363 research outputs found
Sparse spectral methods for integral equations and equilibrium measures
In this thesis, we introduce new numerical approaches to two important types of integral equation problems using sparse spectral methods. First, linear as well as nonlinear Volterra integral and integro-differential equations and second, power-law integral equations on d-dimensional balls involved in the solution of equilibrium measure problems.
These methods are based on ultraspherical spectral methods and share key properties and advantages as a result of their joint starting point: By working in appropriately weighted orthogonal Jacobi polynomial bases, we obtain recursively generated banded operators allowing us to obtain high precision solutions at low computational cost.
This thesis consists of three chapters in which the background of the above-mentioned problems and methods are respectively introduced in the context of their mathematical theory and applications, the necessary results to construct the operators and obtain solutions are proved and the method's applicability and efficiency are showcased by comparing them with current state-of-the-art approaches and analytic results where available. The first chapter gives a general scope introduction to sparse spectral methods using Jacobi polynomials in one and higher dimensions. The second chapter concerns the numerical solution of Volterra integral equations. The introduced method achieves exponential convergence and works for general kernels, a major advantage over comparable methods which are limited to convolution kernels.
The third chapter introduces an approximately banded method to solve power law kernel equilibrium measures in arbitrary dimensional balls. This choice of domain is suggested by the radial symmetry of the problem and analytic results on the supports of the resulting measures. For our method, we obtain the crucial property of computational cost independent of the dimension of the domain, a major contrast to particle simulations which are the current standard approach to these problems and scale extremely poorly with both the dimension and the number of particles.Open Acces
Computation of power law equilibrium measures on balls of arbitrary dimension
We present a numerical approach for computing attractive-repulsive power law equilibrium measures in arbitrary dimension. We prove new recurrence relationships for radial Jacobi polynomials on d-dimensional ball domains, providing a substantial generalization of the work started in Gutleb et al. (Math Comput 9:2247â2281, 2022) for the one-dimensional case based on recurrence relationships of Riesz potentials on arbitrary dimensional balls. Among the attractive features of the numerical method are good efficiency due to recursively generated banded and approximately banded Riesz potential operators and computational complexity independent of the dimension d, in stark constrast to the widely used particle swarm simulation approaches for these problems which scale catastrophically with the dimension. We present several numerical experiments to showcase the accuracy and applicability of the method and discuss how our method compares with alternative numerical approaches and conjectured analytical solutions which exist for certain special cases. Finally, we discuss how our method can be used to explore the analytically poorly understood gap formation boundary to spherical shell support
A static memory sparse spectral method for time-fractional PDEs in arbitrary dimensions
We introduce a method which provides accurate numerical solutions to
fractional-in-time partial differential equations posed on with without the excessive memory
requirements associated with the nonlocal fractional derivative operator
operator. Our approach combines recent advances in the development and
utilization of multivariate sparse spectral methods as well as fast methods for
the computation of Gauss quadrature nodes with recursive non-classical methods
for the Caputo fractional derivative of general fractional order .
An attractive feature of the method is that it has minimal theoretical overhead
when using it on any domain on which an orthogonal polynomial basis is
already available. We discuss the memory requirements of the method, present
several numerical experiments demonstrating the method's performance in solving
time-fractional PDEs on intervals, triangles and disks and derive error bounds
which suggest sensible convergence strategies. As an important model problem
for this approach we consider a type of wave equation with time-fractional
dampening related to acoustic waves in viscoelastic media with applications in
the physics of medical ultrasound and outline future research steps required to
use such methods for the reverse problem of image reconstruction from sensor
data.Comment: 28 pages, 13 figure
Explicit fractional Laplacians and Riesz potentials of classical functions
We prove and collect numerous explicit and computable results for the
fractional Laplacian with as well as its whole space
inverse, the Riesz potential, with
. Choices of include weighted classical
orthogonal polynomials such as the Legendre, Chebyshev, Jacobi, Laguerre and
Hermite polynomials, or first and second kind Bessel functions with or without
sinusoid weights. Some higher dimensional fractional Laplacians and Riesz
potentials of generalized Zernike polynomials on the unit ball and its
complement as well as whole space generalized Laguerre polynomials are also
discussed. The aim of this paper is to aid in the continued development of
numerical methods for problems involving the fractional Laplacian or the Riesz
potential in bounded and unbounded domains -- both directly by providing useful
basis or frame functions for spectral method approaches and indirectly by
providing accessible ways to construct computable toy problems on which to test
new numerical methods.Comment: 37 pages, 7 tables, 2 figure
Atomic Cluster Expansion without Self-Interaction
The Atomic Cluster Expansion (ACE) (Drautz, Phys. Rev. B 99, 2019) has been
widely applied in high energy physics, quantum mechanics and atomistic modeling
to construct many-body interaction models respecting physical symmetries.
Computational efficiency is achieved by allowing non-physical self-interaction
terms in the model. We propose and analyze an efficient method to evaluate and
parameterize an orthogonal, or, non-self-interacting cluster expansion model.
We present numerical experiments demonstrating improved conditioning and more
robust approximation properties than the original expansion in regression tasks
both in simplified toy problems and in applications in the machine learning of
interatomic potentials.Comment: Typo fix and minor changes in wording in v
Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations
We describe fast algorithms for approximating the connection coefficients
between a family of orthogonal polynomials and another family with a
polynomially or rationally modified measure. The connection coefficients are
computed via infinite-dimensional banded matrix factorizations and may be used
to compute the modified Jacobi matrices all in linear complexity with respect
to the truncation degree. A family of orthogonal polynomials with modified
classical weights is constructed that support banded differentiation matrices,
enabling sparse spectral methods with modified classical orthogonal
polynomials
Nanoparticles in the environment: assessment using the causal diagram approach
Nanoparticles (NPs) cause concern for health and safety as their impact on the environment and humans is not known. Relatively few studies have investigated the toxicological and environmental effects of exposure to naturally occurring NPs (NNPs) and man-made or engineered NPs (ENPs) that are known to have a wide variety of effects once taken up into an organism
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