10,804 research outputs found
On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients
In this paper, we propose a novel non-standard Local Fourier Analysis (LFA)
variant for accurately predicting the multigrid convergence of problems with
random and jumping coefficients. This LFA method is based on a specific basis
of the Fourier space rather than the commonly used Fourier modes. To show the
utility of this analysis, we consider, as an example, a simple cell-centered
multigrid method for solving a steady-state single phase flow problem in a
random porous medium. We successfully demonstrate the prediction capability of
the proposed LFA using a number of challenging benchmark problems. The
information provided by this analysis helps us to estimate a-priori the time
needed for solving certain uncertainty quantification problems by means of a
multigrid multilevel Monte Carlo method
Fast reliable interrogation of procedurally defined implicit surfaces using extended revised affine arithmetic.
Techniques based on interval and previous termaffine arithmetic next term and their modifications are shown to provide previous term reliable next term function range evaluation for the purposes of previous termsurface interrogation.next term In this paper we present a technique for the previous termreliable interrogation of implicit surfacesnext term using a modification of previous termaffine arithmeticnext term called previous term revised affine arithmetic.next term We extend the range of functions presented in previous termrevised affine arithmeticnext term by introducing previous termaffinenext term operations for arbitrary functions such as set-theoretic operations with R-functions, blending and conditional operators. The obtained previous termaffinenext term forms of arbitrary functions provide previous termfasternext term and tighter function range evaluation. Several case studies for operations using previous termaffinenext term forms are presented. The proposed techniques for previous termsurface interrogationnext term are tested using ray-previous termsurfacenext term intersection for ray-tracing and spatial cell enumeration for polygonisation. These applications with our extensions provide previous termfast and reliablenext term rendering of a wide range of arbitrary previous termprocedurally defined implicit surfacesnext term (including polynomial previous termsurfaces,next term constructive solids, pseudo-random objects, previous termprocedurally definednext term microstructures, and others). We compare the function range evaluation technique based on previous termextended revised affine arithmeticnext term with other previous termreliablenext term techniques based on interval and previous termaffine arithmeticnext term to show that our technique provides the previous termfastestnext term and tightest function range evaluation for previous termfast and reliable interrogation of procedurally defined implicit surfaces.next term
Research Highlights
The main contributions of this paper are as follows. âș The widening of the scope of previous termreliablenext term ray-tracing and spatial enumeration algorithms for previous termsurfacesnext term ranging from algebraic previous termsurfaces (definednext term by polynomials) to general previous termimplicit surfaces (definednext term by function evaluation procedures involving both previous termaffinenext term and non-previous termaffinenext term operations based on previous termrevised affine arithmetic)next term. âș The introduction of a technique for representing procedural models using special previous termaffinenext term forms (illustrated by case studies of previous termaffinenext term forms for set-theoretic operations in the form of R-functions, blending operations and conditional operations). âș The detailed derivation of special previous termaffinenext term forms for arbitrary operators
Quasilinear Structures in Stochastic Arithmetic and their Application
Stochastic arithmetic has been developed as a model for computing
with imprecise numbers. In this model, numbers are represented
by independent Gaussian variables with known mean value and standard
deviation and are called stochastic numbers.
The algebraic properties of stochastic numbers have already been studied by
several authors. Anyhow, in most life problems the variables are not independent
and a direct application of the model to estimate the standard deviation on
the result of a numerical computation may lead to some overestimation of
the correct value.
In this work âquasilinearâ algebraic structures based on standard stochastic arithmetic
are studied and, from pure abstract algebraic considerations, new arithmetic operations
called âinner stochastic addition and subtractionâ are introduced.
They appear to be stochastic analogues to the inner interval addition and subtraction
used in interval arithmetic. The algebraic properties of these operations and
the involved algebraic structures are then studied.
Finally, the connection of these inner operations to the correlation coefficient of
the variables is developed and it is shown that they allow the computation with
non-independent variables. The corresponding methodology for the practical
application of the new structures in relation to problems analogous to âdependency problemsâ
in interval arithmetic is given and some numerical experiments showing the interest of
these new operations are presented.
ACM Computing Classification System (1998): D.2.4, G.3, G.4
Research and Education in Computational Science and Engineering
Over the past two decades the field of computational science and engineering
(CSE) has penetrated both basic and applied research in academia, industry, and
laboratories to advance discovery, optimize systems, support decision-makers,
and educate the scientific and engineering workforce. Informed by centuries of
theory and experiment, CSE performs computational experiments to answer
questions that neither theory nor experiment alone is equipped to answer. CSE
provides scientists and engineers of all persuasions with algorithmic
inventions and software systems that transcend disciplines and scales. Carried
on a wave of digital technology, CSE brings the power of parallelism to bear on
troves of data. Mathematics-based advanced computing has become a prevalent
means of discovery and innovation in essentially all areas of science,
engineering, technology, and society; and the CSE community is at the core of
this transformation. However, a combination of disruptive
developments---including the architectural complexity of extreme-scale
computing, the data revolution that engulfs the planet, and the specialization
required to follow the applications to new frontiers---is redefining the scope
and reach of the CSE endeavor. This report describes the rapid expansion of CSE
and the challenges to sustaining its bold advances. The report also presents
strategies and directions for CSE research and education for the next decade.Comment: Major revision, to appear in SIAM Revie
Infinite Volume Relaxation in the Sherrington-Kirkpatrick Model
In a recent work (Eissfeller and Opper, 1992) a numerical method has been
proposed to simulate off-equilibrium zero-temperature parallel dynamics for the
SK model without finite size effects. We study the extension of the method to
non-zero temperature and sequential dynamics, and analyze more carefully the
involved computational problems. We find evidence, in the glassy phase, for an
algebraic relaxation of the energy density to its equilibrium value, at least
at large enough temperatures, and for an algebraic relaxation of the
magnetization to zero at non-zero temperatures, with an exponent directly
proportional to the temperature.Comment: 20 pages, Plain TeX with macros included, 11 PostScript figures in a
separate file, included via EPSF and ROTATE macro packages for DVIPS driver,
internal report n. 1039 Dipartimento di Fisica dell'Univ. di Roma La Sapienza
e INFN sezione di Rom
p-Adic Mathematical Physics
A brief review of some selected topics in p-adic mathematical physics is
presented.Comment: 36 page
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