Search results for "Computational Mathematic"
showing 10 items of 987 documents
Grapham: Graphical models with adaptive random walk Metropolis algorithms
2008
Recently developed adaptive Markov chain Monte Carlo (MCMC) methods have been applied successfully to many problems in Bayesian statistics. Grapham is a new open source implementation covering several such methods, with emphasis on graphical models for directed acyclic graphs. The implemented algorithms include the seminal Adaptive Metropolis algorithm adjusting the proposal covariance according to the history of the chain and a Metropolis algorithm adjusting the proposal scale based on the observed acceptance probability. Different variants of the algorithms allow one, for example, to use these two algorithms together, employ delayed rejection and adjust several parameters of the algorithm…
Alignment-free Genomic Analysis via a Big Data Spark Platform
2021
Abstract Motivation Alignment-free distance and similarity functions (AF functions, for short) are a well-established alternative to pairwise and multiple sequence alignments for many genomic, metagenomic and epigenomic tasks. Due to data-intensive applications, the computation of AF functions is a Big Data problem, with the recent literature indicating that the development of fast and scalable algorithms computing AF functions is a high-priority task. Somewhat surprisingly, despite the increasing popularity of Big Data technologies in computational biology, the development of a Big Data platform for those tasks has not been pursued, possibly due to its complexity. Results We fill this impo…
Quantum GestART: Identifying and Applying Correlations between Mathematics, Art, and Perceptual Organization
2020
Mathematics can help analyze the arts and inspire new artwork. Mathematics can also help make transformations from one artistic medium to another, considering exceptions and choices, as well as artists' individual and unique contributions. We propose a method based on diagrammatic thinking and quantum formalism. We exploit decompositions of complex forms into a set of simple shapes, discretization of complex images, and Dirac notation, imagining a world of "prototypes" that can be connected to obtain a fine or coarse-graining approximation of a given visual image. Visual prototypes are exchanged with auditory ones, and the information (position, size) characterizing visual prototypes is con…
Coherent Quantum Tomography
2016
We discuss a quantum mechanical indirect measurement method to recover a position dependent Hamilton matrix from time evolution of coherent quantum mechanical states through an object. A mathematical formulation of this inverse problem leads to weighted X-ray transforms where the weight is a matrix. We show that such X-ray transforms are injective with very rough weights. Consequently, we can solve our quantum mechanical inverse problem in several settings, but many physically relevant problems we pose also remain open. We discuss the physical background of the proposed imaging method in detail. We give a rigorous mathematical treatment of a neutrino tomography method that has been previous…
Free boundary methods and non-scattering phenomena
2021
We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from t…
On differences and similarities in the analysis of Lorenz, Chen, and Lu systems
2015
Currently it is being actively discussed the question of the equivalence of various Lorenz-like systems and the possibility of universal consideration of their behavior (Algaba et al., 2013a,b, 2014b,c; Chen, 2013; Chen and Yang, 2013; Leonov, 2013a), in view of the possibility of reduction of such systems to the same form with the help of various transformations. In the present paper the differences and similarities in the analysis of the Lorenz, the Chen and the Lu systems are discussed. It is shown that the Chen and the Lu systems stimulate the development of new methods for the analysis of chaotic systems. Open problems are discussed.
A fully adaptive wavelet algorithm for parabolic partial differential equations
2001
We present a fully adaptive numerical scheme for the resolution of parabolic equations. It is based on wavelet approximations of functions and operators. Following the numerical analysis in the case of linear equations, we derive a numerical algorithm essentially based on convolution operators that can be efficiently implemented as soon as a natural condition on the space of approximation is satisfied. The algorithm is extended to semi-linear equations with time dependent (adapted) spaces of approximation. Numerical experiments deal with the heat equation as well as the Burgers equation.
Diffusion front capturing schemes for a class of Fokker–Planck equations: Application to the relativistic heat equation
2010
In this research work we introduce and analyze an explicit conservative finite difference scheme to approximate the solution of initial-boundary value problems for a class of limited diffusion Fokker-Planck equations under homogeneous Neumann boundary conditions. We show stability and positivity preserving property under a Courant-Friedrichs-Lewy parabolic time step restriction. We focus on the relativistic heat equation as a model problem of the mentioned limited diffusion Fokker-Planck equations. We analyze its dynamics and observe the presence of a singular flux and an implicit combination of nonlinear effects that include anisotropic diffusion and hyperbolic transport. We present numeri…
Growth of two-dimensional Au patches in graphene pores: A density-functional study
2017
Inspired by recent studies of various two-dimensional (2D) metals such as Au, Fe and Ag, we study the growth of two-dimensional gold patches in graphene pores by density-functional theory. We find that at room temperature gold atoms diffuse readily on top of both graphene and two-dimensional gold with energy barriers less than $0.5$ eV. Furthermore, gold atoms move without barriers from the top of graphene to its edge and from the top of 2D gold to its edge. The energy barriers are absent even at the interface of 2D gold and graphene, so that the gold atoms move effortlessly across the interface. We hope our demonstration for the propensity of diffusing gold atoms to grow 2D gold patches in…
A second-order sparse factorization method for Poisson's equation with mixed boundary conditions
1992
Abstract We propose an algorithm for solving Poisson's equation on general two-dimensional regions with an arbitrary distribution of Dirichlet and Neumann boundary conditions. The algebraic system, generated by the five-point star discretization of the Laplacian, is solved iteratively by repeated direct sparse inversion of an approximating system whose coefficient matrix — the preconditioner — is second-order both in the interior and on the boundary. The present algorithm for mixed boundary value problems generalizes a solver for pure Dirichlet problems (proposed earlier by one of the authors in this journal (1989)) which was found to converge very fast for problems with smooth solutions. T…