9,098 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
A Jacobian-free Newton-Krylov method for time-implicit multidimensional hydrodynamics
This work is a continuation of our efforts to develop an efficient implicit
solver for multidimensional hydrodynamics for the purpose of studying important
physical processes in stellar interiors, such as turbulent convection and
overshooting. We present an implicit solver that results from the combination
of a Jacobian-Free Newton-Krylov method and a preconditioning technique
tailored to the inviscid, compressible equations of stellar hydrodynamics. We
assess the accuracy and performance of the solver for both 2D and 3D problems
for Mach numbers down to . Although our applications concern flows in
stellar interiors, the method can be applied to general advection and/or
diffusion-dominated flows. The method presented in this paper opens up new
avenues in 3D modeling of realistic stellar interiors allowing the study of
important problems in stellar structure and evolution.Comment: Accepted for publication in A&
Development and testing of Parabolic Dish Concentrator No. 1
Parabolic Dish Concentrator No. 1 (PDC-1) is a 12-m-diameter prototype concentrator with low life-cycle costs for use with thermal-to-electric energy conversion devices. The concentrator assembly features panels made of a resin transfer molded balsa core/fiberglass sandwich with plastic reflective film as the reflective surface and a ribbed framework to hold the panels in place. The concentrator assembly tracks in azimuth and elevation on a base frame riding on a circular track. It is shown that the panels do not exhibit the proper parabolic contour. However, thermal gradients were discovered in the panels with daily temperature changes. The PDC-1 has sufficient optical quality to operate satisfactorily in a dish-electric system. The PDC-1 development provides the impetus for creating innovative optical testing methods and valuable information for use in designing and fabricating concentrators of future dish-electric systems
Simultaneous Reduced Basis Approximation of Parameterized Elliptic Eigenvalue Problems
The focus is on a model reduction framework for parameterized elliptic
eigenvalue problems by a reduced basis method. In contrast to the standard
single output case, one is interested in approximating several outputs
simultaneously, namely a certain number of the smallest eigenvalues. For a fast
and reliable evaluation of these input-output relations, we analyze a
posteriori error estimators for eigenvalues. Moreover, we present different
greedy strategies and study systematically their performance. Special attention
needs to be paid to multiple eigenvalues whose appearance is
parameter-dependent. Our methods are of particular interest for applications in
vibro-acoustics
- …