8,010 research outputs found
Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
In this paper, we discuss a general multiscale model reduction framework
based on multiscale finite element methods. We give a brief overview of related
multiscale methods. Due to page limitations, the overview focuses on a few
related methods and is not intended to be comprehensive. We present a general
adaptive multiscale model reduction framework, the Generalized Multiscale
Finite Element Method. Besides the method's basic outline, we discuss some
important ingredients needed for the method's success. We also discuss several
applications. The proposed method allows performing local model reduction in
the presence of high contrast and no scale separation
Polynomial-Sized Topological Approximations Using The Permutahedron
Classical methods to model topological properties of point clouds, such as
the Vietoris-Rips complex, suffer from the combinatorial explosion of complex
sizes. We propose a novel technique to approximate a multi-scale filtration of
the Rips complex with improved bounds for size: precisely, for points in
, we obtain a -approximation with at most simplices of dimension or lower. In conjunction with dimension
reduction techniques, our approach yields a -approximation of size for Rips filtrations on arbitrary metric
spaces. This result stems from high-dimensional lattice geometry and exploits
properties of the permutahedral lattice, a well-studied structure in discrete
geometry.
Building on the same geometric concept, we also present a lower bound result
on the size of an approximate filtration: we construct a point set for which
every -approximation of the \v{C}ech filtration has to contain
features, provided that for .Comment: 24 pages, 1 figur
Introduction to Regularity Structures
These are short notes from a series of lectures given at the University of
Rennes in June 2013, at the University of Bonn in July 2013, at the XVIIth
Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich
in September 2013. We give a concise overview of the theory of regularity
structures as exposed in Hairer (2014). In order to allow to focus on the
conceptual aspects of the theory, many proofs are omitted and statements are
simplified. We focus on applying the theory to the problem of giving a solution
theory to the stochastic quantisation equations for the Euclidean
quantum field theory.Comment: 33 page
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