Volume 15, Issue 3

We consider a least squares inverse problem with a model for inducer-mediated reactivation of latent viruses. We illustrate the difficulties associated with the lack of sufficient data (quantity as well as form) in such problems. The modeling and estimation efforts described suggest new experiments as well as needed model extensions.

In this paper we consider a parameter estimation problem for delay differential equations that are typical in the bioscience. The problem is motivated by a model of the dynamics of the intracellular development cycle of Chlamydia Trachomatis and can be formulated as a inverse problem for a system of delay equations. We derive the continuous sensitivity equations and show that the corresponding initial value problem is well-posed. The sensitivity equations are used to compute gradients in the optimization loop for the parameter estimation problem. Numerical examples are given to illustrate the key ideas and we close with a note about the use of continuous sensitivity equations in the design of experiments.

We study nonlinear inverse problems of the form F ( x ) = y , and their stable solution via iterative regularization methods, in particular by Newton-type methods, which are well-known for their fast convergence for well-posed problems. A basic step of Newton’s method consists of calculating the update Δ x k via solution of a linearized equation , which will in general be ill-posed if the nonlinear problem is. Thus, for ill-posed problems, the linearized equations have to be solved by some regularization method. In particular for large scale problems, e.g., inverse problems in partial differential equations, where F ( x ) is only defined implicitly via the solution of a PDE, iterative methods have to be used for this purpose. In order to keep the overall effort, i.e., the overall number of iterations, as small as possible, appropriate preconditioning has to be applied. We propose and analyse a general preconditioning strategy in Hilbert scales, and show that the overall number of iterations can be reduced to about the square root by preconditioning. Moreover, in many examples differential operators can be used as preconditioners, and thus preconditioning is almost for free. The theoretical results are illustrated in numerical examples. A comparison with preconditioned Landweber iteration shows that the iteration numbers can be further reduced, if fast iterative methods, e.g., the v -methods, are used for the solution of the linearized problems.

Extrapolation is a well-known tool for increasing the accuracy of approximation methods. We consider extrapolation of Tikhonov and Lavrentiev methods and iterated variants of these methods for solving linear ill-posed problems in Hilbert space. For extrapolated approximation we take the linear combination of n ≥ 2 approximations of the original method with different parameters and with proper coefficients, guaranteeing for extrapolated method higher qualification than in original method. Extrapolated approximation can be used for approximation to solution of the equation or for construction of a posteriori rules for choice of the regularization parameter in original method. As shown, extrapolating n Tikhonov or Lavrentiev approximations gives the same approximation as one gets in nonstationary implicit iterative method after n iterations with the same parameters. If the solution is smooth and δ is noise level of data, a proper choice of n = n (δ) guarantees for extrapolated Tikhonov approximation accuracy versus accuracy of Tikhonov approximation. Note that in a posteriori parameter choice in Tikhonov method often several approximations with different parameters are computed and then computation of their linear combination is an easy task.

We investigate continuous regularization methods for linear inverse problems of static and dynamic type. These methods are based on dynamic programming approaches for linear quadratic optimal control problems. We prove regularization properties and also obtain rates of convergence for our methods. A numerical example concerning a dynamical electrical impedance tomography (EIT) problem is used to illustrate the theoretical results.

In this paper we improve existing convergence and convergence rate results for the iteratively regularized Gauss-Newton method in two respects: First we show optimal rates of convergence under general source conditions, and second we assume that the linearized equations are solved only approximately in each Newton step. The latter point is important for large scale problems where the linearized equation can often only be solved iteratively, e.g. by the conjugate gradient method.