274 research outputs found
Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations
Computational costs of numerically solving multidimensional partial
differential equations (PDEs) increase significantly when the spatial
dimensions of the PDEs are high, due to large number of spatial grid points.
For multidimensional reaction-diffusion equations, stiffness of the system
provides additional challenges for achieving efficient numerical simulations.
In this paper, we propose a class of Krylov implicit integration factor (IIF)
discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion
equations on high spatial dimensions. The key ingredient of spatial DG
discretization is the multiwavelet bases on nested sparse grids, which can
significantly reduce the numbers of degrees of freedom. To deal with the
stiffness of the DG spatial operator in discretizing reaction-diffusion
equations, we apply the efficient IIF time discretization methods, which are a
class of exponential integrators. Krylov subspace approximations are used to
evaluate the large size matrix exponentials resulting from IIF schemes for
solving PDEs on high spatial dimensions. Stability and error analysis for the
semi-discrete scheme are performed. Numerical examples of both scalar equations
and systems in two and three spatial dimensions are provided to demonstrate the
accuracy and efficiency of the methods. The stiffness of the reaction-diffusion
equations is resolved well and large time step size computations are obtained
MADNESS: A Multiresolution, Adaptive Numerical Environment for Scientific Simulation
MADNESS (multiresolution adaptive numerical environment for scientific
simulation) is a high-level software environment for solving integral and
differential equations in many dimensions that uses adaptive and fast harmonic
analysis methods with guaranteed precision based on multiresolution analysis
and separated representations. Underpinning the numerical capabilities is a
powerful petascale parallel programming environment that aims to increase both
programmer productivity and code scalability. This paper describes the features
and capabilities of MADNESS and briefly discusses some current applications in
chemistry and several areas of physics
Performance study of the multiwavelet discontinuous Galerkin approach for solving the GreenâNaghdi equations
This paper presents a multiresolution discontinuous Galerkin scheme for the adaptive solution of Boussinesqâtype equations. The model combines multiwaveletâbased grid adaptation with a discontinuous Galerkin (DG) solver based on the system of fully nonlinear and weakly dispersive GreenâNaghdi (GN) equations. The key feature of the adaptation procedure is to conduct a multiresolution analysis using multiwavelets on a hierarchy of nested grids to improve the efficiency of the reference DG scheme on a uniform grid by computing on a locally refined adapted grid. This way the local resolution level will be determined by manipulating multiwavelet coefficients controlled by a single userâdefined threshold value. The proposed adaptive multiwavelet discontinuous Galerkin solver for GN equations (MWDGâGN) is assessed using several benchmark problems related to wave propagation and transformation in nearshore areas. The numerical results demonstrate that the proposed scheme retains the accuracy of the reference scheme, while significantly reducing the computational cost
Construction of interpolating and orthonormal multigenerators and multiwavelets on the interval
In den letzten Jahren haben sich Wavelets zu einem hochwertigen Hilfsmittel in der angewandten
Mathematik entwickelt. Eine Waveletbasis ist im Allgemeinen ein System von
Funktionen, das durch die Skalierung, Translation und Dilatation einer endlichen Menge
von Funktionen, den sogenannten Mutterwavelets, entsteht. Wavelets wurden sehr erfolgreich
in der digitalen Signal- und Bildanalyse, z. B. zur Datenkompression verwendet.
Ein weiteres wichtiges Anwendungsfeld ist die Analyse und die numerische Behandlung
von Operatorgleichungen. Insbesondere ist es gelungen, adaptive numerische Algorithmen
basierend auf Wavelets fĂŒr eine riesige Klasse von Operatorgleichungen, einschlieĂlich
Operatoren mit negativer Ordnung, zu entwickeln. Der Erfolg der Wavelet-
Algorithmen ergibt sich als Konsequenz der folgenden Fakten:
- Gewichtete Folgennormen von Wavelet-Expansionskoeffizienten sind in einem bestimmten
Bereich (abhÀngig von der RegularitÀt der Wavelets) Àquivalent zu
GlÀttungsnormen wie Besov- oder Sobolev-Normen.
- FĂŒr eine breite Klasse von Operatoren ist ihre Darstellung in Wavelet-Koordinaten
nahezu diagonal.
- Die verschwindenden Momente von Wavelets entfernen den glatten Teil einer Funktion
und fĂŒhren zu sehr effizienten Komprimierungsstrategien.
Diese Fakten können z. B. verwendet werden, um adaptive numerische Strategien mit
optimaler Konvergenzgeschwindigkeit zu konstruieren, in dem Sinne, dass diese Algorithmen
die Konvergenzordnung der besten N-Term-Approximationsschemata realisieren.
Die maĂgeblichen Ergebnisse lassen sich fĂŒr lineare, symmetrische, elliptische Operatorgleichungen
erzielen. Es existiert auch eine Verallgemeinerung fĂŒr nichtlineare elliptische
Gleichungen. Hier verbirgt sich jedoch eine ernste Schwierigkeit: Jeder numerische Algorithmus
fĂŒr diese Gleichungen erfordert die Auswertung eines nichtlinearen Funktionals,
welches auf eine Wavelet-Reihe angewendet wird. Obwohl einige sehr ausgefeilte Algorithmen
existieren, erweisen sie sich als ziemlich langsam in der Praxis. In neueren Studien
wurde gezeigt, dass dieses Problem durch sogenannte Interpolanten verbessert werden
kann. Dabei stellt sich heraus, dass die meisten bekannten Basen der Interpolanten
keine stabilen Basen in L2[a,b] bilden.
In der vorliegenden Arbeit leisten wir einen wesentlichen Beitrag zu diesem Problem
und konstruieren neue Familien von Interpolanten auf beschrÀnkten Gebieten, die nicht
nur interpolierend, sondern auch stabil in L2[a,b] sind. Da dies mit nur einem Generator
schwer (oder vielleicht sogar unmöglich) zu erreichen ist, werden wir mit Multigeneratoren
und Multiwavelets arbeiten.In recent years, wavelets have become a very powerful tools in applied
mathematics. In general,
a wavelet basis is a system of functions that is generated by scaling, translating and dilating a
finite set of functions, the so-called mother wavelets. Wavelets have been very successfully
applied in image/signal analysis, e.g., for denoising and compression purposes. Another
important field of applications is the analysis and the numerical treatment of operator
equations. In particular, it has been possible to design adaptive numerical algorithms based on
wavelets for a huge class of operator equations including operators of negative order. The
success of wavelet algorithms is an ultimative consequence of the following facts:
- Weighted sequence norms of wavelet expansion coefficients are equivalent in a certain
range (depending on the regularity of the wavelets) to smoothness norms such as Besov
or Sobolev norms.
- For a wide class of operators their representation in wavelet coordinates is nearly
diagonal.
-The vanishing moments of wavelets remove the smooth part of a function.
These facts can,
e.g., be used to construct adaptive numerical strategies that are guaranteed to
converge with optimal order, in the sense that these algorithms realize the convergence order
of best N-term approximation schemes. The most far-reaching results have been obtained for
linear, symmetric elliptic operator equations. Generalization to nonlinear elliptic equations also
exist. However, then one is faced with a serious bottleneck: every numerical algorithm for these
equations requires the evaluation of a nonlinear functional applied to a wavelet series.
Although some very sophisticated algorithms exist, they turn out to perform quite slowly in
practice. In recent studies, it has been shown that this problem can be ameliorated by means of
so called interpolants. However, then the problem occurs that most of the known bases of
interpolants do not form stable bases in L2[a,b].
In this PhD project, we intend to provide a significant
contribution to this problem. We want to
construct new families of interpolants on domains that are not only interpolating, but also
stable in L2[a,b]or even orthogonal. Since this is hard to achieve (or maybe even impossible)
with just one generator, we worked with multigenerators and multiwavelets
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