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The boundary of the Krein space tracial numerical range, an algebraic approach and a numerical algorithm

Bebiano, N.; Nakazato, H.; Nata, A.; Providência, J. da
Tipo: Pré-impressão
Português
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56.01%
In this article, tracial numerical ranges associated with matrices in an inde nite inner product space are investigated. The boundary equations of these sets are obtained and the case of the boundary being a polygon is studied. As an application, a numerical algorithm for plotting the tracial numerical range of an arbitrary complex matrix, is presented. Our approach uses the elementary idea that the boundary may be traced by computing the supporting lines.

Integração das equações diferenciais do filtro digital de Butterworth mediante algoritmo de quadratura numérica de ordem elevada; Integration of the Butterworth digital filters differential equations using numerical algorithm of high order integrator

Noronha Neto, Celso de Carvalho
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
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46.03%
Neste trabalho se apresenta o desenvolvimento de algoritmos hermitianos de integração das equações diferenciais do filtro digital de Butterworth mediante operadores de integração numérica de ordem elevada com passo único. A teoria do filtro de Butterworth é apresentada mediante o emprego da transformada de Fourier. Exemplos de aplicação apresentados através destes algoritmos mostram que os resultados são, como esperado, mais precisos que os resultantes dos métodos usuais presentes na literatura especializada; In this work is presented the development of hermitian algorithm for integration of the Butterworth digital filters differential equations by means of high order numerical one step operators. The Butterworth filters theory is presented based on the Fourier transform. Numerical examples show that the results of the developed hermitian algorithm are more accurate than the usual methods present in the specialized literature, as expected

A New Numerical Algorithm for Thermoacoustic and Photoacoustic Tomography with Variable Sound Speed

Qian, Jianliang; Stefanov, Plamen; Uhlmann, Gunther; Zhao, Hongkai
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.07%
We present a new algorithm for reconstructing an unknown source in Thermoacoustic and Photoacoustic Tomography based on the recent advances in understanding the theoretical nature of the problem. We work with variable sound speeds that might be also discontinuous across some surface. The latter problem arises in brain imaging. The new algorithm is based on an explicit formula in the form of a Neumann series. We present numerical examples with non-trapping, trapping and piecewise smooth speeds, as well as examples with data on a part of the boundary. These numerical examples demonstrate the robust performance of the new algorithm.

A Numerical Algorithm for Zero Counting. III: Randomization and Condition

Cucker, Felipe; Krick, Teresa; Malajovich, Gregorio; Wschebor, Mario
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
56.03%
In a recent paper (Cucker, Krick, Malajovich and Wschebor, A Numerical Algorithm for Zero Counting. I: Complexity and accuracy, J. Compl.,24:582-605, 2008) we analyzed a numerical algorithm for computing the number of real zeros of a polynomial system. The analysis relied on a condition number kappa(f) for the input system f. In this paper, we look at kappa(f) as a random variable derived from imposing a probability measure on the space of polynomial systems and give bounds for both the tail P{kappa(f) > a} and the expected value E(log kappa(f)).

A boundary integral algorithm for the Laplace Dirichlet-Neumann mixed eigenvalue problem

Akhmetgaliyev, Eldar; Bruno, Oscar; Nigam, Nilima
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.06%
We present a novel integral-equation algorithm for evaluation of Zaremba eigenvalues and eigenfunctions}, that is, eigenvalues and eigenfunctions of the Laplace operator with mixed Dirichlet-Neumann boundary conditions; of course, (slight modifications of) our algorithms are also applicable to the pure Dirichlet and Neumann eigenproblems. Expressing the eigenfunctions by means of an ansatz based on the single layer boundary operator, the Zaremba eigenproblem is transformed into a nonlinear equation for the eigenvalue $\mu$. For smooth domains the singular structure at Dirichlet-Neumann junctions is incorporated as part of our corresponding numerical algorithm---which otherwise relies on use of the cosine change of variables, trigonometric polynomials and, to avoid the Gibbs phenomenon that would arise from the solution singularities, the Fourier Continuation method (FC). The resulting numerical algorithm converges with high order accuracy without recourse to use of meshes finer than those resulting from the cosine transformation. For non-smooth (Lipschitz) domains, in turn, an alternative algorithm is presented which achieves high-order accuracy on the basis of graded meshes. In either case, smooth or Lipschitz boundary, eigenvalues are evaluated by searching for zero minimal singular values of a suitably stabilized discrete version of the single layer operator mentioned above. (The stabilization technique is used to enable robust non-local zero searches.) The resulting methods...

A new numerical method for inverse Laplace transforms used to obtain gluon distributions from the proton structure function

Block, Martin M.; Durand, Loyal
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.1%
We recently derived a very accurate and fast new algorithm for numerically inverting the Laplace transforms needed to obtain gluon distributions from the proton structure function $F_2^{\gamma p}(x,Q^2)$. We numerically inverted the function $g(s)$, $s$ being the variable in Laplace space, to $G(v)$, where $v$ is the variable in ordinary space. We have since discovered that the algorithm does not work if $g(s)\rightarrow 0$ less rapidly than $1/s$ as $s\rightarrow\infty$, e.g., as $1/s^\beta$ for $0<\beta<1$. In this note, we derive a new numerical algorithm for such cases, which holds for all positive and non-integer negative values of $\beta$. The new algorithm is {\em exact} if the original function $G(v)$ is given by the product of a power $v^{\beta-1}$ and a polynomial in $v$. We test the algorithm numerically for very small positive $\beta$, $\beta=10^{-6}$ obtaining numerical results that imitate the Dirac delta function $\delta(v)$. We also devolve the published MSTW2008LO gluon distribution at virtuality $Q^2=5$ GeV$^2$ down to the lower virtuality $Q^2=1.69$ GeV$^2$. For devolution, $\beta$ is negative, giving rise to inverse Laplace transforms that are distributions and not proper functions. This requires us to introduce the concept of Hadamard Finite Part integrals...

A numerical algorithm for a class of BSDEs via branching process

Henry-Labordere, Pierre; Tan, Xiaolu; Touzi, Nizar
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
56.04%
We generalize the algorithm for semi-linear parabolic PDEs in Henry-Labord\ere (2012) to the non-Markovian case for a class of Backward SDEs (BSDEs). By simulating the branching process, the algorithm does not need any backward regression. To prove that the numerical algorithm converges to the solution of BSDEs, we use the notion of viscosity solution of path dependent PDEs introduced by Ekren, Keller, Touzi and Zhang (2012) and extended in Ekren, Touzi and Zhang (2013).; Comment: 31 pages

Harmonic Shears and Numerical Conformal Mappings

Quach, Tri
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.01%
In this article we introduce a numerical algorithm for finding harmonic mappings by using the shear construction introduced by Clunie and Sheil-Small in 1984. The MATLAB implementation of the algorithm is based on the numerical conformal mapping package, the Schwarz-Christoffel toolbox, by T. Driscoll. Several numerical examples are given. In addition, we discuss briefly the minimal surfaces associated with harmonic mappings and give a numerical example of minimal surfaces.; Comment: 15 pages, 6 figures

A numerical algorithm for $L_2$ semi-discrete optimal transport in 3D

Levy, Bruno
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.06%
This paper introduces a numerical algorithm to compute the $L_2$ optimal transport map between two measures $\mu$ and $\nu$, where $\mu$ derives from a density $\rho$ defined as a piecewise linear function (supported by a tetrahedral mesh), and where $\nu$ is a sum of Dirac masses. I first give an elementary presentation of some known results on optimal transport and then observe a relation with another problem (optimal sampling). This relation gives simple arguments to study the objective functions that characterize both problems. I then propose a practical algorithm to compute the optimal transport map between a piecewise linear density and a sum of Dirac masses in 3D. In this semi-discrete setting, Aurenhammer et.al [\emph{8th Symposium on Computational Geometry conf. proc.}, ACM (1992)] showed that the optimal transport map is determined by the weights of a power diagram. The optimal weights are computed by minimizing a convex objective function with a quasi-Newton method. To evaluate the value and gradient of this objective function, I propose an efficient and robust algorithm, that computes at each iteration the intersection between a power diagram and the tetrahedral mesh that defines the measure $\mu$. The numerical algorithm is experimented and evaluated on several datasets...

On convergence of numerical algorithm of a class of the spatial segregation of reaction-diffusion system with two population densities

Arakelyan, Avetik
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
56%
Recently, much interest has gained the numerical approximation of equations of the Spatial Segregation of Reaction-diffusion systems with m population densities. These problems are governed by a minimization problem subject to the closed but non-convex set. In the present work we deal with the numerical approximation of equations of stationary states for a certain class of the Spatial Segregation of Reaction-diffusion system with two population densities having disjoint support. We prove the convergence of the numerical algorithm for two competing populations with non-negative internal dynamics $f_i(x)\geq 0.$ At the end of the paper we present computational tests.; Comment: 13 pages, 8 figures, Free boundary, Two-phase membrane problem, Reaction-diffusion systems, Finite difference

A Stable Numerical Algorithm for the Brinkman Equations by Weak Galerkin Finite Element Methods

Mu, Lin; Wang, Junping; Ye, Xiu
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.01%
This paper presents a stable numerical algorithm for the Brinkman equations by using weak Galerkin (WG) finite element methods. The Brinkman equations can be viewed mathematically as a combination of the Stokes and Darcy equations which model fluid flow in a multi-physics environment, such as flow in complex porous media with a permeability coefficient highly varying in the simulation domain. In such applications, the flow is dominated by Darcy in some regions and by Stokes in others. It is well known that the usual Stokes stable elements do not work well for Darcy flow and vise versa. The challenge of this study is on the design of numerical schemes which are stable for both the Stokes and the Darcy equations. This paper shows that the WG finite element method is capable of meeting this challenge by providing a numerical scheme that is stable and accurate for both Darcy and the Stokes dominated flows. Error estimates of optimal order are established for the corresponding WG finite element solutions. The paper also presents some numerical experiments that demonstrate the robustness, reliability, flexibility and accuracy of the WG method for the Brinkman equations.; Comment: 20 pages, 21 plots and figures

Particle velocity based universal algorithm for numerical simulation of hydraulic fractures

Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.1%
In the paper, we propose a new effective mathematical formulation and resulting universal numerical algorithm capable of tackling various HF models in the framework of a unified approach. The presented numerical scheme is not limited to any particular elasticity model or crack propagation regime. Its basic assumptions are: i) proper choice of independent and dependent variables (with the direct utilization of a new one - the reduced particle velocity), ii) tracing the fracture front by use of the speed equation which can be integrated in a closed form and sets an explicit relation between the crack propagation speed and the coefficients in the asymptotic expansion of the crack opening, iii) proper regularization techniques, iv) improved temporal approximation, v) modular algorithm architecture. The application of the new dependent variable, the reduced particle velocity, instead of the usual fluid flow rate, facilitates the computation of the crack propagation speed from the local relation based on the speed equation. As a result, the position of the crack front is accurately determined from an explicit formula derived from the speed equation. The underlying ideas employed in the algorithm are combined together producing a robust and efficient numerical scheme. Its performance is demonstrated using classical examples of 1D models for hydraulic fracturing: PKN and KGD under various fracture propagation regimes. Solution accuracy is verified against dedicated analytical benchmarks and other solutions available in the literature. Most of the ideas developed here...

Revisiting the method of characteristics via a convex hull algorithm

LeFloch, Philippe G.; Mercier, Jean-Marc
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.04%
We revisit the method of characteristics for shock wave solutions to nonlinear hyperbolic problems and we describe a novel numerical algorithm - the convex hull algorithm (CHA) - in order to compute, both, entropy dissipative solutions (satisfying all relevant entropy inequalities) and entropy conservative (or multivalued) solutions to nonlinear hyperbolic conservation laws. Our method also applies to Hamilton-Jacobi equations and other problems endowed with a method of characteristics. From the multivalued solutions determined by the method of characteristic, our algorithm "extracts" the entropy dissipative solutions, even after the formation of shocks. It applies to, both, convex or non-convex flux/Hamiltonians. We demonstrate the relevance of the proposed approach with a variety of numerical tests including a problem from fluid dynamics.; Comment: 14 pages

A conservation formulation and a numerical algorithm for the double-gyre nonlinear shallow-water model

Kuang, Dongyang; Lee, Long
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
56.08%
We present a conservation formulation and a numerical algorithm for the reduced-gravity shallow-water equations on a beta plane, subjected to a constant wind forcing that leads to the formation of double-gyre circulation in a closed ocean basin. The novelty of the paper is that we reformulate the governing equations into a nonlinear hyperbolic conservation law plus source terms. A second-order fractional-step algorithm is used to solve the reformulated equations. In the first step of the fractional-step algorithm, we solve the homogeneous hyperbolic shallow-water equations by the wave-propagation finite volume method. The resulting intermediate solution is then used as the initial condition for the initial-boundary value problem in the second step. As a result, the proposed method is not sensitive to the choice of viscosity and gives high-resolution results for coarse grids, as long as the Rossby deformation radius is resolved. We discuss the boundary conditions in each step, when no-slip boundary conditions are imposed to the problem. We validate the algorithm by a periodic flow on an f-plane with exact solutions. The order-of-accuracy for the proposed algorithm is tested numerically. We illustrate a quasi-steady-state solution of the double-gyre model via the height anomaly and the contour of stream function for the formation of double-gyre circulation in a closed basin. Our calculations are highly consistent with the results reported in the literature. Finally...

A numerical algorithm for singular optimal LQ control systems

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.1%
A numerical algorithm to obtain the consistent conditions satisfied by singular arcs for singular linear-quadratic optimal control problems is presented. The algorithm is based on the presymplectic constraint algorithm (PCA) by Gotay-Nester \cite{Go78,Vo99} that allows to solve presymplectic hamiltonian systems and that provides a geometrical framework to the Dirac-Bergmann theory of constraints for singular Lagrangian systems \cite{Di49}. The numerical implementation of the algorithm is based on the singular value decomposition that, on each step allows to construct a semi-explicit system. Several examples and experiments are discussed, among them a family of arbitrary large singular LQ systems with index 3 and a family of examples of arbitrary large index, all of them exhibiting stable behaviour.; Comment: An old paper (2009) posted for archival purposes

An efficient numerical algorithm for the L2 optimal transport problem with applications to image processing

Saumier, Louis-Philippe; Agueh, Martial; Khouider, Boualem
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.14%
We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method is based on a numerical resolution of the corresponding Monge-Amp\ere equation. We extend the damped Newton algorithm of Loeper and Rapetti \cite{LR} to the more general case of a non uniform density which is relevant to the optimal transport problem, and we show that our algorithm converges for sufficiently large damping coefficients. The main idea consists of designing an iterative scheme where the fully nonlinear equation is approximated by a non-constant coefficient linear elliptic PDE that we solve numerically. We introduce several improvements and some new techniques for the numerical resolution of the corresponding linear system. Namely, we use a Fast Fourier Transform (FFT) method by Strain \cite{St}, which allows to increase the efficiency of our algorithm against the standard finite difference method. Moreover, we use a fourth order finite difference scheme to approximate the partial derivatives involved in the nonlinear terms of the Newton algorithm, which are evaluated once at each iteration; this leads to a significant improvement of the accuracy of the method...

A Numerical Algorithm for Zero Counting. I: Complexity and Accuracy

Cucker, Felipe; Krick, Teresa; Malajovich, Gregorio; Wschebor, Mario
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.08%
We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials' degree, and kappa(f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a polynomial bound for the precision required to ensure the returned output is correct is exhibited. This bound is a major feature of our algorithm since it is in contrast with the exponential precision required by the existing (symbolic) algorithms for counting real zeros. The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time with an exponential number of processors.; Comment: We made minor but necessary improvements in the presentation

Physical Formulation and Numerical Algorithm for Simulating N Immiscible Incompressible Fluids Involving General Order Parameters

Dong, Suchuan
Tipo: Artigo de Revista Científica
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56.03%
We present a physical formulation, and a numerical algorithm, based on a class of general order parameters for simulating the motion of a mixture of $N$ ($N\geqslant 2$) immiscible incompressible fluids with given densities, dynamic viscosities, and pairwise surface tensions. The introduction of general order parameters leads to a more strongly coupled system of phase field equations, in contrast to that with certain special choice of the order parameters. However, the general form enables one to compute the N-phase mixing energy density coefficients in an explicit fashion in terms of the pairwise surface tensions. From the simulation perspective, the increased complexity in the form of the phase field equations with general order parameters in actuality does not cause essential computational difficulties. Our numerical algorithm reformulates the ($N-1$) strongly-coupled phase field equations for general order parameters into $2(N-1)$ Helmholtz-type equations that are completely de-coupled from one another, leading to a computational complexity essentially the same as that of the simpler phase field equations associated with special choice of order parameters. We demonstrate the capabilities of the method developed herein using several test problems involving multiple fluid phases and large contrasts in densities and viscosities among the multitude of fluids. In particular...

Second-order hyperbolic Fuchsian systems. Gowdy spacetimes and the Fuchsian numerical algorithm

Beyer, Florian; LeFloch, Philippe G.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.03%
This is the second part of a series devoted to the singular initial value problem for second-order hyperbolic Fuchsian systems. In the first part, we defined and investigated this general class of systems, and we established a well-posedness theory in weighted Sobolev spaces. This theory is applied here to the vacuum Einstein equations for Gowdy spacetimes admitting, by definition, two Killing fields satisfying certain geometric conditions. We recover, by more direct and simpler arguments, the well-posedness results established earlier by Rendall and collaborators. In addition, in this paper we introduce a natural approximation scheme, which we refer to as the Fuchsian numerical algorithm and is directly motivated by our general theory. This algorithm provides highly accurate, numerical approximations of the solution to the singular initial value problem. In particular, for the class of Gowdy spacetimes under consideration, various numerical experiments are presented which show the interest and efficiency of the proposed method. Finally, as an application, we numerically construct Gowdy spacetimes containing a smooth, incomplete, non-compact Cauchy horizon.; Comment: 22 pages. A shortened version is included in: F. Beyer and P.G. LeFloch...

Numerical investigation into the existence of limit cycles in two-dimensional predator-prey systems

van der Hoff,Quay; Greeff,Johanna C.; Kloppers,P Hendrik
Fonte: South African Journal of Science Publicador: South African Journal of Science
Tipo: Artigo de Revista Científica Formato: text/html