38,719 research outputs found
Multiscale Finite Element Methods for Nonlinear Problems and their Applications
In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities
Optimal Local Multi-scale Basis Functions for Linear Elliptic Equations with Rough Coefficient
This paper addresses a multi-scale finite element method for second order
linear elliptic equations with arbitrarily rough coefficient. We propose a
local oversampling method to construct basis functions that have optimal local
approximation property. Our methodology is based on the compactness of the
solution operator restricted on local regions of the spatial domain, and does
not depend on any scale-separation or periodicity assumption of the
coefficient. We focus on a special type of basis functions that are harmonic on
each element and have optimal approximation property. We first reduce our
problem to approximating the trace of the solution space on each edge of the
underlying mesh, and then achieve this goal through the singular value
decomposition of an oversampling operator. Rigorous error estimates can be
obtained through thresholding in constructing the basis functions. Numerical
results for several problems with multiple spatial scales and high contrast
inclusions are presented to demonstrate the compactness of the local solution
space and the capacity of our method in identifying and exploiting this compact
structure to achieve computational savings
q-deformed Supersymmetric t-J Model with a Boundary
The q-deformed supersymmetric t-J model on a semi-infinite lattice is
diagonalized by using the level-one vertex operators of the quantum affine
superalgebra . We give the bosonization of the boundary
states. We give an integral expression of the correlation functions of the
boundary model, and derive the difference equations which they satisfy.Comment: LaTex file 18 page
Self-similar Singularity of a 1D Model for the 3D Axisymmetric Euler Equations
We investigate the self-similar singularity of a 1D model for the 3D
axisymmetric Euler equations, which is motivated by a particular singularity
formation scenario observed in numerical computation. We prove the existence of
a discrete family of self-similar profiles for this model and analyze their
far-field properties. The self-similar profiles we find agree with direct
simulation of the model and seem to have some stability
Data-Driven Time-Frequency Analysis
In this paper, we introduce a new adaptive data analysis method to study
trend and instantaneous frequency of nonlinear and non-stationary data. This
method is inspired by the Empirical Mode Decomposition method (EMD) and the
recently developed compressed (compressive) sensing theory. The main idea is to
look for the sparsest representation of multiscale data within the largest
possible dictionary consisting of intrinsic mode functions of the form , where , consists of the
functions smoother than and . This problem can
be formulated as a nonlinear optimization problem. In order to solve this
optimization problem, we propose a nonlinear matching pursuit method by
generalizing the classical matching pursuit for the optimization problem.
One important advantage of this nonlinear matching pursuit method is it can be
implemented very efficiently and is very stable to noise. Further, we provide a
convergence analysis of our nonlinear matching pursuit method under certain
scale separation assumptions. Extensive numerical examples will be given to
demonstrate the robustness of our method and comparison will be made with the
EMD/EEMD method. We also apply our method to study data without scale
separation, data with intra-wave frequency modulation, and data with incomplete
or under-sampled data
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