SPP 1324:  Extraction of Quantifiable Information from Complex Systems

• Subject Area: Mathematics
• Term: from 2008 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 42881441

Mathematical models of complex systems form the foundation for further technological developments in science, engineering and computational finance; particularly important but also particularly challenging problems arise in connection with high-dimensional parameter spaces.
Motivated by the continuously increasing computer power ever more realistic models have been developed in recent years. These models have also become increasingly complex, and their numerical treatment is facing serious challenges.
To overcome these difficulties, a possible approach would be to start with the application at hand and to concentrate on this particular case. However, within this Priority Programme, we propose a different strategy. Recent developments in mathematics suggest that in the long run, much more powerful solution strategies can be derived if the interconnections between the different fields of research are systematically exploited at a conceptual level. In order to do so, a deeper understanding of the mathematical foundations as well as the development of new and efficient numerical algorithms will be the main goals of this Priority Programme.
The treatment of high-dimensional systems is clearly one of the most challenging tasks in applied mathematics. Since the problem of high-dimensionality appears in many fields of applications, the above mentioned synergy and cross-fertilisation effects are very likely to make a great impact. To be really successful, the following issues have to be kept in mind: theoretical research and practical applications must be developed hand in hand; moreover, it is necessary to combine different fields of mathematics, such as numerical analysis and stochastics. To keep the whole programme sufficiently focussed, we will concentrate on specific, but related fields of applications such as data mining, high-dimensional problems in physics, computational finance and differential equations with random coefficients. All these applications share common characteristics and, therefore, they allow for closely related approaches.

• DFG Programme: Priority Programmes
• International Connection: Netherlands, Russia, Switzerland, United Kingdom

Projects

• Adaptive Approximation Algorithms for Sparse Data Representation (Applicants Iske, Armin ;  Plonka-Hoch, Gerlind )
• Adaptive Hierarchical Low Rank Formats of High-dimensional Tensors with Applications in PDEs with Stochastic Parameters (Applicant Grasedyck, Lars )
• Adaptive wavelet frame methods for operator equations: Sparse grids, vector-valued spaces and applications to nonlinear inverse parabolic problems (Applicants Dahlke, Stephan ;  Maaß, Peter )
• Adaptive Wavelet Methods for SPDEs (Applicants Dahlke, Stephan ;  Ritter, Klaus ;  Schilling, René Leander )
• Adaptive Wavelet Methods for Structured Financial Products (Applicant Urban, Karsten )
• Application of rough path theory for filtering and numerical integration methods (Applicant Friz, Peter Karl )
• Constructive Quantization and Multilevel Algorithms for Quadrature of SDEs (Applicants Dereich, Steffen ;  Müller-Gronbach, Thomas ;  Neuenkirch, Andreas )
• Efficient and reliable numerical methods for energy markets (Applicants Kiesel, Rüdiger ;  Urban, Karsten )
• Fast computation of expectations and integrals by Markov chain Monte Carlo methods; error bounds and burn-in (Applicant Novak, Erich )
• High-dimensional stochastic differential equations under sparsity constraints (Applicant Rohde, Angelika )
• Koordination des Schwerpunktprogramms "Mathematische Methoden zur Extraktion quantifizierbarer Information aus komplexen Systemen" (Applicant Dahlke, Stephan )
• Lower-dimensional principal manifold learning in higher-dimensional data spaces by sparse grid methods (Applicant Griebel, Michael )
• Nonlinear eigenproblems for high-dimensional data analysis (Applicant Hein, Matthias )
• Numerical and harmonic analysis of problems with anisotropic features, directional representation systems and the solution of transport dominated problems, in particular, for parameter dependent high dimensional versions (Applicants Dahmen, Wolfgang ;  Kutyniok, Gitta )
• Numerical methods for high-dimensional stochastic reaction networks (Applicant Jahnke, Tobias )
• Numerical methods in quantum dynamics (Applicant Lubich, Christian )
• Optimal approximation of tensor products of linear operators (Applicant Sickel, Winfried )
• Regularity, complexity, and approximability of electronic wavefunctions (Applicant Yserentant, Harry )
• Reinforcement learing in a continuous state space (Applicant Garcke, Jochen )
• Solving optimal stopping problems and reflected backward stochastic differential equations by convex optimization and penalization (Applicants Belomestny, Denis ;  Bender, Christian )
• Sparse Fast Fourier Transforms (Applicants Kunis, Stefan ;  Potts, Daniel )
• Sparsity and compressed sensing in inverse problems (Applicants Lorenz, Dirk A. ;  Teschke, Gerd )
• Stochastic Galerkin Methods: Fundamentals and Algorithms (Applicants Ernst, Ph.D., Oliver ;  Starkloff, Hans-Jörg )
• Tensor methods in multi-dimensional spectral problems with particular application in electronic structure calculations (Applicants Hackbusch, Wolfgang ;  Schneider, Reinhold )
• The linear algebra of compressed sensing, with applications to PDEs (Applicant Holtz, Olga )
• Validating numerical solutions of high-dimensional backward SDEs arising from finance (Applicants Bender, Christian ;  Bollhöfer, Matthias )

• Spokesperson: Professor Dr. Stephan Dahlke