stability in multi-dimensional inverse problems

Recent results on uniqueness and stability of identification of coefficients and right sides of partial differential equations from overdetermined boundary data are described. Elliptic, hyperbolic, and parabolic equations and scattering theory are considered. Proofs are given or outlined whenever they contain a new and fruitful idea and are sufficiently short. This review is supposed to be quite comprehensive. In fact, we do not cover only inverse spectral theory. Some interesting numerical methods are mentioned, but numerics is also beyond the scope of this paper. A significant part is dedicated to so-called many boundary measurements (equations with given Dirichlet-to-Neumann map), but we also discuss results about single boundary measurements. An extensive bibliography contains basic papers in the field.

On uniqueness in th invese transmission scattering problem

We give uniqueness results for the inverse scattering problem where the unknown scatterer D is a bounded open set and some coefficients of an elliptic equation are unknown as well. On the boundary of D the transmission conditions are prescribed, so we consider penetrable medium. Our data is the scattering amplitude A given for one or two frequences and for all directions

The inverse problem of option pricing

Valuation of options and other financial derivatives critically depends on the underlying stochastic process specified for a particular market. An inverse problem of option pricing is to determine the nature of this stochastic process, namely, the distribution of expected asset returns implied by current market prices of options with different strikes. We give a rigorous mathematical formulation of this inverse problem, establish uniqueness, and suggest an efficient numerical solution. We apply the method to the S&P 500 Index and conclude that the index is negatively skewed with a higher probability of the sudden decline of the US stock market.

The detection of surface vibrations from interior acoustical pressure

We consider the problem of detecting the source of acoustical noise inside the cabin of a midsize aircraft from measurements of the acoustical pressure field inside the cabin. Mathematically this field satisfies the Helmholtz equation. In this paper we consider the three-dimensional case. We show that any regular solution of this equation admits a unique representation by a single-layer potential, so that the problem is equivalent to the solution of a linear integral equation of the first kind. We study uniqueness of reconstruction and obtain a sharp stability estimate and convergence rates for some regularization algorithms when the domain is a sphere. We have developed a boundary element code to solve the integral equation. We report numerical results with this code applied to three geometries: a sphere, a cylinder with spherical endcaps and a cylinder with a floor modelling the interior of an aircraft cabin. The exact test solution is given by a point source exterior to the surfaces with about 1% random noise added. Regularization methods using the truncated singular value decomposition with generalized cross validation and the conjugate gradient (cg) method with a stopping rule due to Hanke and Raus are compared. An interesting feature of the three-dimensional problem is the relative insensitivity of the optimal regularization parameter (number of iterations) for the cg method to the wavenumber and the multiplicity of the singular values of the integral operator.

The Detection of the Source of Acoustical Noise in Two Dimensions

We consider the problem of detecting the source of acoustical noise inside the cabin of a midsize aircraft from measurements of the acoustical pressure field inside the cabin. Mathematically this field satisfies the Helmholtz equation. In this paper we consider the model two-dimensional case. We show that any regular solution of this equation admits a unique representation by a single layer potential, so that the problem is reduced to the solution of a linear integral equation of the first kind. We prove uniqueness of reconstruction and obtain a sharp stability estimate. Finally, for two geometries and sources of noise simulating the cabin of the aircraft and two engines, we give results of the numerical solution of this integral equation, comparing regularization by the truncated singular value decomposition and the conjugate gradient method.

Completeness of products of solutions and some inverse problems for PDE

Abstract In this paper we give conditions which guarantee that products of solutions of partial differential equations Pu + au = 0 are complete in L 2 (Ω) . Here P is a linear partial differential operator with constant coefficients and a is a function in L ∞ (Ω) . We check these conditions for elliptic, parabolic, and hyperbolic equations of second order and give applications to inverse problems for related equations where a is to be found.

On the inverse conductivity problem with one measurement

The authors are looking for an open set D entering the coefficient of the elliptic equation div((1+chi (D)) Del u)=0 in a domain Omega when for one given non-zero Neumann data on delta Omega they know the Dirichlet data on a part of delta Omega (a single boundary measurement). Here chi (D) is the indicator function of D. They prove uniqueness for sets D which are convex cylinders or unions of discs whose centres are extreme points of the convex hull of those centres.

Global uniqueness for a two-dimensional semilinear elliptic inverse problem

For a general class of nonlinear Schrodinger equations -Au+a(x, u) = 0 in a bounded planar domain £2 we show that the function a(x, u) can be recovered from knowledge of the corresponding Dirichlet-to-Neumann map on the boundary dQ .

Stability estimates for hyperbolic inverse problems with local boundary data

The authors obtain stability estimates of recovery of two coefficients of a hyperbolic partial differential equation from all possible measurements implemented at a part of the lateral boundary. These estimates are of logarithmic type in the plane case and of Holder type in the three-dimensional case. As an important auxiliary result they have proved stability estimates in (attenuated) integral geometry with incomplete data.

Increased stability in the continuation of solutions to the Helmholtz equation

In this paper we give an analytical derivation and numerical evidence of how stability in the Cauchy problem for the Helmholtz equation grows with frequency. This effect depends on convexity properties of the surface where the Cauchy data are given. Proofs use Carleman estimates and the theory of elliptic boundary value problems in Sobolev spaces. Our theory is illustrated by numerical experiments, including an example in the nearfield acoustical holography.