Achievements of Research Area J

High-dimensional problems.

A famous theorem by Kolmogorov states that any high-dimensional function can be represented as a superposition of one-dimensional functions. While the previous proofs were non-constructive, Griebel and Braun gave in [J:BG09] the first constructive proof, which promises a direct usability in numerical algorithms. Griebel and Wozniakowski showed in [J:GW06] that for special function classes there exists no curse of dimension at all. Such functions have widespread applications in economy and finance and researchers from the Hoover Institute, Stanford, USA frequently refer to these results. The concept of sparse grids has been substantially advanced by Griebel and coworkers and is nowadays widely accepted as one of the key technologies in high dimensional data approximation. Griebel and Knapek showed in [J:GK09] under which prerequisites on the error norms and smoothness assumptions the curse of dimensionality can be completely avoided, and they constructed corresponding optimal sparse grid discretisations for these function spaces. Based on the regularity theory of Yserentant, which shows that the solution of the electronic Schrödinger equation exhibits bounded mixed derivatives (depending on the spins either of order 1/2 or even of order 1), Griebel and Hamaekers developed an efficient adaptive sparse grid method for the direct numerical treatment of the electronic Schrödinger equation [J:GH06, J:GH07, J:Ham09]. This way, it is now possible to run accurate numerical simulations for atoms up to lithium. For the efficient treatment of high-dimensional integrals, a new dimension-adaptive sparse grid quadrature method was developed which is based on the so-called anchor ANOVA  decomposition. Moreover, Gerstner developed blockwise sparse grid quadrature methods. They enable the efficient treatment of complex high-dimensional tasks in option pricing and asset liability management in finance and insurance [J:GH10b]. HCM-Postdoc Rieger derived sampling inequalities for high-dimensional data approximation and Bonn Junior Fellow Chernov studied quadrature rules and optimal convergence estimates for the trace of the L2-polynomial projection operator in high dimensions and the numerical solution of pseudo-differential equations on the sphere using spherical splines [J:Che, J:CvPS11, J:CH11]. Recently, Eberle derived new Lp estimates and Lp=Lq bounds for a class of  time-dependent Feynman-Kac propagators [J:EM10a]. In contrast to previously known bounds, these results lead to tractable estimates in high dimensions due to the factorisation properties of the constants in the applied logarithmic Sobolev and generalised Poincaré estimates. In the rapidly developing field of compressed sensing, one asks for sparse measurement strategies to recover complex signals at high precision. In [J:RRT] Bonn Junior Fellow Rauhut derived the so far best known bounds on the minimal amount of measurements for sparse recovery  using partial random circulant matrices. The proofs use concentration of measure inequalities as a crucial tool. Furthermore, he showed that certain functions can be reconstructed via compressive sensing with small error from a number of samples that scales only logarithmically in the spatial dimension.

Multiscale methods.

Many models in the sciences exhibit effects on different scales and their interaction. A particular challenge for both the analytical treatment and the efficient numerical simulation are multiscale problems with models of different mathematical type on the different scales: discrete–continuous models, quantum–continuum mechanics, diffusive–sharp interface propagation.
Schlein worked on the rigorous derivation of effective evolution equations for the dynamics of a certain class of quantum systems. In particular, in a series of joint papers with Erdös and Yau, a mathematically rigorous derivation was given for the Gross-Pitaevskii equation, which describes the time evolution of initially trapped Bose-Einstein condensates [J:ES10, J:ESY09]. Krause developed novel multiscale methods for friction and contact and related domain decomposition techniques with applications to biomechanics. Griebel, Scherer, and Schweitzer derived a new theory for an additive multilevel method tailored to the robust treatment of perturbed diffusion problems [J:GSS07]. Furthermore, particle-based multilevel methods were investigated by Schweitzer. Bartels explored new numerical multilevel techniques for phasefield problems. A new scaling law for the principal eigenvalue of the linearised higher order evolution operator near generic singularities led to robust estimates and cover the case of nonsmooth potentials relevant in the deep quench regime for temperatures far below the critical one [J:BM11, J:BMO11]. Multiphase flow and tailored fast multi-grid solvers were studied by HCM-Postdoc Gross.
Bebendorf developed so-called adaptive cross approximations for the efficient treatment of systems of linear PDEs on microstructured geometries [J:Beb08, J:Beb].