17,050 research outputs found
A mixed regularization approach for sparse simultaneous approximation of parameterized PDEs
We present and analyze a novel sparse polynomial technique for the
simultaneous approximation of parameterized partial differential equations
(PDEs) with deterministic and stochastic inputs. Our approach treats the
numerical solution as a jointly sparse reconstruction problem through the
reformulation of the standard basis pursuit denoising, where the set of jointly
sparse vectors is infinite. To achieve global reconstruction of sparse
solutions to parameterized elliptic PDEs over both physical and parametric
domains, we combine the standard measurement scheme developed for compressed
sensing in the context of bounded orthonormal systems with a novel mixed-norm
based regularization method that exploits both energy and sparsity. In
addition, we are able to prove that, with minimal sample complexity, error
estimates comparable to the best -term and quasi-optimal approximations are
achievable, while requiring only a priori bounds on polynomial truncation error
with respect to the energy norm. Finally, we perform extensive numerical
experiments on several high-dimensional parameterized elliptic PDE models to
demonstrate the superior recovery properties of the proposed approach.Comment: 23 pages, 4 figure
Approximation systems
We introduce the notion of an approximation system as a generalization of
Taylor approximation, and we give some first examples. Next we develop the
general theory, including error bounds and a sufficient criterion for
convergence. More examples follow. We conclude the article with a description
of numerical implementation and directions for future research. Prerequisites
are mostly elementary complex analysis.Comment: 27 pages; in v3 minor change
Small Winding-Number Expansion: Vortex Solutions at Critical Coupling
We study an axially symmetric solution of a vortex in the Abelian-Higgs model
at critical coupling in detail. Here we propose a new idea for a perturbative
expansion of a solution, where the winding number of a vortex is naturally
extended to be a real number and the solution is expanded with respect to it
around its origin. We test this idea on three typical constants contained in
the solution and confirm that this expansion works well with the help of the
Pad\'e approximation. For instance, we analytically reproduce the value of the
scalar charge of the vortex with an error of . This expansion is
also powerful even for large winding numbers.Comment: 38 pages,48 figure
On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations
We describe a method for calculating the roots of special functions
satisfying second order linear ordinary differential equations. It exploits the
recent observation that the solutions of a large class of such equations can be
represented via nonoscillatory phase functions, even in the high-frequency
regime. Our algorithm achieves near machine precision accuracy and the time
required to compute one root of a solution is independent of the frequency of
oscillations of that solution. Moreover, despite its great generality, our
approach is competitive with specialized, state-of-the-art methods for the
construction of Gaussian quadrature rules of large orders when it used in such
a capacity. The performance of the scheme is illustrated with several numerical
experiments and a Fortran implementation of our algorithm is available at the
author's website
Invariant Discretization Schemes Using Evolution-Projection Techniques
Finite difference discretization schemes preserving a subgroup of the maximal
Lie invariance group of the one-dimensional linear heat equation are
determined. These invariant schemes are constructed using the invariantization
procedure for non-invariant schemes of the heat equation in computational
coordinates. We propose a new methodology for handling moving discretization
grids which are generally indispensable for invariant numerical schemes. The
idea is to use the invariant grid equation, which determines the locations of
the grid point at the next time level only for a single integration step and
then to project the obtained solution to the regular grid using invariant
interpolation schemes. This guarantees that the scheme is invariant and allows
one to work on the simpler stationary grids. The discretization errors of the
invariant schemes are established and their convergence rates are estimated.
Numerical tests are carried out to shed some light on the numerical properties
of invariant discretization schemes using the proposed evolution-projection
strategy
Wave polynomials, transmutations and Cauchy's problem for the Klein-Gordon equation
We prove a completeness result for a class of polynomial solutions of the
wave equation called wave polynomials and construct generalized wave
polynomials, solutions of the Klein-Gordon equation with a variable
coefficient. Using the transmutation (transformation) operators and their
recently discovered mapping properties we prove the completeness of the
generalized wave polynomials and use them for an explicit construction of the
solution of the Cauchy problem for the Klein-Gordon equation. Based on this
result we develop a numerical method for solving the Cauchy problem and test
its performance.Comment: 31 pages, 8 figures (16 graphs
Algebraic structure of stochastic expansions and efficient simulation
We investigate the algebraic structure underlying the stochastic Taylor
solution expansion for stochastic differential systems.Our motivation is to
construct efficient integrators. These are approximations that generate strong
numerical integration schemes that are more accurate than the corresponding
stochastic Taylor approximation, independent of the governing vector fields and
to all orders. The sinhlog integrator introduced by Malham & Wiese (2009) is
one example. Herein we: show that the natural context to study stochastic
integrators and their properties is the convolution shuffle algebra of
endomorphisms; establish a new whole class of efficient integrators; and then
prove that, within this class, the sinhlog integrator generates the optimal
efficient stochastic integrator at all orders.Comment: 19 page
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