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Learn about a randomized Greedy algorithm designed for constructing and certifying local approximation spaces in multiscale partial differential equations (PDEs) through this 41-minute mathematical lecture. Discover how this innovative algorithm employs strategic random sampling of local problem data—including boundary conditions, source functions, and PDE coefficients—from carefully designed probability distributions to build effective training sets at each iteration. Explore the theoretical foundations that enable this algorithm to provide high-probability certification across entire parameter sets, drawing on sampling discretization theory and concentration of measure phenomena. Examine the algorithm's favorable sampling complexity properties that potentially overcome the curse of dimensionality that challenges deterministic Greedy algorithms in training set selection. Review numerical results demonstrating the algorithm's effectiveness in building reduced approximation spaces for benchmark PDE problems, and investigate its application to constructing local approximation spaces for solutions to the p-Laplace equation—a nonlinear PDE—through sampling of local boundary conditions.
