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This monograph is devoted to random walk based stochastic
algorithms for solving high-dimensional boundary value problems of
mathematical physics and chemistry. It includes Monte Carlo methods
where the random walks live not only on the boundary, but also
inside the domain. A variety of examples from capacitance
calculations to electron dynamics in semiconductors are discussed
to illustrate the viability of the approach. The book is written
for mathematicians who work in the field of partial differential
and integral equations, physicists and engineers dealing with
computational methods and applied probability, for students and
postgraduates studying mathematical physics and numerical
mathematics. Contents: Introduction Random walk algorithms for
solving integral equations Random walk-on-boundary algorithms for
the Laplace equation Walk-on-boundary algorithms for the heat
equation Spatial problems of elasticity Variants of the random walk
on boundary for solving stationary potential problems Splitting and
survival probabilities in random walk methods and applications A
random WOS-based KMC method for electron-hole recombinations Monte
Carlo methods for computing macromolecules properties and solving
related problems Bibliography
The book presents integral formulations for partial differential
equations, with the focus on spherical and plane integral
operators. The integral relations are obtained for different
elliptic and parabolic equations, and both direct and inverse mean
value relations are studied. The derived integral equations are
used to construct new numerical methods for solving relevant
boundary value problems, both deterministic and stochastic based on
probabilistic interpretation of the spherical and plane integral
operators.
The book presents advanced stochastic models and simulation methods
for random flows and transport of particles by turbulent velocity
fields and flows in porous media. Two main classes of models are
constructed: (1) turbulent flows are modeled as synthetic random
fields which have certain statistics and features mimicing those of
turbulent fluid in the regime of interest, and (2) the models are
constructed in the form of stochastic differential equations for
stochastic Lagrangian trajectories of particles carried by
turbulent flows. The book is written for mathematicians,
physicists, and engineers studying processes associated with
probabilistic interpretation, researchers in applied and
computational mathematics, in environmental and engineering
sciences dealing with turbulent transport and flows in porous
media, as well as nucleation, coagulation, and chemical reaction
analysis under fluctuation conditions. It can be of interest for
students and post-graduates studying numerical methods for solving
stochastic boundary value problems of mathematical physics and
dispersion of particles by turbulent flows and flows in porous
media.
This is the proceedings of the "8th IMACS Seminar on Monte Carlo
Methods" held from August 29 to September 2, 2011 in Borovets,
Bulgaria, and organized by the Institute of Information and
Communication Technologies of the Bulgarian Academy of Sciences in
cooperation with the International Association for Mathematics and
Computers in Simulation (IMACS). Included are 24 papers which cover
all topics presented in the sessions of the seminar: stochastic
computation and complexity of high dimensional problems,
sensitivity analysis, high-performance computations for Monte Carlo
applications, stochastic metaheuristics for optimization problems,
sequential Monte Carlo methods for large-scale problems,
semiconductor devices and nanostructures. The history of the IMACS
Seminar on Monte Carlo Methods goes back to April 1997 when the
first MCM Seminar was organized in Brussels: 1st IMACS Seminar,
1997, Brussels, Belgium 2nd IMACS Seminar, 1999, Varna, Bulgaria
3rd IMACS Seminar, 2001, Salzburg, Austria 4th IMACS Seminar, 2003,
Berlin, Germany 5th IMACS Seminar, 2005, Tallahassee, USA 6th IMACS
Seminar, 2007, Reading, UK 7th IMACS Seminar, 2009, Brussels,
Belgium 8th IMACS Seminar, 2011, Borovets, Bulgaria
This book deals with Random Walk Methods for solving
multidimensional boundary value problems. Monte Carlo algorithms
are constructed for three classes of problems: (1) potential
theory, (2) elasticity, and (3) diffusion. Some of the advantages
of our new methods as compared to conventional numerical methods
are that they cater for stochasticities in the boundary value
problems and complicated shapes of the boundaries.
The Monte Carlo method is based on the munerical realization of
natural or artificial models of the phenomena under considerations.
In contrast to classical computing methods the Monte Carlo
efficiency depends weakly on the dimen sion and geometric details
of the problem. The method is used for solving complex problems of
the radiation transfer theory, turbulent diffusion, chemi cal
kinetics, theory of rarefied gases, diffraction of waves on random
surfaces, etc. The Monte Carlo method is especially effective when
using multi-processor computing systems which allow many
independent statistical experiments to be simulated simultaneously.
The weighted Monte Carlo estimates are constructed in order to
diminish errors and to obtain dependent estimates for the
calculated functionals for different values of parameters of the
problem, i.e., to improve the functional dependence. In addition,
the weighted estimates make it possible to evaluate special
functionals, for example, the derivatives with respect to the
parameters. There are many works concerned with the development of
the weighted estimates. In Chap. 1 we give the necessary
information about these works and present a set of illustrations.
The rest of the book is devoted to the solution of a series of
mathematical problems related to the optimization of the weighted
Monte Carlo estimates."
This monographs presents new spherical mean value relations for
classical boundary value problems of mathematical physics. The
derived spherical mean value relations provide equivalent integral
formulations of original boundary value problems. Direct and
converse mean value theorems are proved for scalar elliptic
equations (the Laplace, Helmholtz and diffusion equations),
parabolic equations, high-order elliptic equations (biharmonic and
metaharmonic equations), and systems of elliptic equations (the
Lami equation, systems of diffusion and elasticity equations). In
addition, applications to the random walk on spheres method are
given.
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