In April 2007, the Deutsche Forschungsgemeinschaft (DFG) approved the Priority Program 1324 “Mathematical Methods for Extracting Quantifiable Information from Complex Systems.” This volume presents a comprehensive overview of the most important results obtained over the course of the program. Mathematical models of complex systems provide the foundation for further technological developments in science, engineering and computational finance. Motivated by the trend toward steadily increasing computer power, ever more realistic models have been developed in recent years. These models have also become increasingly complex, and their numerical treatment poses serious challenges. Recent developments in mathematics suggest that, in the long run, much more powerful numerical solution strategies could be derived if the interconnections between the different fields of research were systematically exploited at a conceptual level. Accordingly, a deeper understanding of the mathematical foundations as well as the development of new and efficient numerical algorithms were among the main goals of this Priority Program. The treatment of high-dimensional systems is clearly one of the most challenging tasks in applied mathematics today. Since the problem of high-dimensionality appears in many fields of application, the above-mentioned synergy and cross-fertilization effects were expected to make a great impact. To be truly successful, the following issues had to be kept in mind: theoretical research and practical applications had to be developed hand in hand; moreover, it has proven necessary to combine different fields of mathematics, such as numerical analysis and computational stochastics. To keep the whole program sufficiently focused, we concentrated on specific but related fields of application that share common characteristics and as such, they allowed us to use closely related approaches.
D. Belomestny, C. Bender, F. Dickmann, and N. Schweizer: Solving
Stochastic Dynamic Programs by Convex Optimization and Simulation.- W.
Dahmen, C. Huang, G. Kutyniok, W.- Q Lim, C. Schwab, and G. Welper: Efficient
Resolution of Anisotropic Structures.- R. Ressel, P. Dülk, S. Dahlke, K. S.
Kazimierski, and P. Maass: Regularity of the Parameter-to-state Map of a
Parabolic Partial Differential Equation.- N. Chegini, S. Dahlke, U.
Friedrich, and R. Stevenson: Piecewise Tensor Product Wavelet Bases by
Extensions and Approximation Rates.- P. A. Cioica, S. Dahlke, N. Döhring, S.
Kinzel, F. Lindner, T. Raasch, K. Ritter, and R. Schilling: Adaptive Wavelet
Methods for SPDEs.- M. Altmayer, S. Dereich, S. Li, T. Müller-Gronbach, A.
Neuenkirch, K. Ritter and L. Yaroslavtseva: Constructive Quantization and
Multilevel Algorithms for Quadrature of Stochastic Differential Equations.-
O. G. Ernst, B. Sprungk, and H.- J. Starkloff: Bayesian Inverse Problems and
Kalman Filters.- J. Diehl, P. Friz, H. Mai, H. Oberhauser, S. Riedel, and W.
Stannat: Robustness in Stochastic Filtering and Maximum Likelihood Estimation
for SDEs.- J. Garcke and I. Klompmaker: Adaptive Sparse Grids in
Reinforcement Learning.- J. Ballani, L. Grasedyck, and M. Kluge: A Review on
Adaptive Low-Rank Approximation Techniques in the Hierarchical Tensor
Format.- M. Griebel, J. Hamaekers, and F. Heber: A Bond Order Dissection
ANOVA Approach for Efficient Electronic Structure Calculations.- W. Hackbusch
and R. Schneider: Tensor Spaces and Hierarchical Tensor Representations.- L.
Jost, S. Setzer, and M. Hein: Nonlinear Eigenproblems in Data Analysis -
Balanced Graph Cuts and the Ratio DCA-Prox.- M. Guillemard, D. Heinen, A.
Iske, S. Krause-Solberg, and G. Plonka: Adaptive Approximation Algorithms for
Sparse Data Representation.- T. Jahnke and V. Sunkara: Error Bound for Hybrid
Models of Two-scaled Stochastic Reaction Systems.- R. Kiesel, A. Rupp, and K.
Urban: Valuation of Structured Financial Products byAdaptive Multi wavelet
Methods in High Dimensions.- L Kämmerer, S. Kunis, I. Melzer, D. Potts, and
T. Volkmer: Computational Methods for the Fourier Analysis of Sparse
High-Dimensional Functions.- E. Herrholz, D. Lorenz, G. Teschke, and D.
Trede: Sparsity and Compressed Sensing in Inverse Problems.- C. Lubich:
Low-Rank Dynamics.- E. Novak and D. Rudolf: Computation of Expectations by
Markov Chain Monte Carlo Methods.- H. Yserentant: Regularity, Complexity, and
Approximability of Electronic Wave functions.- Index.
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