Schnelle Löser für Partielle Differentialgleichungen

This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds

Adaptive Numerical Methods for PDEs

This collection contains the extended abstracts of the talks given at the Oberwolfach Conference on “Adaptive Numerical Methods for PDEs”, June 10th - June 16th, 2007. These talks covered various aspects of a posteriori error estimation and mesh as well as model adaptation in solving partial differential equations. The topics ranged from the theoretical convergence analysis of self-adaptive methods, over the derivation of a posteriori error estimates for the finite element Galerkin discretization of various types of problems to the practical implementation and application of adaptive methods

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

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 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

Wavelet and Multiscale Methods

[no abstract available

Wavelet and Multiscale Methods

Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers for partial differential equations. These have substantially advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines

Adaptive Wavelet Methods for Variational Formulations of Nonlinear Elliptic PDES on Tensor-Product Domains

This thesis is concerned with the numerical solution of boundary value problems (BVPs) governed by semilinear elliptic partial differential equations (PDEs). Semilinearity here refers to a special case of nonlinearity, i.e., the case of a linear operator combined with a nonlinear operator acting as a perturbation. In general, such BVPs are solved in an iterative fashion. It is, therefore, of primal importance to develop efficient schemes that guarantee convergence of the numerically approximated PDE solutions towards the exact solution. Unlike the typical finite element method (FEM) theory for the numerical solution of the nonlinear operators, the new adaptive wavelet theory proposed in [Cohen.Dahmen.DeVore:2003:a, Cohen.Dahmen.DeVore:2003:b] guarantees convergence of adaptive schemes with fixed approximation rates. Furthermore, optimal, i.e., linear, complexity estimates of such adaptive solution methods have been established. These achievements are possible since wavelets allow for a completely new perspective to attack BVPs: namely, to represent PDEs in their original infinite dimensional realm. Wavelets are the ideal candidate for this purpose since they allow to represent functions in infinite-dimensional general Banach or Hilbert spaces and operators on these. The purpose of adaptivity in the solution process of nonlinear PDEs is to invest extra degrees of freedom (DOFs) only where necessary, i.e., where the exact solution requires a higher number of function coefficients to represent it accurately. Wavelets in this context represent function bases with special analytical properties, e.g., the wavelets considered herein are piecewise polynomials, have compact support and norm equivalences between certain function spaces and the l_2 sequence spaces of expansion coefficients exist. This new paradigm presents nevertheless some problems in the design of practical algorithms. Imposing a certain structure, a tree structure, remedies these problems completely while restricting the applicability of the theoretical scheme only very slightly. It turns out that the considered approach naturally fits the theoretical background of nonlinear PDEs. The practical realization on a computer, however, requires to reduce the relevant ingredients to finite-dimensional quantities. It is this particular aspect that is the guiding principle of this thesis. This theoretical framework is implemented in the course of this thesis in a truly dimensionally unrestricted adaptive wavelet program code, which allows one to harness the proven theoretical results for the first time when numerically solving the above mentioned BVPs. In the implementation, great emphasis is put on speed, i.e., overall execution speed and convergence speed, while not sacrificing on the freedom to adapt many important numerical details at runtime and not at the compilation stage. This means that the user can test and choose wavelets perfectly suitable for any specific task without having to rebuild the software. The computational overhead of these freedoms is minimized by caching any interim data, e.g., values for the preconditioners and polynomial representations of wavelets in multiple dimensions. Exploiting the structure in the construction of wavelet spaces prevents this step from becoming a burden on the memory requirements while at the same time providing a huge performance boost because necessary computations are only executed as needed and then only once. The essential BVP boundary conditions are enforced using trace operators, which leads to a saddle point problem formulation. This particular treatment of boundary conditions is very flexible, which especially useful if changing boundary conditions have to be accommodated, e.g., when iteratively solving control problems with Dirichlet boundary control based upon the herein considered PDE operators. Another particular feature is that saddle point problems allow for a variety of different geometrical setups, including fictitious domain approaches. Numerical studies of 2D and 3D PDEs and BVPs demonstrate the feasibility and performance of the developed schemes. Local transformations of the wavelet basis are employed to lower the absolute condition number of the already optimally preconditioned operators. The effect of these basis transformations can be seen in the absolute runtimes of solution processes, where the semilinear PDEs are solved as fast as in fractions of a second. This task can be accomplished using simple Richardson-style solvers, e.g., the method of steepest descent, or more involved solvers like the Newton's method. The BVPs are solved using an adaptive Uzawa algorithm, which requires repeated solution of semilinear PDE sub-problems. The efficiency of different numerical methods is compared and the theoretical optimal convergence rates and complexity estimates are verified. In summary, this thesis presents for the first time a numerically competitive implementation of a new theoretical paradigm to solve semilinear elliptic PDEs in arbitrary space dimensions with a complete convergence and complexity theory