@PhDThesis{ itlic:2012-003,
author = {Magnus Gustafsson},
title = {Towards an Adaptive Solver for High-Dimensional {PDE}
Problems on Clusters of Multicore Processors},
school = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2012},
number = {2012-003},
type = {Licentiate thesis},
month = mar,
day = {4},
note = {Included papers available at
\url{http://dx.doi.org/10.1007/978-3-642-11795-4_44},
\url{http://dx.doi.org/10.1007/978-3-642-28145-7_36},
\url{http://www.it.uu.se/research/publications/reports/2011-022}
and
\url{http://www.it.uu.se/research/publications/reports/2012-001}.}
,
abstract = {Accurate numerical simulation of time-dependent phenomena
in many spatial dimensions is a challenging computational
task apparent in a vast range of application areas, for
instance quantum dynamics, financial mathematics, systems
biology and plasma physics. Particularly problematic is
that the number of unknowns in the governing equations (the
number of grid points) grows exponentially with the number
of spatial dimensions introduced, often referred to as the
curse of dimensionality. This limits the range of problems
that we can solve, since the computational effort and
requirements on memory storage directly depend on the
number of unknowns for which to solve the equations.
In order to push the limit of tractable problems, we are
developing an implementation framework, HAParaNDA, for
high-dimensional PDE-problems. By using high-order accurate
schemes and adaptive mesh refinement (AMR) in space, we aim
at reducing the number of grid points used in the
discretization, thereby enabling the solution of larger and
higher-dimensional problems. Within the framework, we use
structured grids for spatial discretization and a
block-decomposition of the spatial domain for
parallelization and load balancing. For integration in
time, we use exponential integration, although the
framework allows the flexibility of other integrators to be
implemented as well. Exponential integrators using the
Lanzcos or the Arnoldi algorithm has proven a succesful and
efficient approach for large problems. Using a truncation
of the Magnus expansion, we can attain high levels of
accuracy in the solution.
As an example application, we have implemented a solver for
the time-dependent Schr{\"o}dinger equation using this
framework. We provide scaling results for small and medium
sized clusters of multicore nodes, and show that the solver
fulfills the expected rate of convergence.}
}