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Explore advanced mathematical concepts through this comprehensive lecture series from the Hausdorff Research Institute for Mathematics' trimester program on multiscale problems. Delve into cutting-edge research presentations covering algorithms, numerical analysis, and computational methods for complex multiscale phenomena. Learn about mesoscale crystal plasticity through continuum dislocation dynamics, nonlocal theories for crack propagation in brittle materials, and finite deformation constitutive models in peridynamic mixtures. Examine stochastic homogenization, adaptive finite element methods, and convergence theory for numerical solutions of partial differential equations. Discover innovative approaches to strong advection problems, QTT FEM solvers for elliptic multiscale problems, and numerical homogenization techniques for fast multiscale PDE solvers. Study fracture-to-damage multiscale mechanics, coupling strategies for nonlocal and local models, and large random complex networks. Investigate localization techniques for multiscale problems, adaptive wavelet methods, and semi-Lagrangian approaches for the Monge-Ampère equation on unstructured grids. Explore two-scale finite element methods for non-variational elliptic PDEs, anisotropic linearized peridynamic models, and homogenization methods for weakly compressible elastic materials. Gain insights into perturbation-method-based post-processing of planewave approximations, variational multiscale stabilization for convection problems, and bilevel learning approaches in variational image processing. Master spectral upscaling for graph Laplacian problems, localized radial basis functions for pseudo-spectral methods, and asymptotically compatible discretization techniques for nonlocal models through quadrature-based approaches.
