60 research outputs found
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Numerical multiscale methods: from homogenization to milestoning
The dissertation focuses on addressing the challenges posed by multiscale problems in applied mathematics, which stem from the intricate interplay between microscales and the computational demands of resolving fine details. To alleviate this burden, numerical homogenization and averaging methods are favored. This study explores three interconnected topics related to numerical techniques for handling multiscale problems in both spatial and temporal domains. In the first part, we establish the equivalence principle between time averaging and space homogenization. This principle facilitates the application of various numerical averaging techniques, such as FLAVORS, Seamless, and HMM, to boundary value problems. Moreover, we introduce the dilation operator as a decomposition-free approach for numerical homogenization in higher dimensions. Additionally, we utilize the Synchrosqueezing transform as a preprocessing step to extract oscillatory components, crucial for the structure-aware dilation method. The second part extends the Deep Ritz method to multiscale problems. We delve into the scale convergence theory to derive the [Gamma]-limit of energy functionals exhibiting oscillatory behavior. The resulting limit object, formulated as a minimization problem, captures spatial oscillations and can be tackled using existing neural network architectures. In the third part, we lay the groundwork for the milestoning algorithm, a successful tool in computational chemistry for molecular dynamics simulations. We adapt this algorithm to a domain-decomposition-based framework for coarse-grained descriptions and establish the well-posedness of primal and dual PDEs. Additionally, we investigate the convergence rate and optimal milestone placements. We illustrate this framework through the understanding of the Forward Flux algorithm as a specific example.Mathematic
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
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