108 research outputs found
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
Coordinate-Space Hartree-Fock-Bogoliubov Solvers for Superfluid Fermi Systems in Large Boxes
The self-consistent Hartree-Fock-Bogoliubov problem in large boxes can be
solved accurately in the coordinate space with the recently developed solvers
HFB-AX (2D) and MADNESS-HFB (3D). This is essential for the description of
superfluid Fermi systems with complicated topologies and significant spatial
extend, such as fissioning nuclei, weakly-bound nuclei, nuclear matter in the
neutron star rust, and ultracold Fermi atoms in elongated traps. The HFB-AX
solver based on B-spline techniques uses a hybrid MPI and OpenMP programming
model for parallel computation for distributed parallel computation, within a
node multi-threaded LAPACK and BLAS libraries are used to further enable
parallel calculations of large eigensystems. The MADNESS-HFB solver uses a
novel multi-resolution analysis based adaptive pseudo-spectral techniques to
enable fully parallel 3D calculations of very large systems. In this work we
present benchmark results for HFB-AX and MADNESS-HFB on ultracold trapped
fermions.Comment: Conference on Computational Physics (CCP 2011) Proceedin
Tricomi's composition formula and the analysis of multiwavelet approximation methods for boundary integral equations
The present paper is mainly concerned with the convergence analysis of Galerkin-Petrov methods for the numerical solution of periodic pseudodifferential equations using wavelets and multiwavelets as trial functions and test functionals. Section 2 gives an overview on the symbol calculus of multidimensional singular integrals using Tricomi's composition formula. In Section 3 we formulate necessary and sufficient stability conditions in terms of the so-called numerical symbols and demonstrate applications to the Dirchlet problem for the Laplace equation
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 für eine riesige Klasse von Operatorgleichungen, einschließlich
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 (abhängig von der Regularität der Wavelets) äquivalent zu
Glättungsnormen wie Besov- oder Sobolev-Normen.
- Für 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 führen 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 maßgeblichen Ergebnisse lassen sich für lineare, symmetrische, elliptische Operatorgleichungen
erzielen. Es existiert auch eine Verallgemeinerung für nichtlineare elliptische
Gleichungen. Hier verbirgt sich jedoch eine ernste Schwierigkeit: Jeder numerische Algorithmus
für 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 beschränkten 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
Kinetic energy-free Hartree–Fock equations: an integral formulation
We have implemented a self-consistent feld solver for Hartree–Fock calculations,
by making use of Multiwavelets and Multiresolution Analysis. We show how such a
solver is inherently a preconditioned steepest descent method and therefore a good
starting point for rapid convergence. A distinctive feature of our implementation is
the absence of any reference to the kinetic energy operator. This is desirable when
Multiwavelets are employed, because diferential operators such as the Laplacian in
the kinetic energy are challenging to represent correctly. The theoretical framework
is described in detail and the implemented algorithm is both presented in the paper
and made available as a Python notebook. Two simple examples are presented, highlighting the main features of our implementation: arbitrary predefned precision,
rapid and robust convergence, absence of the kinetic energy operator
A multiwavelet approach to the direct solution of the Poisson equation: implementation and optimization
This work is presenting a fully numerical approach for the direct solution of the Poisson equation for the electrostatic potential given by an arbitrary charge density. Efficient solution of this equation is important in many fields of science, where the current work is dealing with nuclear and electronic potential calculations, used in the field of computational chemistry. The equation is solved using the mathematical framework of multiwavelets, which is a theory that enables us to represent functions and operators with rigorous error control. The multiwavelet basis is well suited to treat the multiple length scales present in the calculation of electronic structure.
A prototype implementation of the solution of the Poisson equation using the multiwavelet formalism has previously been obtained by the computational chemistry group at the University of Tromso¸, and the current work is dealing with optimization of the existing code and with the development of the code into new areas of applicability, specifically to the calculation of electronic structure using the framework of density functional theory
Multiwavelet approximation methods for pseudodifferential equations on curves. Stability and convergence analysis
We develop a stability and convergence analysis of Galerkin-Petrov schemes based on a general setting of multiresolution generated by several refinable functions for the numerical solution of pseudodifferential equations on smooth closed curves. Particular realizations of such a multiresolution analysis are trial spaces generated by biorthogonal wavelets or by splines with multiple knots. The main result presents necessary and sufficient conditions for the stability of the numerical method in terms of the principal symbol of the pseudodifferential operator and the Fourier transforms of the generating multiscaling functions as well as of the test functionals. Moreover, optimal convergence rates for the approximate solutions in a range of Sobolev spaces are established
Gröbner bases and wavelet design
AbstractIn this paper, we detail the use of symbolic methods in order to solve some advanced design problems arising in signal processing. Our interest lies especially in the construction of wavelet filters for which the usual spectral factorization approach (used for example to construct the well-known Daubechies filters) is not applicable. In these problems, we show how the design equations can be written as multivariate polynomial systems of equations and accordingly how Gröbner algorithms offer an effective way to obtain solutions in some of these cases
- …