2,718 research outputs found
On the numerical stability of Fourier extensions
An effective means to approximate an analytic, nonperiodic function on a
bounded interval is by using a Fourier series on a larger domain. When
constructed appropriately, this so-called Fourier extension is known to
converge geometrically fast in the truncation parameter. Unfortunately,
computing a Fourier extension requires solving an ill-conditioned linear
system, and hence one might expect such rapid convergence to be destroyed when
carrying out computations in finite precision. The purpose of this paper is to
show that this is not the case. Specifically, we show that Fourier extensions
are actually numerically stable when implemented in finite arithmetic, and
achieve a convergence rate that is at least superalgebraic. Thus, in this
instance, ill-conditioning of the linear system does not prohibit a good
approximation.
In the second part of this paper we consider the issue of computing Fourier
extensions from equispaced data. A result of Platte, Trefethen & Kuijlaars
states that no method for this problem can be both numerically stable and
exponentially convergent. We explain how Fourier extensions relate to this
theoretical barrier, and demonstrate that they are particularly well suited for
this problem: namely, they obtain at least superalgebraic convergence in a
numerically stable manner
Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems
We present a method for approximating the solution of a parameterized,
nonlinear dynamical system using an affine combination of solutions computed at
other points in the input parameter space. The coefficients of the affine
combination are computed with a nonlinear least squares procedure that
minimizes the residual of the governing equations. The approximation properties
of this residual minimizing scheme are comparable to existing reduced basis and
POD-Galerkin model reduction methods, but its implementation requires only
independent evaluations of the nonlinear forcing function. It is particularly
appropriate when one wishes to approximate the states at a few points in time
without time marching from the initial conditions. We prove some interesting
characteristics of the scheme including an interpolatory property, and we
present heuristics for mitigating the effects of the ill-conditioning and
reducing the overall cost of the method. We apply the method to representative
numerical examples from kinetics - a three state system with one parameter
controlling the stiffness - and conductive heat transfer - a nonlinear
parabolic PDE with a random field model for the thermal conductivity.Comment: 28 pages, 8 figures, 2 table
An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons
In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included
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