Extraction of Quantifiable Information from Complex Systems

1. Front Matter

   Pages i-xix

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2. Solving Stochastic Dynamic Programs by Convex Optimization and Simulation

   • Denis Belomestny, Christian Bender, Fabian Dickmann, Nikolaus Schweizer

   Pages 1-23

3. Efficient Resolution of Anisotropic Structures

   • Wolfgang Dahmen, Chunyan Huang, Gitta Kutyniok, Wang-Q Lim, Christoph Schwab, Gerrit Welper

   Pages 25-51

4. Regularity of the Parameter-to-State Map of a Parabolic Partial Differential Equation

   • Rudolf Ressel, Patrick Dülk, Stephan Dahlke, Kamil S. Kazimierski, Peter Maass

   Pages 53-67

5. Piecewise Tensor Product Wavelet Bases by Extensions and Approximation Rates

   • Nabi G. Chegini, Stephan Dahlke, Ulrich Friedrich, Rob Stevenson

   Pages 69-81

6. Adaptive Wavelet Methods for SPDEs

   • Petru A. Cioica, Stephan Dahlke, Nicolas Döhring, Stefan Kinzel, Felix Lindner, Thorsten Raasch et al.

   Pages 83-107

7. Constructive Quantization and Multilevel Algorithms for Quadrature of Stochastic Differential Equations

   • Martin Altmayer, Steffen Dereich, Sangmeng Li, Thomas Müller-Gronbach, Andreas Neuenkirch, Klaus Ritter et al.

   Pages 109-132

8. Bayesian Inverse Problems and Kalman Filters

   • Oliver G. Ernst, Björn Sprungk, Hans-Jörg Starkloff

   Pages 133-159

9. Robustness in Stochastic Filtering and Maximum Likelihood Estimation for SDEs

   • Joscha Diehl, Peter K. Friz, Hilmar Mai, Harald Oberhauser, Sebastian Riedel, Wilhelm Stannat

   Pages 161-178

10. Adaptive Sparse Grids in Reinforcement Learning

    • Jochen Garcke, Irene Klompmaker

    Pages 179-194

11. A Review on Adaptive Low-Rank Approximation Techniques in the Hierarchical Tensor Format

    • Jonas Ballani, Lars Grasedyck, Melanie Kluge

    Pages 195-210

12. A Bond Order Dissection ANOVA Approach for Efficient Electronic Structure Calculations

    • Michael Griebel, Jan Hamaekers, Frederik Heber

    Pages 211-235

13. Tensor Spaces and Hierarchical Tensor Representations

    • Wolfgang Hackbusch, Reinhold Schneider

    Pages 237-261

14. Nonlinear Eigenproblems in Data Analysis: Balanced Graph Cuts and the RatioDCA-Prox

    • Leonardo Jost, Simon Setzer, Matthias Hein

    Pages 263-279

15. Adaptive Approximation Algorithms for Sparse Data Representation

    • Mijail Guillemard, Dennis Heinen, Armin Iske, Sara Krause-Solberg, Gerlind Plonka

    Pages 281-302

16. Error Bound for Hybrid Models of Two-Scaled Stochastic Reaction Systems

    • Tobias Jahnke, Vikram Sunkara

    Pages 303-319

17. Valuation of Structured Financial Products by Adaptive Multiwavelet Methods in High Dimensions

    • Rüdiger Kiesel, Andreas Rupp, Karsten Urban

    Pages 321-345

18. Computational Methods for the Fourier Analysis of Sparse High-Dimensional Functions

    • Lutz Kämmerer, Stefan Kunis, Ines Melzer, Daniel Potts, Toni Volkmer

    Pages 347-363

19. Sparsity and Compressed Sensing in Inverse Problems

    • Evelyn Herrholz, Dirk Lorenz, Gerd Teschke, Dennis Trede

    Pages 365-379

20. Low-Rank Dynamics

    • Christian Lubich

    Pages 381-396

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.

• complex systems
• computational stochastics
• high-dimensional problems
• numerical analysis
• partial differential equations

• FB 12 Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

  Stephan Dahlke

• Inst. Geometrie & Praktische Mathemathik, RWTH Aachen University, Aachen, Germany

  Wolfgang Dahmen

• Universität Bonn Institut für Numerische Simulation, Bonn, Germany

  Michael Griebel

• Max-Planck-Inst. f. Mathem. in d. Naturwissenschaften, Leipzig, Germany

  Wolfgang Hackbusch

• Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

  Klaus Ritter

• Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

  Reinhold Schneider, Harry Yserentant

• Seminar für Angewandte Mathematik, ETH Zürich, Zürich, Switzerland

  Christoph Schwab

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