Adaptive Wavelet Methods for Inverse Problems: Acceleration Strategies, Adaptive Rothe Method and Generalized Tensor Wavelets
In general, inverse problems can be described as the task of inferring conclusions about the cause u from given observations y of its effect. This can be described as the inversion of an operator equation K(u) = y, which is assumed to be ill-posed or ill-conditioned. To arrive at a meaningful soluti...
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|In general, inverse problems can be described as the task of inferring conclusions about the cause u from given observations y of its effect. This can be described as the inversion of an operator equation K(u) = y, which is assumed to be ill-posed or ill-conditioned. To arrive at a meaningful solution in this setting, regularization schemes need to be applied. One of the most important regularization methods is the so called Tikhonov regularization. As an approximation to the unknown truth u it is possible to consider the minimizer v of the sum of the data error K(v)-y (in a certain norm) and a weighted penalty term F(v). The development of efficient schemes for the computation of the minimizers is a field of ongoing research and a central Task in this thesis. Most computation schemes for v are based on some generalized gradient descent approach. For problems with weighted lp-norm penalty terms this typically leads to iterated soft shrinkage methods. Without additional assumptions the convergence of these iterations is only guaranteed for subsequences, and even then only to stationary points. In general, stationary points of the minimization problem do not have any regularization properties. Also, the basic iterated soft shrinkage algorithm is known to converge very poorly in practice. This is critical as each iteration step includes the application of the nonlinear operator K and the adjoint of its derivative. This in itself may already be numerically demanding.
This thesis is concerned with the development of strategies for the fast computation of the solution of inverse problems with provable convergence rates. In particular, the application and generalization of efficient numerical schemes for the treatment of the arising nonlinear operator equations is considered.
The first result of this thesis is a general acceleration strategy for the iterated soft thresholding iteration to compute the solution of the inverse problem. It is based on a decreasing strategy for the weights of the penalty term. The new method converges with linear rate to a global minimizer.
A very important class of inverse problems are parameter identification problems for partial differential equations. As a prototype for this class of problems the identification of parameters in a specific parabolic partial differential equation is investigated. The arising operators are analyzed, the applicability of Tikhonov Regularization is proven and the parameters in a simplified test equation are reconstructed.
The parabolic differential equations are solved by means of the so called horizontal method of lines, also known as Rothes method. Here the parabolic problem is interpreted as an abstract Cauchy problem. It is discretized in time by means of an implicit scheme. This is combined with a discretization of the resulting system of spatial problems. In this thesis the application of adaptive discretization schemes to solve the spatial subproblems is investigated. Such methods realize highly nonuniform discretizations. Therefore, they tend to require much less degrees of freedom than classical discretization schemes. To ensure the convergence of the resulting inexact Rothe method, a rigorous convergence proof is given. In particular, the application of implementable asymptotically optimal adaptive methods, based on wavelet bases, is considered. An upper bound for the degrees of freedom of the overall scheme that are needed to adaptively approximate the solution up to a prescribed tolerance is derived. As an important case study, the complexity of the approximate solution of the heat equation is investigated. To this end a regularity result for the spatial equations that arise in the Rothe method is proven.
The rate of convergence of asymptotically optimal adaptive methods deteriorates with the spatial dimension of the problem. This is often called the curse of dimensionality. One way to avoid this problem is to consider tensor wavelet discretizations. Such discretizations lead to dimension independent convergence rates. However, the classical tensor wavelet construction is limited to domains with simple product geometry. Therefor, in this thesis, a generalized tensor wavelet basis is constructed. It spans a range of Sobolev spaces over a domain with a fairly general geometry. The construction is based on the application of extension operators to appropriate local bases on subdomains that form a non-overlapping domain decomposition. The best m-term approximation of functions with the new generalized tensor product basis converges with a rate that is independent of the spatial dimension of the domain. For two- and three-dimensional polytopes it is shown that the solution of Poisson type problems satisfies the required regularity condition. Numerical tests show that the dimension independent rate is indeed realized in practice.