8,326 research outputs found
A fractional wavelet Galerkin method for the fractional diffusion problem
The aim of this paper is to solve some fractional differential problems hav-
ing time fractional derivative by means of a wavelet Galerkin method that
uses the fractional scaling functions introduced in a previpous paper as approximating
functions. These refinable functions, which are a generalization of the
fractional B-splines, have many interesting approximation properties.
In particular, their fractional derivatives have a closed form that involves
just the fractional difference operator. This allows us to construct accurate
and efficient numerical methods to solve fractional differential problems.
Some numerical tests on a fractional diffusion problem will be given
SpECTRE: A Task-based Discontinuous Galerkin Code for Relativistic Astrophysics
We introduce a new relativistic astrophysics code, SpECTRE, that combines a
discontinuous Galerkin method with a task-based parallelism model. SpECTRE's
goal is to achieve more accurate solutions for challenging relativistic
astrophysics problems such as core-collapse supernovae and binary neutron star
mergers. The robustness of the discontinuous Galerkin method allows for the use
of high-resolution shock capturing methods in regions where (relativistic)
shocks are found, while exploiting high-order accuracy in smooth regions. A
task-based parallelism model allows efficient use of the largest supercomputers
for problems with a heterogeneous workload over disparate spatial and temporal
scales. We argue that the locality and algorithmic structure of discontinuous
Galerkin methods will exhibit good scalability within a task-based parallelism
framework. We demonstrate the code on a wide variety of challenging benchmark
problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the
code's scalability including its strong scaling on the NCSA Blue Waters
supercomputer up to the machine's full capacity of 22,380 nodes using 671,400
threads.Comment: 41 pages, 13 figures, and 7 tables. Ancillary data contains
simulation input file
Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States
We resume the recent successes of the grid-based tensor numerical methods and
discuss their prospects in real-space electronic structure calculations. These
methods, based on the low-rank representation of the multidimensional functions
and integral operators, led to entirely grid-based tensor-structured 3D
Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core
Hamiltonian and two-electron integrals (TEI) in complexity using
the rank-structured approximation of basis functions, electron densities and
convolution integral operators all represented on 3D
Cartesian grids. The algorithm for calculating TEI tensor in a form of the
Cholesky decomposition is based on multiple factorizations using algebraic 1D
``density fitting`` scheme. The basis functions are not restricted to separable
Gaussians, since the analytical integration is substituted by high-precision
tensor-structured numerical quadratures. The tensor approaches to
post-Hartree-Fock calculations for the MP2 energy correction and for the
Bethe-Salpeter excited states, based on using low-rank factorizations and the
reduced basis method, were recently introduced. Another direction is related to
the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for
finite lattice-structured systems, where one of the numerical challenges is the
summation of electrostatic potentials of a large number of nuclei. The 3D
grid-based tensor method for calculation of a potential sum on a lattice manifests the linear in computational work, ,
instead of the usual scaling by the Ewald-type approaches
Reproducibility, accuracy and performance of the Feltor code and library on parallel computer architectures
Feltor is a modular and free scientific software package. It allows
developing platform independent code that runs on a variety of parallel
computer architectures ranging from laptop CPUs to multi-GPU distributed memory
systems. Feltor consists of both a numerical library and a collection of
application codes built on top of the library. Its main target are two- and
three-dimensional drift- and gyro-fluid simulations with discontinuous Galerkin
methods as the main numerical discretization technique. We observe that
numerical simulations of a recently developed gyro-fluid model produce
non-deterministic results in parallel computations. First, we show how we
restore accuracy and bitwise reproducibility algorithmically and
programmatically. In particular, we adopt an implementation of the exactly
rounded dot product based on long accumulators, which avoids accuracy losses
especially in parallel applications. However, reproducibility and accuracy
alone fail to indicate correct simulation behaviour. In fact, in the physical
model slightly different initial conditions lead to vastly different end
states. This behaviour translates to its numerical representation. Pointwise
convergence, even in principle, becomes impossible for long simulation times.
In a second part, we explore important performance tuning considerations. We
identify latency and memory bandwidth as the main performance indicators of our
routines. Based on these, we propose a parallel performance model that predicts
the execution time of algorithms implemented in Feltor and test our model on a
selection of parallel hardware architectures. We are able to predict the
execution time with a relative error of less than 25% for problem sizes between
0.1 and 1000 MB. Finally, we find that the product of latency and bandwidth
gives a minimum array size per compute node to achieve a scaling efficiency
above 50% (both strong and weak)
Poisson inverse problems
In this paper we focus on nonparametric estimators in inverse problems for
Poisson processes involving the use of wavelet decompositions. Adopting an
adaptive wavelet Galerkin discretization, we find that our method combines the
well-known theoretical advantages of wavelet--vaguelette decompositions for
inverse problems in terms of optimally adapting to the unknown smoothness of
the solution, together with the remarkably simple closed-form expressions of
Galerkin inversion methods. Adapting the results of Barron and Sheu [Ann.
Statist. 19 (1991) 1347--1369] to the context of log-intensity functions
approximated by wavelet series with the use of the Kullback--Leibler distance
between two point processes, we also present an asymptotic analysis of
convergence rates that justifies our approach. In order to shed some light on
the theoretical results obtained and to examine the accuracy of our estimates
in finite samples, we illustrate our method by the analysis of some simulated
examples.Comment: Published at http://dx.doi.org/10.1214/009053606000000687 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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