62,555 research outputs found
A boundary integral formalism for stochastic ray tracing in billiards
Determining the flow of rays or non-interacting particles driven by a force or velocity field is fundamental to modelling many physical processes. These include particle flows arising in fluid mechanics and ray flows arising in the geometrical optics limit of linear wave equations. In many practical applications, the driving field is not known exactly and the dynamics are determined only up to a degree of uncertainty. This paper presents a boundary integral framework for propagating flows including uncertainties, which is shown to systematically interpolate between a deterministic and a completely random description of the trajectory propagation. A simple but efficient discretisation approach is applied to model uncertain billiard dynamics in an integrable rectangular domain
An introduction to Multitrace Formulations and Associated Domain Decomposition Solvers
Multitrace formulations (MTFs) are based on a decomposition of the problem
domain into subdomains, and thus domain decomposition solvers are of interest.
The fully rigorous mathematical MTF can however be daunting for the
non-specialist. We introduce in this paper MTFs on a simple model problem using
concepts familiar to researchers in domain decomposition. This allows us to get
a new understanding of MTFs and a natural block Jacobi iteration, for which we
determine optimal relaxation parameters. We then show how iterative multitrace
formulation solvers are related to a well known domain decomposition method
called optimal Schwarz method: a method which used Dirichlet to Neumann maps in
the transmission condition. We finally show that the insight gained from the
simple model problem leads to remarkable identities for Calderon projectors and
related operators, and the convergence results and optimal choice of the
relaxation parameter we obtained is independent of the geometry, the space
dimension of the problem{\color{black}, and the precise form of the spatial
elliptic operator, like for optimal Schwarz methods. We illustrate our analysis
with numerical experiments
Smolyak's algorithm: A powerful black box for the acceleration of scientific computations
We provide a general discussion of Smolyak's algorithm for the acceleration
of scientific computations. The algorithm first appeared in Smolyak's work on
multidimensional integration and interpolation. Since then, it has been
generalized in multiple directions and has been associated with the keywords:
sparse grids, hyperbolic cross approximation, combination technique, and
multilevel methods. Variants of Smolyak's algorithm have been employed in the
computation of high-dimensional integrals in finance, chemistry, and physics,
in the numerical solution of partial and stochastic differential equations, and
in uncertainty quantification. Motivated by this broad and ever-increasing
range of applications, we describe a general framework that summarizes
fundamental results and assumptions in a concise application-independent
manner
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
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