Convergence of computational homogenization methods based on the fast Fourier transform
Tuesday, 6.2.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Since their inception in the mid 1990s, computational methods based on the fast Fourier transform (FFT) have been established as efficient and powerful tools for computing the effective mechanical properties of composite materials. These methods operate on a regular grid, employ periodic boundary conditions for the displacement fluctuation and utilize the FFT to design matrix-free iterative schemes whose iteration count is (most often) bounded independently of the grid spacing.\nIn the talk at hand, we will take a look both at the convergence of the used iterative schemes and the convergence of the underlying spectral discretization. Remarkably, despite the presence of discontinuous coefficients, the spectral discretization enjoys the same convergence rate as a finite-element discretization on a regular grid. Moreover, the convergence behavior of the effective stresses profit from a superconvergence phenomenon apparently inherent to computational homogenization problems.\n---------------------------------------------------------
On strong approximation of SDEs with a discontinuous drift coefficient
Tuesday, 13.2.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The classical assumption in the literature on numerical approximation of stochastic differential equations (SDEs) is global Lipschitz continuity of the coefficients of the equation.\nHowever, many SDEs arising in applications fail to have globally Lipschitz continuous coefficients.\nIn the last decade an intensive study of numerical approximation of SDEs with nonglobally Lipschitz continuous coefficients has begun. In particular, strong approximation\nof SDEs with a drift coefficient that is discontinuous in space has recently gained a lot of\ninterest. Such SDEs arise e.g. in mathematical finance, insurance and stochastic control\nproblems. Classical techniques of error analysis are not applicable to such SDEs and well\nknown convergence results for standard methods do not carry over in general.\nIn this talk I will present recent results on strong approximation of such SDEs.\nThe talk is based on joint work with Thomas M¨uller-Gronbach (University of Passau)