Volume 13, Issue 2

We consider a class of probability measure dependent dynamical systems which arise in the study of multiscale phenomena in diverse fields such as immunological population dynamics, viscoelasticity of polymers and rubber, and polarization in dielectric materials. We develop an inverse problem framework for studying systems with distributed temporal delays. In particular, we establish conditions for existence and uniqueness of the forward problem and well-posedness (including method stability under numerical approximations) for the inverse problem of estimating the probability measures. We show that a motivating class of models of HIV infection dynamics satisfies all the conditions of our framework, thereby providing a theoretical foundation for inverse problem computations with these models.

In this paper we deal with the degree of ill-posedness of linear operator equations Ax = y , x ∈ X , y ∈ Y , in the Hilbert space X = Y = L 2 (0, 1), where A = M о J is a compact operator that may be decomposed into the simple integration operator J with a well-known decay rate of singular values and a multiplication operator M determined by the multiplier function m . This case occurs for example for nonlinear operator equations F ( x ) = y with a forward operator F = N о J where N is a Nemytskii operator. Then the local degree of ill-posedness of the nonlinear equation at a point x 0 of the domain of F is investigated via the Fréchet derivative of F which has the form F ' ( x 0 ) = M о J . We show the restricted influence of such multiplication operators M mapping in L 2 (0,1). If the multiplier function m has got zeros, the determination of the degree of ill-posedness is not trivial. We are going to investigate this situation and provide analytical tools as well as their limitations. For power and exponential type multiplier functions with essential zeros we will show by using several numerical approaches that the unbounded inverse of the injective multiplication operator does not influence the local degree of ill-posedness. We provide a conjecture, verified by several numerical studies, how these multiplication operators influence the singular values of A = M о J . Finally we analyze the influence of those multiplication operators M on the possibilities of Tikhonov regularization and corresponding convergence rates. We investigate the role of approximate source conditions in the method of Tikhonov regularization for linear and nonlinear ill-posed operator equations. Based on the studies on approximate source conditions we indicate that only integrals of m and not the decay of multiplier functions near zero determines the convergence behavior of the regularized solution.

Let Ω ⊂ be a bounded domain and {( x 1 , x 2 , x 3 ) | ( x 1 , x 3 ) ∈ Ω, x 2 ∈ (−∞, ∞)} be a cylinder. Assume that the stationary electric field u induced by a pointwise current modelled by the function f ( x 2 ) S ( x 1 , x 3 ) is governed by an elliptic differential equation ∇ · (σ∇ u ) = − f ( x 2 ) S ( x 1 , x 3 ). Also, assume that a pair of the Dirichlet and Neumann conditions is given on an arbitrary part of the lateral surface of the cylinder. The problem of determining the 2D electric conductivity σ( x 1 , x 3 ) in this cylinder from the boundary incomplete data is considered. Using the direct Fourier and inverse Laplace transforms, the original inverse problem is reduced to an auxiliary inverse problem for a hyperbolic equation . The main difficulty is due to the presence of derivatives of the function σ ( x 1 , x 3 ). To overcome this difficulty, the method of Carleman estimates is significantly modified. This allows for establishing the Lipschitz stability for the auxiliary hyperbolic inverse problem. In turn, the stability result implies the global uniqueness theorem for the original inverse conductivity problem. The proposed formulation and uniqueness theorem extend both the formulation of the 1D inverse problem of electrical prospecting and uniqueness result established by Tikhonov in 1949.

We report on a new iterative method for regularizing a nonlinear operator equation in Hilbert spaces. The proposed algorithm is a combination of Tikhonov regularization and a fixed point algorithm for the minimization of the Tikhonov functional. Under the assumptions that the operator F is twice continuous Fréchet-differentiable with Lipschitz-continuous first derivative and that the solution of the equation F ( x ) = y fulfills a smoothness condition we will give a convergence rate result. Numerical results with data from Single Photon Emission Computed Tomography (SPECT) show the rapid convergence of the proposed algorithms.