Ian Knowles

Parameter identification for elliptic problems

Abstract The problem of computing a principal coefficient function P in the differential equation −∇·(P(x)∇u)=f, x∈Ω⊂ R M , M⩾1, on a bounded region Ω from a knowledge of the solution function u and the right-hand side f, where u, f are known only approximately and P may have mild discontinuities, is solved by minimization of an associated functional.

Uniqueness for an Elliptic Inverse Problem

For the elliptic equation

A variational algorithm for electrical impedance tomography

-\nabla\cdot(p(x)\nabla v)+\lambda q(x)v=f,\ x\in\Omega\subset\bbR^n,

On the Inverse Resonance Problem

the problem of determining when one or more of the coefficient functions p, q, and f are defined uniquely by a knowledge of one or more of the solution functions

A variational approach to an elastic inverse problem

v=v_{p,q,f,\lambda}

On the boundary conditions characterizing J-selfadjoint extensions of J-symmetric operators

is considered.

On the extension problem for accretive differential operators

The problem of computing the coefficient function p in the elliptic differential equation , , , over a bounded region , from a knowledge of the Dirichlet-Neumann map for this equation, is of interest in electrical impedance tomography. A new approach to the computation of p involving the minimization of an associated functional is presented. The algorithm is simple to implement and robust in the presence of noise in the Dirichlet-Neumann data.

Characterization of chemical gradients in the collagen gel-visual chemotactic assay

An ew technique is presented which gives conditions under which perturbations of certain base potentials are uniquely determined from the location of eigenvalues and resonances in the context of a Schr¨ odinger operator on a half line. The method extends to complex-valued potentials and certain potentials whose first moment is not integrable.

On the point spectra of complex Sturm—Liouville operators

We present a variational approach to the seismic inverse problem of determining the coefficients C and ρ of the hyperbolic system of partial differential equations from traction and displacement data measured on the surface. A crucial point of our approach will be a transformation of the above system to an elliptic system of partial differential equations Thus, we transform the inverse problem for a hyperbolic system to an inverse problem for an elliptic system. We give a definition of the direct and inverse seismic problem, where we distinguish between the isotropic and anisotropic cases. Further, we develop the theoretical results that we need for a successful recovery procedure of the coefficients C and ρ in the isotropic case. Our approach consists of a minimization procedure based on a conjugate gradient descent algorithm. Finally, we present various numerical results that show the effectiveness of our approach.

On stability conditions for second order linear differential equations

The study of boundary value problems involving linear differentiai equations with real-valued coefficients is by now a well-established area of analysis. On the other hand, much less is known about the solvability of such problems when the coefficients (or boundary conditions) are known to be complex. Examples of this latter type arise naturally, for example, in nuclear physics (e.g., the so-called optical mode1 for low energy scattering [ 1, p. IjO]), electromagnetic field theory (dielectric waveguides with heat loss (c.f. 19 I), or the propagation of radio waves through inhomogeneous media [S]), and elsewhere. In cases like these, the relevant differential expressions are no longer formally symmetric, and hence the powerful methods associated with the spectral theory of selfadjoint operators are not available. To facilitate the study of such problems, Glazman introduced in [4] the concept of a J-symmetric operator: In a complex Hilbert space R, let J be a given conjugation operator on 2’ (i.e., J is a conjugate-linear involution with (Jx, 3~) = (u, x) for all x and y in 2). A closed, densely defined linear operator T in 3’ is said to be J-symmetric if