Victor Isakov

• Wichita State University

In this paper we study the uniqueness and the increasing stability in the inverse source problem for electromagnetic waves in homogeneous and inhomogeneous media from boundary data at multiple wave numbers. For the unique determination of sources, we consider inhomogeneous media and use tangential components of the electric field and magnetic field...

We study increasing stability in the interior inverse source problem for the Helmholtz equation from boundary Cauchy data for multiple wave numbers. By using the Fourier transform with respect to the wave number, explicit bounds for the analytic continuation, uniqueness of the continuation results, and exact observability bounds for the wave equati...

In this work we consider stability of recovery of the conductivity and attenuation coefficients of the stationary Maxwell and Schr\"odinger equations from a complete set of (Cauchy) boundary data. By using complex geometrical optics solutions we derive some bounds which can be viewed as an evidence of increasing stability in these inverse problems...

We study the problem of reconstruction of special time dependent local volatility from market prices of options with different strikes at two expiration times. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an operator W which...

We obtain stability estimates (with explicit constants) for the near field from the far field of a radiating solution of the Helmholtz equation. These estimates are based on new bounds for Hankel functions and quantify increasing stability when the wave number grows.

We consider the Cauchy problem for general second partial differential equations of elliptic type containing large parameter k (like in the Helmholtz equation). By using energy estimates and splitting solution in “low” and “high” frequency parts we obtain bounds of a solution in Sobolev spaces indicating increasing stability in the Cauchy problem w...

We derive some bounds which can be viewed as an evidence of increasing stability in the problem of recovering the potential coefficient in the Schr\"odinger equation from the Dirichlet-to-Neumann map in the presence of attenuation, when energy level/frequency is growing. These bounds hold under certain a-priori regularity constraints on the unknown...

We study the problem of reconstruction of the asset price dependent local volatility from market prices of options with different strikes. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an explicit functional which is linear in...

We propose a model for the gravitational field of a floating iceberg D with snow on its top. The inverse problem of interest in geophysics is to find D and snow thickness g on its known (visible) top from remote measurements of derivatives of the gravitational potential. By modifying the Novikov’s orthogonality method we prove uniqueness of recover...

In this work we study the phenomenon of increasing stability in the inverse boundary value problem for the Schr\"odinger equation. This problem was previously considered by Isakov in which he discussed the phenomenon in different ranges of the wave number (or energy). The main contribution of this work is to provide a unified and easier approach to...

We obtain new analytic results for the problem of the recovery of a doped region D in semiconductor devices from the total flux of electrons/holes through a part of the boundary for various applied potentials on some complementary part of the boundary. We consider the stationary two-dimensional case and we use the index of the gradient of solutions...

We consider the Cauchy problem for harmonic functions outside some disc in the plane with the Cauchy data on an interval. We obtain simple formulae for singular values of the operator solving this Cauchy problem and explicit bounds on the difference between the exact and truncated operators. For a typical particular geometry we compute numerically...

We show that under some conditions one can obtain Carleman type estimates for the transversely isotropic elasticity system with residual stress. We consider both time dependent and static cases. The main idea is to reduce this system to a principally upper triangular one and the main technical tool is Carleman estimates with two large parameters fo...

We prove uniqueness of the term c(u,p) of partial differential equations −Δu + c(u, u) = 0 and ∂ t u − Δu + c(u, u) = 0 with the Dirichlet-to-Neumann map given on a part of the lateral boundary. We use a linearization method and singular solutions in the boundary reconstruction of the linearized equations

We propose a fast local level setmethod for the inverse problemof gravimetry. The theoretical foundation for our approach is based on the following uniqueness result: if an open set D is star-shaped or x3-convex with respect to its center of gravity, then its exterior potential uniquely determines the open set D. To achieve this purpose constructiv...

We derive some bounds which can be viewed as an evidence of increas- ing stability in the problem of recovery of the potential coecient in the Schrodinger equation from the Dirichlet-to-Neumann map, when frequency(energy level) is growing. These bounds hold under certain a-priori bounds on the unknown coecient. Proofs use complex- and real-valued g...

We study the stability in the Cauchy Problem for the Helmholtz equation in de-pendence of the wave number k. For simple geometries, we show analytically that this problem is getting more stable with increasing k. In more detail, there is a subspace of the data space on which the Cauchy Problem is well posed, and this subspace grows with larger k. W...

We establish a probe type reconstruction scheme for identifying an inclusion inside a heat conductive medium by nondestructive testing called thermography. For the one space dimension, this has been already achieved by Y. Daido, H. Kang and G. Nakamura. The present paper shows that their result can be generalized to higher space dimension.

We derive weak Carleman estimates with two large parameters for a general partial differential operator of second order under pseudo-convexity conditions on the weight function. We use these estimates to derive most natural Carleman type estimates for the (anisotropic) system of elasticity with residual stress and give applications to uniqueness an...

In this paper, we obtain bounds showing increasing stability of the continuation for solutions of the stationary Maxwell system when the wave number k is growing. We reduce this system to a new system with the Helmholtz operator in the principal part and use hyperbolic energy and Carleman estimates with k-independent constants in the Cauchy problem...

This paper concerns the identification of a so-called doping profile (source term) in the system of elliptic equations modeling a semiconductor device. We give several simplification of the model and find useful adjoint problems and asymptotics motivated by applications to inverse problems and by a presence of certain small physical parameters As a...

In this review, we describe the main results on the uniqueness and stability of penetrable or impenetrable obstacles from various boundary data. We present basic ideas of proofs in the case of single and many boundary measurements. We discuss some methods of reconstruction. We consider the inverse problem of gravimetry, inverse conductivity and sca...

We consider a hyperbolic differential operator with variable principal term. We first give a sufficient condition for the pseudoconvexity which yields a Carleman estimate and a necessary condition. Our sufficient condition implies that level sets generated by the weight function in the Carleman estimate are convex with respect to the set of rays gi...

We give some recent results on Carleman type estimates for systems of partial differential equations with emphasis on applications to continuum mechanics. In particular we discuss isotropic elasticity with residual stress. We show how to derive stability estimates in the Cauchy problem and we study increased stability for the Helmholtz equation. We...

In this paper, we consider an elasticity system with residual stress. We are interested in recovery of the residual stress from one boundary measurement. We obtain uniqueness and the (best possible) Lipschitz stability results without any smallness assumption.


We prove Carleman type estimates with two large parameters for general linear partial differential operators of second order. Using the second large parameter, from results for scalar equations we derive Carleman estimates for dynamical Lamé system with residual stress. These estimates are used to prove the Hölder and Lipschitz stability for the co...

In this paper we demonstrate uniqueness of a transparent obstacle, of coefficients of rather general boundary transmission condition, and of a potential coefficient inside obstacle from partial Dirichlet-to Neumann map or from complete scattering data at fixed frequency. The proposed transmission problem includes in particular the isotropic ellipti...

We show uniqueness of a (time independent) domain D and of an impedance type boundary condition from either dynamical or scattering data at many frequencies. We assume that the additional boundary (scattering) data are given for one set of boundary data or for one incident direction. In case of general domain and finite (sharp) observation time we...

We derive weak Carleman estimates with two large parameters for a general partial dierential operator of second order under pseudo-convexity conditions on the weight function. We use these estimates to derive most nat- ural Carleman type estimates for the (anisotropic) system of elasticity with residual stress and give applications to uniqueness an...

We derive some bounds which can be viewed as an evidence of increasing stability in the Cauchy problem for the Helmholtz equation with lower order terms when frequency is growing. These bounds hold under certain (pseudo-)convexity properties of the surface, where the Cauchy data are given, and of variable zero order coefficient of the Helmholtz equ...

In this paper we prove a Holder and Lipschitz stability estimates of de- termining all coefficients of a dynamical Lame system with residual stress, including the density, Lame parameters, and the residual stress, by three pairs of observations from the whole boundary or from a part of it. These estimates imply first uniqueness results for determin...

We review simple models of oil reservoirs and suggest some ideas for theoretical and numerical study of this important inverse problem. These models are formed by a system of an elliptic and a parabolic (or first-order hyperbolic) quasilinear partial differential equations. There are and probably there will be serious theoretical and computational...

This paper describes recent uniqueness results in inverse problems for semiconductor devices and in the inverse conductivity problem. We remind basic inverse probelsm in semiconductor theory and outline use of an adjoint equation and a proof of uniqueness of piecewise constant doping profile. For the inverse conductivity problem we give a first uni...

In this paper, we give analytical and numerical evidence of increasing stability in the Cauchy problem for the Helmholtz equation in the whole domain when frequency is growing. This effect depends upon the convexity properties of the surface where the Cauchy data are given. Proofs use previously obtained estimates in subdomains and the theory of So...

We show that the Dirichlet-to-Neumann map given on an arbitrary part of the boundary of a three-dimensional domain with zero Dirichlet (or Neumann) data on the remaining (spherical or plane) part of the boundary uniquely determines conductivity or potential coefficients. This is the first uniqueness result for the Calderon problem with zero data on...

In this paper we give analytical evidence of increasing stability in the Cauchy Problem for the Helmholtz equation when frequency is growing. This eect depends on convexity properties of the surface where the Cauchy Data are given and on some monotonicity properties of the variable coecient of the Helmholtz equation. Proofs use Carleman estimates a...

We show that the Dirichlet-to-Neumann map given on an arbitrary part of the boundary of a three-dimensional domain with zero Dirichlet (or Neumann) data on the remaining (spherical or plane) part of the boundary uniquely determines conductivity or potential coefficients. This is the first uniqueness result for the Calderon problem with zero data on...

In this paper we prove a Hölder and Lipschitz stability estimates of de-termining the residual stress by a single pair of observations from a part of the lateral boundary or from the whole boundary. These estimates imply first uniqueness results for determination of residual stress from few boundary mea-surements.

We consider the problem of recovering surface vibrations from acoustic pressure measurements taken in the interior or the exterior of a region. We give two formulations of the problem. One is based on a representation of the pressure as layer potentials and the other is based on approximation by a class of specific solutions to the Helmholtz equati...

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The paper deals with the problem of recovering the parameters (functions) $\varepsilon$ and $\mu$ of the Maxwell dynamical system $$\varepsilon E_t = {\text{rot}}\,H,\quad \mu H_t = - {\text{rot}}\,E\quad in\quad \Omega \times \left( {0,T} \right);$$ $$E\left| {_{t = 0} = 0,\quad H} \right|_{t = 0} = 0\quad in\quad \Omega ;$$ $$E_{\tan }...

Introduction The Black-Scholes formula [6] provides with an elegant and simple method to price financial derivatives under the assumption that the stock price is log-normally distributed. However, the actual distribution of most assets is rarely log-normal, and theoretical prices of options with di#erent strikes generated by the Black-Scholes formu...

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....

In these lecture notes we derive Carleman type estimates for second order linear partial differential operators and show some of their applications to the uniqueness in the Cauchy problem, approximate and exact controllability, and inverse problems. In section 2 we discuss pseudo-convexity and derive Carleman type estimates with boundary terms. In...

We consider the isotropic elasticity system: \[ \begin{array}{cclcc} \rho \partial _{t}^{2}\mathbf{u}&-&\mu (\Delta\mathbf{u}+\nabla (\nabla ^{T}\mathbf{u})-\nabla (\lambda\nabla ^{T}\mathbf{u})&&\\ &-&\sum _{j=1}^{3}\nabla\mu\cdot (\nabla u_{j}+\partial _{j}\mathbf{u})\mathbf{e}_{j}=0 &\text{in}& \Omega\times (0,T) \end{array} \] for the displacem...

We consider an initial value problem with the homogeneous Neumann boundary condition: y '' (x,t)=Δy(x,t)+λ(t)f(x),x∈Ω,0<t<T, y(x,0)=y ' (x,0)=0,x∈Ω, ∂y ∂ν(x,t)=0,x∈∂Ω,0<t<T· Here Ω⊂ℝ n is a bounded domain whose boundary ∂Ω is of piecewise C 2 -class. Let λ¬≡0 be given and let Γ⊊Ω be fixed suitably. We discuss an inverse problem of determining f fro...

We prove uniqueness and a Hölder-type stability of reconstruction of all three time-independent elastic parameters in the dynamical isotropic system of elasticity from two special sets of boundary measurements. In proofs we use Carleman-type estimates in Sobolev spaces of negative order. © 2003 Wiley Periodicals, Inc.

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...

This paper presents a rigorous mathematical justification of the Helmholtz equation least squares (HELS) method for reconstructing acoustic radiation from an arbitrary source. It is shown that the acoustic pressure radiated from a non-spherical structure can be approximated using the spherical wavefunctions and spherical harmonics, and errors invol...

We obtain new uniqueness of the continuation results for the thermoelasticity system on the plane. The crucial ingredient of the proofs is the use of Carleman-type estimates with two large parameters for basic second order partial differential operators with constant coefficients. We derive these estimates by applying differential quadratic forms....

We prove uniqueness of the term c(x, t, u) of systems of partial differential equations ∂tu - Δu + c(x, t, u) = 0 and - Δu + c(x, u) = 0 with the Dirichlet-to-Neumann map given on a part of the lateral boundary. We use a linearization method, singular solutions in the boundary reconstruction, and almost exponential solutions in the interior reconst...

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 admit...

We consider a hyperbolic equation with a damping term in a bounded domain Ω ⊂ ℝN : $$\frac{{{\partial ^2}u}}{{\partial {t^2}}}\left( {x,t} \right) = \Delta u\left( {x,t} \right) - q\left( x \right)\frac{{\partial u}}{{\partial t}}\left( {x,t} \right),x \in \Omega ,0 < t < T$$ where \(u\left( {\cdot ,0} \right),\frac{{\partial u}}{{\partial t}}\left...

Computational methods for the inverse problem of detecting the source of acoustical noise in an interior region from pressure measurements in the nearfield are discussed. The methods are based on a single layer potential representation of solutions to the Helmholtz equation. Regularization is peformed using the singular value decomposition and the...

This paper deals with the problem of the identification of the diffusion coefficient in a parabolic equation. This inverse problem is formulated as a nonlinear operator equation in Hilbert spaces. Continuity and differentiability of the corresponding operator is shown. In the one-dimensional case uniqueness and conditional stability results are obt...

Computational methods for the inverse problem of detecting the source of acoustical noise in an interior region from pressure measurements in the nearfield are discussed. The methods are based on a single layer potential representation of solutions to the Helmholtz equation. Regularization is peformed using the singular value decomposition and the...

The problem of computing normal velocities on the boundary of a region from pressure measurements on an interior surface is considered. The pressure satisfies the Helmholtz equation and is represented by a single layer potential. Once the density function is found, the normal velocities can be easily computed by applying a second kind of integral o...

1.75> # ) oriented compact 3-dimensional Riemannian manifold with the boundary #. Define the eikonal # :# # R, # (x) = dist (x, #); denote # # = {x ## | # (x) < #} [ #, # # = {x ## | # (x) = #} for # # 0, (# 0 = #), and # T = # × [0, T ). Let T# be the supremum of such T > 0 that the map exp# : # T ## T is a di#eomorphism. We always assume that T <...

Market prices of financial derivatives such as options are directly observable. This information can be used to recover an unobservable local volatility function for the underlying stochastic process. We give a rigorous mathematical formulation of this inverse problem, provide available uniqueness and stability results using the dual equation and r...

. In many applications, such as the heat conduction and hydrology, there is a need to recover the (possibly discontinuous) di#usion coe#cient a from boundary measurements of solutions of a parabolic equation. The complete inverse problem is ill posed and nonlinear, so numerical solution is quite di#cult, and we linearize the problem around constant...

In this chapter we consider the second-order parabolic equation $$ {a_0}{\partial _t}u - div(a\nabla u) + b\nabla u + cu = f{\mkern 1mu} in{\mkern 1mu} Q = \Omega \times (0,{\mkern 1mu} T), $$ (9.0.1) where Ω is a bounded domain the space ℝn with the C 2-smooth boundary ∂Ω. In Section 9.5 we study the nonlinear equation $$ {a_0}(x,u){u_t} - \Delta...

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 prov...

We prove the uniqueness of the determination of a surface crack from one special boundary measurement of an electrical or elastic field. Then we suggest and test a numerical algorithm for identification of a polygonal plane crack based on the Schwarz-Christoffel formula. The numerical experiments with cracks consisting of one or two intervals show...

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 suffici...

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 s...

We seek to recover the interior electrical conductivity of an inhomogeneous object by linearizing the inverse conductivity problem as suggested by Calderon. First, we reduce the Dirichlet-to-Neumann data to the data of the problem in the whole with point sources and suggest a linearization of this new, simpler linear inverse problem. Then, we study...

We study stability of recovery of an obstacle D from its scattering amplitude corresponding to the Helmholtz equation Delta u+K2u=0 in the three-dimensional space outside D. We assume u is zero on the boundary of D (soft obstacle). We consider smooth star-shaped and analytic D and obtain double logarithmic type and logarithmic type estimates.

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 stri...

The numerical results on the uniqueness, stability and existence of a second order parabolic equation were presented. The problem had applications in the nondestructive evaluation of physical bodies from measurements of their temperature fields. The problem was reduced to a lateral problem of Cauchy type and the conditional logarithmic stability es...

Consider the second order hyperbolic equation $${u_{{tt}}} - div\left( {a\nabla u} \right) + b{u_{t}} = 0{\text{ }}in{\text{ }}Q = \times \left( {0,T} \right)$$ (1.1) with zero initial conditions $$u = {u_{t}} = 0{\text{ }}on{\text{ }}\Omega \left\{ 0 \right\}$$ (1.2) and the lateral Neumann boundary condition $$a{\partial _{v}}u = h{\text{ }}on{\t...

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 stri...

We prove uniqueness of a discontinuous principal coefficient of a second-order parabolic equation of the form a0 + χ(Q*)b with known smooth a0 and unknown b = b(x) from all possible lateral boundary measurements of solutions of this equation. In the proofs, we make use of singular solutions of parabolic equations.

In this chapter we formulate and in many cases prove results on uniqueness and stability of solutions of the Cauchy problem for general partial differential equations. One of the basic tools is Carleman-type estimates. In Section 3.1 we describe the results for a simplest problem of this kind (the backward parabolic equation), where a choice of the...

We deal with recovering the discontinuity surface of the speed of the threedimensional wave propagation from (single) boundary observation. We prove a sharp uniqueness result assuming that the surface is the graph of a function of the depth and that the reference domain is a half-space. We make use of (new) exact results on the uniqueness of the co...

We describe some recent results on recovery of the principal coefficient of a second order partial differential equation of parabolic type, given one or all possible sets of the lateral Cauchy data of its solution. We outline ideas of proofs referring for details to other publications. The results are expected to be of importance in the inverse hea...

We treat the inverse problem of the determination of the conductivity coefficient a = l + μχD, D ⊂⊂ Ω, μ = constant, in the elliptic equation div(a∇u) = Ω in il, when overdetermined boundary data for one nontrivial solution u are assigned . We show that nonuniqueness in the determination of the domain D would imply that a part Γ of ∂D is a solution...

For a general class of nonlinear Schrödinger 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 .

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

We prove local uniqueness of a domain D entering the conductivity equation $\operatorname{div}((1 + \chi(D))\nabla u) = 0$ in a bounded planar domain Ω given the Cauchy data for u on a part of ∂Ω. The main assumption is that ∇ u has zero index on ∂Ω which is easy to guarantee by choosing special boundary data for u. To achieve our goals we study in...

We describe a specific measurement process that works well in practice for locating steel reinforcement bars in concrete. For the case that the total volume of these bars is small, we derive an approximate linear, but sufficiently accurate, mathematical model for which we can prove a uniqueness result.

In this paper we estimate distances between two soft obstacles in terms of corresponding scattering amplitudes. The estimate is of logarithmic type and it is obtained by using estimates for continuation of solutions of elliptic equations and some methods of potential theory. Scattering amplitudes are given for one frequency comparable with size of...

Let D be a subdomain of a bounded domain OMEGA in R(n). The conductivity coefficient of D is a positive constant k not-equal 1 and the conductivity of OMEGA/D is equal to 1. For a given current density g on partial derivative OMEGA, we compute the resulting potential u and denote by f the value of u on partial derivative OMEGA. The general inverse...

In this paper we give conditions which guarantee that products of solutions of partial differential equations Pu + au = 0 are complete in . HereP is a linear partial differential operator with constant coefficients and a is a function in . We check these conditions for elliptic, parabolic, and hyperbolic equations of second order and give applicati...

An explicit stability estimate for the two-dimensional wave equation when the Cauchy data are prescribed on a part of the lateral boundary is derived. Our result is obtained using a combination of the Friedrichs-Leray energy integrals and Carleman type estimates of Hörmander [“Linear Partial Differential Operators,” Springer-Verlag, New York/Berlin...