Smoothness of Nonlinear and Non-Separable Subdivision Schemes

We study in this paper nonlinear subdivision schemes in a multivariate setting allowing arbitrary dilation matrix. We investigate the convergence of such iterative process to some limit function. Our analysis is based on some conditions on the contractivity of the associated scheme for the differences. In particular, we show the regularity of the limit function, in $L^p$ and Sobolev spaces

Multiresolution approximation of the vector fields on T^3

Multiresolution approximation (MRA) of the vector fields on T^3 is studied. We introduced in the Fourier space a triad of vector fields called helical vectors which derived from the spherical coordinate system basis. Utilizing the helical vectors, we proved the orthogonal decomposition of L^2(T^3) which is a synthesis of the Hodge decomposition of the differential 1- or 2-form on T^3 and the Beltrami decomposition that decompose the space of solenoidal vector fields into the eigenspaces of curl operator. In the course of proof, a general construction procedure of the divergence-free orthonormal complete basis from the basis of scalar function space is presented. Applying this procedure to MRA of L^2(T^3), we discussed the MRA of vector fields on T^3 and the analyticity and regularity of vector wavelets. It is conjectured that the solenoidal wavelet basis must break r-regular condition, i.e. some wavelet functions cannot be rapidly decreasing function because of the inevitable singularities of helical vectors. The localization property and spatial structure of solenoidal wavelets derived from the Littlewood-Paley type MRA (Meyer's wavelet) are also investigated numerically.Comment: LaTeX, 33 Pages, 3 figures. submitted to J. Math. Phy

A multiscale collocation method for fractional differential problems

We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale collocation method are proved and some numerical results are shown.We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale collocation method are proved and some numerical results are shown

Wavelets and their use

This review paper is intended to give a useful guide for those who want to apply discrete wavelets in their practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to corresponding literature. The multiresolution analysis and fast wavelet transform became a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for achievement of a goal. Analysis of various functions with the help of wavelets allows to reveal fractal structures, singularities etc. Wavelet transform of operator expressions helps solve some equations. In practical applications one deals often with the discretized functions, and the problem of stability of wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves by some examples only. The authors would be grateful for any comments which improve this review paper and move us closer to the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh

Vector Subdivision Schemes for Arbitrary Matrix Masks

Employing a matrix mask, a vector subdivision scheme is a fast iterative averaging algorithm to compute refinable vector functions for wavelet methods in numerical PDEs and to produce smooth curves in CAGD. In sharp contrast to the well-studied scalar subdivision schemes, vector subdivision schemes are much less well understood, e.g., Lagrange and (generalized) Hermite subdivision schemes are the only studied vector subdivision schemes in the literature. Because many wavelets used in numerical PDEs are derived from refinable vector functions whose matrix masks are not from Hermite subdivision schemes, it is necessary to introduce and study vector subdivision schemes for any general matrix masks in order to compute wavelets and refinable vector functions efficiently. For a general matrix mask, we show that there is only one meaningful way of defining a vector subdivision scheme. Motivated by vector cascade algorithms and recent study on Hermite subdivision schemes, we shall define a vector subdivision scheme for any arbitrary matrix mask and then we prove that the convergence of the newly defined vector subdivision scheme is equivalent to the convergence of its associated vector cascade algorithm. We also study convergence rates of vector subdivision schemes. The results of this paper not only bridge the gaps and establish intrinsic links between vector subdivision schemes and vector cascade algorithms but also strengthen and generalize current known results on Lagrange and (generalized) Hermite subdivision schemes. Several examples are provided to illustrate the results in this paper on various types of vector subdivision schemes with convergence rates

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

We study tensor product multiscale many-particle spaces with finite-order weights and their application for the electronic Schrödinger equation. Any numerical solution of the electronic Schrödinger equation using conventional discretization schemes is impossible due to its high dimensionality. Therefore, typically Monte Carlo methods (VMC/DMC) or nonlinear model approximations like Hartree-Fock (HF), coupled cluster (CC) or density functional theory (DFT) are used. In this work we develop and implement in parallel a numerical method based on adaptive sparse grids and a particle-wise subspace splitting with respect to one-particle functions which stem from a nonlinear rank-1 approximation. Sparse grids allow to overcome the exponential complexity exhibited by conventional discretization procedures and deliver a convergent numerical approach with guaranteed convergence rates. In particular, the introduced weighted many-particle tensor product multiscale approximation spaces include the common configuration interaction (CI) spaces as a special case. To realize our new approach, we first introduce general many-particle Sobolev spaces, which particularly include the standard Sobolev spaces as well as Sobolev spaces of dominated mixed smoothness. For this novel variant of sparse grid spaces we show estimates for the approximation and complexity orders with respect to the smoothness and decay parameters. With known regularity properties of the electronic wave function it follows that, up to logarithmic terms, the convergence rate is independent of the number of electrons and almost the same as in the two-electron case. However, besides the rate, also the dependence of the complexity constants on the number of electrons plays an important role for a truly practical method. Based on a splitting of the one-particle space we construct a subspace splitting of the many-particle space, which particularly includes the known ANOVA decomposition, the HDMR decomposition and the CI decomposition as special cases. Additionally, we introduce weights for a restriction of this subspace splitting. In this way weights of finite order q lead to many-particle spaces in which the problem of an approximation of an N-particle function reduces to the problem of the approximation of q-particle functions. To obtain as small as possible constants with respect to the cost complexity, we introduce a heuristic adaptive scheme to build a sequence of finite-dimensional subspaces of a weighted tensor product multiscale many-particle approximation space. Furthermore, we construct a multiscale Gaussian frame and apply Gaussians and modulated Gaussians for the nonlinear rank-1 approximation. In this way, all matrix entries of the corresponding discrete eigenvalue problem can be computed in terms of analytic formulae for the one and two particle operator integrals. Finally, we apply our novel approach to small atomic and diatomic systems with up to 6 electrons (18 space dimensions). The numerical results demonstrate that our new method indeed allows for convergence with expected rates.Tensorprodukt-Multiskalen-Mehrteilchenräume mit Gewichten endlicher Ordnung für die elektronische Schrödingergleichung In der vorliegenden Arbeit beschäftigen wir uns mit gewichteten Tensorprodukt-Multiskalen-Mehrteilchen-Approximationsräumen und deren Anwendung zur numerischen Lösung der elektronischen Schrödinger-Gleichung. Aufgrund der hohen Problemdimension ist eine direkte numerische Lösung der elektronischen Schrödinger-Gleichung mit Standard-Diskretisierungsverfahren zur linearen Approximation unmöglich, weshalb üblicherweise Monte Carlo Methoden (VMC/DMC) oder nichtlineare Modellapproximationen wie Hartree-Fock (HF), Coupled Cluster (CC) oder Dichtefunktionaltheorie (DFT) verwendet werden. In dieser Arbeit wird eine numerische Methode auf Basis von adaptiven dünnen Gittern und einer teilchenweisen Unterraumzerlegung bezüglich Einteilchenfunktionen aus einer nichtlinearen Rang-1 Approximation entwickelt und für parallele Rechnersysteme implementiert. Dünne Gitter vermeiden die in der Dimension exponentielle Komplexität üblicher Diskretisierungsmethoden und führen zu einem konvergenten numerischen Ansatz mit garantierter Konvergenzrate. Zudem enthalten unsere zugrunde liegenden gewichteten Mehrteilchen Tensorprodukt-Multiskalen-Approximationsräume die bekannten Configuration Interaction (CI) Räume als Spezialfall. Zur Konstruktion unseres Verfahrens führen wir zunächst allgemeine Mehrteilchen-Sobolevräume ein, welche die Standard-Sobolevräume sowie Sobolevräume mit dominierender gemischter Glattheit beinhalten. Wir analysieren die Approximationseigenschaften und schätzen Konvergenzraten und Kostenkomplexitätsordnungen in Abhängigkeit der Glattheitsparameter und Abfalleigenschaften ab. Mit Hilfe bekannter Regularitätseigenschaften der elektronischen Wellenfunktion ergibt sich, dass die Konvergenzrate bis auf logarithmische Terme unabhängig von der Zahl der Elektronen und fast identisch mit der Konvergenzrate im Fall von zwei Elektronen ist. Neben der Rate spielt allerdings die Abhängigkeit der Konstanten in der Kostenkomplexität von der Teilchenzahl eine wichtige Rolle. Basierend auf Zerlegungen des Einteilchenraumes konstruieren wir eine Unterraumzerlegung des Mehrteilchenraumes, welche insbesondere die bekannte ANOVA-Zerlegung, die HDMR-Zerlegung sowie die CI-Zerlegung als Spezialfälle beinhaltet. Eine zusätzliche Gewichtung der entsprechenden Unterräume mit Gewichten von endlicher Ordnung q führt zu Mehrteilchenräumen, in denen sich das Approximationsproblem einer N-Teilchenfunktion zu Approximationsproblemen von q-Teilchenfunktionen reduziert. Mit dem Ziel, Konstanten möglichst kleiner Größe bezüglich der Kostenkomplexität zu erhalten, stellen wir ein heuristisches adaptives Verfahren zur Konstruktion einer Sequenz von endlich-dimensionalen Unterräumen eines gewichteten Mehrteilchen-Tensorprodukt-Multiskalen-Approximationsraumes vor. Außerdem konstruieren wir einen Frame aus Multiskalen-Gauss-Funktionen und verwenden Einteilchenfunktionen im Rahmen der Rang-1 Approximation in der Form von Gauss- und modulierten-Gauss-Funktionen. Somit können die zur Aufstellung der Matrizen des zugehörigen diskreten Eigenwertproblems benötigten Ein- und Zweiteilchenintegrale analytisch berechnet werden. Schließlich wenden wir unsere Methode auf kleine Atome und Moleküle mit bis zu sechs Elektronen (18 Raumdimensionen) an. Die numerischen Resultate zeigen, dass sich die aus der Theorie zu erwartenden Konvergenzraten auch praktisch ergeben