329 research outputs found
Hybrid PDE solver for data-driven problems and modern branching
The numerical solution of large-scale PDEs, such as those occurring in
data-driven applications, unavoidably require powerful parallel computers and
tailored parallel algorithms to make the best possible use of them. In fact,
considerations about the parallelization and scalability of realistic problems
are often critical enough to warrant acknowledgement in the modelling phase.
The purpose of this paper is to spread awareness of the Probabilistic Domain
Decomposition (PDD) method, a fresh approach to the parallelization of PDEs
with excellent scalability properties. The idea exploits the stochastic
representation of the PDE and its approximation via Monte Carlo in combination
with deterministic high-performance PDE solvers. We describe the ingredients of
PDD and its applicability in the scope of data science. In particular, we
highlight recent advances in stochastic representations for nonlinear PDEs
using branching diffusions, which have significantly broadened the scope of
PDD.
We envision this work as a dictionary giving large-scale PDE practitioners
references on the very latest algorithms and techniques of a non-standard, yet
highly parallelizable, methodology at the interface of deterministic and
probabilistic numerical methods. We close this work with an invitation to the
fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European
Journal of Applied Mathematics (EJAM
Jarzynski's equality, fluctuation theorems, and variance reduction: Mathematical analysis and numerical algorithms
In this paper, we study Jarzynski's equality and fluctuation theorems for
diffusion processes. While some of the results considered in the current work
are known in the (mainly physics) literature, we review and generalize these
nonequilibrium theorems using mathematical arguments, therefore enabling
further investigations in the mathematical community. On the numerical side,
variance reduction approaches such as importance sampling method are studied in
order to compute free energy differences based on Jarzynski's equality.Comment: journal versio
Reconstruction of thermally-symmetrized quantum autocorrelation functions from imaginary-time data
In this paper, I propose a technique for recovering quantum dynamical
information from imaginary-time data via the resolution of a one-dimensional
Hamburger moment problem. It is shown that the quantum autocorrelation
functions are uniquely determined by and can be reconstructed from their
sequence of derivatives at origin. A general class of reconstruction algorithms
is then identified, according to Theorem 3. The technique is advocated as
especially effective for a certain class of quantum problems in continuum
space, for which only a few moments are necessary. For such problems, it is
argued that the derivatives at origin can be evaluated by Monte Carlo
simulations via estimators of finite variances in the limit of an infinite
number of path variables. Finally, a maximum entropy inversion algorithm for
the Hamburger moment problem is utilized to compute the quantum rate of
reaction for a one-dimensional symmetric Eckart barrier.Comment: 15 pages, no figures, to appear in Phys. Rev.
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