Extraction of Quantifiable Information from Complex Systems

✍ Scribed by Stephan Dahlke, Wolfgang Dahmen, Michael Griebel, Wolfgang Hackbusch, Klaus Ritter, Reinhold Schneider, Christoph Schwab, Harry Yserentant (eds.)

• Publisher: Springer International Publishing
• Year: 2014
• Tongue: English
• Leaves: 446
• Series: Lecture Notes in Computational Science and Engineering 102
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xix
Solving Stochastic Dynamic Programs by Convex Optimization and Simulation....Pages 1-23
Efficient Resolution of Anisotropic Structures....Pages 25-51
Regularity of the Parameter-to-State Map of a Parabolic Partial Differential Equation....Pages 53-67
Piecewise Tensor Product Wavelet Bases by Extensions and Approximation Rates....Pages 69-81
Adaptive Wavelet Methods for SPDEs....Pages 83-107
Constructive Quantization and Multilevel Algorithms for Quadrature of Stochastic Differential Equations....Pages 109-132
Bayesian Inverse Problems and Kalman Filters....Pages 133-159
Robustness in Stochastic Filtering and Maximum Likelihood Estimation for SDEs....Pages 161-178
Adaptive Sparse Grids in Reinforcement Learning....Pages 179-194
A Review on Adaptive Low-Rank Approximation Techniques in the Hierarchical Tensor Format....Pages 195-210
A Bond Order Dissection ANOVA Approach for Efficient Electronic Structure Calculations....Pages 211-235
Tensor Spaces and Hierarchical Tensor Representations....Pages 237-261
Nonlinear Eigenproblems in Data Analysis: Balanced Graph Cuts and the RatioDCA-Prox....Pages 263-279
Adaptive Approximation Algorithms for Sparse Data Representation....Pages 281-302
Error Bound for Hybrid Models of Two-Scaled Stochastic Reaction Systems....Pages 303-319
Valuation of Structured Financial Products by Adaptive Multiwavelet Methods in High Dimensions....Pages 321-345
Computational Methods for the Fourier Analysis of Sparse High-Dimensional Functions....Pages 347-363
Sparsity and Compressed Sensing in Inverse Problems....Pages 365-379
Low-Rank Dynamics....Pages 381-396
Computation of Expectations by Markov Chain Monte Carlo Methods....Pages 397-411
Regularity, Complexity, and Approximability of Electronic Wavefunctions....Pages 413-428
Back Matter....Pages 429-440

✦ Subjects

Partial Differential Equations; Applications of Mathematics; Computational Mathematics and Numerical Analysis; Numerical Analysis; Probability Theory and Stochastic Processes; Approximations and Expansions