A heat transfer with a source: the complete set of invariant difference schemes

In this letter we present the set of invariant difference equations and meshes which preserve the Lie group symmetries of the equation u_{t}=(K(u)u_{x})_{x}+Q(u). All special cases of K(u) and Q(u) that extend the symmetry group admitted by the differential equation are considered. This paper completes the paper [J. Phys. A: Math. Gen. 30, no. 23 (1997) 8139-8155], where a few invariant models for heat transfer equations were presented.Comment: arxiv version is already officia

Distributed Dominating Sets on Grids

This paper presents a distributed algorithm for finding near optimal dominating sets on grids. The basis for this algorithm is an existing centralized algorithm that constructs dominating sets on grids. The size of the dominating set provided by this centralized algorithm is upper-bounded by $\lceil\frac{(m+2)(n+2)}{5}\rceil$ for $m\times n$ grids and its difference from the optimal domination number of the grid is upper-bounded by five. Both the centralized and distributed algorithms are generalized for the $k$-distance dominating set problem, where all grid vertices are within distance $k$ of the vertices in the dominating set.Comment: 10 pages, 9 figures, accepted in ACC 201

Greedy vector quantization

We investigate the greedy version of the $L^p$-optimal vector quantization problem for an $\mathbb{R}^d$-valued random vector $X\!\in L^p$. We show the existence of a sequence $(a_N)_{N\ge 1}$ such that $a_N$ minimizes $a\mapsto\big \|\min_{1\le i\le N-1}|X-a_i|\wedge |X-a|\big\|_{L^p}$ ($L^p$-mean quantization error at level $N$ induced by $(a_1,\ldots,a_{N-1},a)$). We show that this sequence produces $L^p$-rate optimal $N$-tuples $a^{(N)}=(a_1,\ldots,a_{_N})$ ($i.e.$ the $L^p$-mean quantization error at level $N$ induced by $a^{(N)}$ goes to $0$ at rate $N^{-\frac 1d}$). Greedy optimal sequences also satisfy, under natural additional assumptions, the distortion mismatch property: the $N$-tuples $a^{(N)}$ remain rate optimal with respect to the $L^q$-norms, $p\le q <p+d$. Finally, we propose optimization methods to compute greedy sequences, adapted from usual Lloyd's I and Competitive Learning Vector Quantization procedures, either in their deterministic (implementable when $d=1$) or stochastic versions.Comment: 31 pages, 4 figures, few typos corrected (now an extended version of an eponym paper to appear in Journal of Approximation

Local bisection refinement for nn-simplicial grids generated by reflection

A simple local bisection refinement algorithm for the adaptive refinement of $n$-simplicial grids is presented. The algorithm requires that the vertices of each simplex be ordered in a special way relative to those in neighboring simplices. It is proven that certain regular simplicial grids on $[0,1]^n$ have this property, and the more general grids to which this method is applicable are discussed. The edges to be bisected are determined by an ordering of the simplex vertices, without local or global computation or communication. Further, the number of congruency classes in a locally refined grid turns out to be bounded above by $n$, independent of the level of refinement. Simplicial grids of higher dimension are frequently used to approximate solution manifolds of parametrized equations, for instance, as in [W. C. Rheinboldt, Numer. Math., 53 (1988), pp. 165–180] and [E. Allgower and K. Georg, Utilitas Math., 16 (1979), pp. 123–129]. They are also used for the determination of fixed points of functions from ${\bf R}^n$ to ${\bf R}^n$, as described in [M. J. Todd, Lecture Notes in Economic and Mathematical Systems, 124, Springer-Verlag, Berlin, 1976]. In two and three dimensions, such grids of triangles, respectively, tetrahedrons, are used for the computation of finite element solutions of partial differential equations, for example, as in [O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems, Academic Press, Orlando, 1984], [R. E. Bank and B. D. Welfert, SIAM J. Numer. Anal., 28 (1991), pp. 591–623], [W. F. Mitchell, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 146–147], and [M. C. Rivara, J. Comput. Appl. Math., 36 (1991), pp. 79–89]. The new method is applicable to any triangular grid and may possibly be applied to many tetrahedral grids using additional closure refinement to avoid incompatibilities

Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

This paper analyses the following question: let $\mathbf{A}_j$, $j=1,2,$ be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations $\nabla\cdot (A_j \nabla u_j) + k^2 n_j u_j= -f$. How small must $\|A_1 -A_2\|_{L^q}$ and $\|{n_1} - {n_2}\|_{L^q}$ be (in terms of $k$-dependence) for GMRES applied to either $(\mathbf{A}_1)^{-1}\mathbf{A}_2$ or $\mathbf{A}_2(\mathbf{A}_1)^{-1}$ to converge in a $k$-independent number of iterations for arbitrarily large $k$? (In other words, for $\mathbf{A}_1$ to be a good left- or right-preconditioner for $\mathbf{A}_2$?). We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation with random coefficients $A$ and $n$. Such a calculation may require the solution of many deterministic Helmholtz problems, each with different $A$ and $n$, and the answer to the question above dictates to what extent a previously-calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices

Functional quantization and metric entropy for Riemann-Liouville processes

We derive a high-resolution formula for the $L^2$-quantization errors of Riemann-Liouville processes and the sharp Kolmogorov entropy asymptotics for related Sobolev balls. We describe a quantization procedure which leads to asymptotically optimal functional quantizers. Regular variation of the eigenvalues of the covariance operator plays a crucial role