The Vlasov equation is a kinetic model describing the evolution of charged
particles, and is coupled with Poisson's equation, which rules the evolution of
the self-consistent electric field. In this paper, we introduce a new class of
forward Semi-Lagrangian schemes for the Vlasov-Poisson system based on a Cauchy
Kovalevsky (CK) procedure for the numerical solution of the characteristic
curves. Exact conservation properties of the first moments of the distribution
function for the schemes are derived and a convergence study is performed that
applies as well for the CK scheme as for a more classical Verlet scheme. The
convergence in L1 norm of the schemes is proved and error estimates are
obtained.
jueves, 31 de diciembre de 2009
Analysis of a new class of Forward Semi-Lagrangian schemes for the 1D Vlasov-Poisson Equations. (arXiv:0912.4952v1 [math.NA])
Periodic reordering
For many networks in nature, science and technology, it is possible to order the nodes so that most links are short-range, connecting near-neighbours, and relatively few long-range links, or shortcuts, are present. Given a network as a set of observed links (interactions), the task of finding an ordering of the nodes that reveals such a range-dependent structure is closely related to some sparse matrix reordering problems arising in scientific computation. The spectral, or Fiedler vector, approach for sparse matrix reordering has successfully been applied to biological data sets, revealing useful structures and subpatterns. In this work we argue that a periodic analogue of the standard reordering task is also highly relevant. Here, rather than encouraging nonzeros only to lie close to the diagonal of a suitably ordered adjacency matrix, we also allow them to inhabit the off-diagonal corners. Indeed, for the classic small-world model of Watts & Strogatz (1998, Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442) this type of periodic structure is inherent. We therefore devise and test a new spectral algorithm for periodic reordering. By generalizing the range-dependent random graph class of Grindrod (2002, Range-dependent random graphs and their application to modeling large small-world proteome datasets. Phys. Rev. E, 66, 066702-1–066702-7) to the periodic case, we can also construct a computable likelihood ratio that suggests whether a given network is inherently linear or periodic. Tests on synthetic data show that the new algorithm can detect periodic structure, even in the presence of noise. Further experiments on real biological data sets then show that some networks are better regarded as periodic than linear. Hence, we find both qualitative (reordered networks plots) and quantitative (likelihood ratios) evidence of periodicity in biological networks.
miércoles, 30 de diciembre de 2009
A modified Newton-Jarratt’s composition
Abstract A reduced composition technique has been used on Newton and Jarratt’s methods in order to obtain an optimal relation between
convergence order, functional evaluations and number of operations. Following this aim, a family of methods is obtained whose
efficiency indices are proved to be better for systems of nonlinear equations.
convergence order, functional evaluations and number of operations. Following this aim, a family of methods is obtained whose
efficiency indices are proved to be better for systems of nonlinear equations.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-009-9359-z
- Authors
- Alicia Cordero, Universidad Politécnica de Valencia Instituto de Matemática Multidisciplinar Camino de Vera, s/n 46022 Valencia Spain
- José L. Hueso, Universidad Politécnica de Valencia Instituto de Matemática Multidisciplinar Camino de Vera, s/n 46022 Valencia Spain
- Eulalia Martínez, Universidad Politécnica de Valencia Instituto de Matemática Pura y Aplicada Camino de Vera, s/n 46022 Valencia Spain
- Juan R. Torregrosa, Universidad Politécnica de Valencia Instituto de Matemática Multidisciplinar Camino de Vera, s/n 46022 Valencia Spain
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
Greedy Drawings of Triangulations
Abstract
Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination
at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and
Ratajczak (Theor. Comput. Sci. 344(1):3–14, 2005) came up with the following conjecture:
Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination
at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and
Ratajczak (Theor. Comput. Sci. 344(1):3–14, 2005) came up with the following conjecture:
Any 3-connected planar graph can be drawn in the plane such that for every pair of vertices s and t a distance decreasing path can be found. A path s=v
1,v
2,…,v
k
=t in a drawing is said to be distance decreasing if ‖v
i
−t‖<‖v
i−1−t‖,2≤i≤k where ‖…‖ denotes the Euclidean distance.
We settle this conjecture in the affirmative for the case of triangulations.
A partitioning of the edges of a triangulation G into 3 trees, called the realizer of G, was first developed by Schnyder who also gave a drawing algorithm based on this. We generalize Schnyder’s algorithm to obtain
a whole class of drawings of any given triangulation G. We show, using the Knaster–Kuratowski–Mazurkiewicz Theorem, that some drawing of G belonging to this class is greedy.
a whole class of drawings of any given triangulation G. We show, using the Knaster–Kuratowski–Mazurkiewicz Theorem, that some drawing of G belonging to this class is greedy.
- Content Type Journal Article
- DOI 10.1007/s00454-009-9235-6
- Authors
- Raghavan Dhandapani, New York University Courant Institute of Mathematical Sciences New York NY 10012 USA
- Journal Discrete and Computational Geometry
- Online ISSN 1432-0444
- Print ISSN 0179-5376
Nonspherical Noble Metal Nanoparticles: Colloid-Chemical Synthesis and Morphology Control
Metal nanoparticles have been the subject of widespread research over the past two decades. In recent years, noble metals have been the focus of numerous studies involving synthesis, characterization, and applications. Synthesis of an impressive range of noble metal nanoparticles with varied morphologies has been reported. Researchers have made a great progress in learning how to engineer materials on a nanometer length scale that has led to the understanding of the fundamental size- and shape-dependent properties of matter and to devising of new applications. In this article, we review the recent progress in the colloid-chemical synthesis of nonspherical nanoparticles of a few important noble metals (mainly Ag, Au, Pd, and Pt), highlighting the factors that influence the particle morphology and discussing the mechanisms behind the nonspherical shape evolution. The article attempts to present a thorough discussion of the basic principles as well as state-of-the-art morphology control in noble metal nanoparticles.
martes, 29 de diciembre de 2009
Space-time finite element computation of complex fluid-structure interactions
New special fluid-structure interaction (FSI) techniques, supplementing the ones developed earlier, are employed with the Stabilized Space-Time FSI (SSTFSI) technique. The new special techniques include improved ways of calculating the equivalent fabric porosity in Homogenized Modeling of Geometric Porosity (HMGP), improved ways of building a starting point in FSI computations, ways of accounting for fluid forces acting on structural components that are not expected to influence the flow, adaptive HMGP, and multiscale sequentially coupled FSI techniques. While FSI modeling of complex parachutes was the motivation behind developing some of these techniques, they are also applicable to other classes of complex FSI problems. We also present new ideas to increase the scope of our FSI and CFD techniques. Copyright © 2009 John Wiley & Sons, Ltd.
A new finite volume method on junction coupling and boundary treatment for flow network system analyses
To adequately analyze the flow in a pipe or duct network system, traditional node-based junction coupling methods require junction losses, which are specified by empirical or analytic correlations. In this paper, a new finite volume junction coupling method using a ghost junction cell is developed by considering the interchange of linear momentum as well as the important wall effect at the junction without requiring any correlation on the junction loss. Also, boundary treatment is modified to preserve the stagnation enthalpy across boundaries, such as the pipe end and the interface between the junction and the branch.The computational accuracy and efficiency of Godunov-type finite volume schemes are investigated by tracing the total mechanical energy of rapid transients due to sudden closure of a valve at the downstream end. Among the approximate Riemann solvers, the proposed RoeM scheme turns out to be more suitable for finite volume junction treatment than the original Roe's approximate Riemann solver because of conservation of the stagnation enthalpy across the geometric discontinuity. From the viewpoint of computational cost, the implicit LU-SGS time integration is appropriate for steady and slow transients, while the explicit third-order TVD Runge-Kutta scheme is advantageous for rapid transients. Copyright © 2009 John Wiley & Sons, Ltd.
A fast universal solver for 1D continuous and discontinuous steady flows in rivers and pipes
Simulation of 1D steady flow covers a wide range of practical applications, such as rivers, pipes and hydraulic structures. Various flow patterns coexist in such situations: free surface flows (supercritical, subcritical and hydraulic jump), pressurized flows as well as mixed flows. As a result, development of a unified 1D model for all the situations of interest in civil engineering remains challenging. In this paper, a fast universal solver for 1D continuous and discontinuous steady flows in rivers and pipes is set up and assessed. Developments are initiated from an original unified mathematical model using the Saint-Venant equations. Application of these equations, originally dedicated to free-surface flow, is extended to pressurized flow by means of the Preissmann slot model. In particular, an original negative slot is developed in order to handle sub-atmospheric pressurized flow. Next, the full unsteady model is simplified under the assumption of steadiness and reformulated into a single pseudo-unsteady differential equation. The derived pseudo-unsteady formulation aims at keeping the hyperbolic feature of the equation. Stability analysis of the differential equation suggests a unique splitting for the finite volume scheme whatever the flow conditions. The numerical scheme obtained is a universal Flux Vector Splitting which shows robustness and simplicity. Accuracy and performance of the new methodology is assessed by comparison with analytical and experimental results. Copyright © 2009 John Wiley & Sons, Ltd.
Residual-based variational multiscale methods for laminar, transitional and turbulent variable-density flow at low Mach number
In the present study, residual-based variational multiscale methods are developed for and applied to variable-density flow at low Mach number. In particular, two different formulations are considered in this study: a standard stabilized formulation featuring SUPG/PSG/grad-div terms and a complete residual-based variational multiscale formulation additionally containing cross- and Reynolds-stress terms as well as subgrid-scale velocity terms in the energy-conservation equation. The proposed methods are tested for various laminar flow test cases as well as a test case at laminar via transitional to turbulent flow stages. Stable and accurate results are obtained for all numerical examples. Substantial differences in the results between the two approaches do not become notable until a high temperature gradient is applied and the flow reaches a turbulent flow stage. The more pronounced influence of adding subgrid-scale velocity terms to the energy-conservation equation on the results than adding analogous terms to the momentum-conservation equation in this situation appears particularly noteworthy. Copyright © 2009 John Wiley & Sons, Ltd.
Equation-free model reduction for complex dynamical systems
This paper presents a reduced model strategy for simulation of complex physical systems. A classical reduced basis is first constructed relying on proper orthogonal decomposition of the system. Then, unlike the alternative approaches, such as Galerkin projection schemes for instance, an equation-free reduced model is constructed. It consists in the determination of an explicit transformation, or mapping, for the evolution over a coarse time-step of the projection coefficients of the system state on the reduced basis. The mapping is expressed as an explicit polynomial transformation of the projection coefficients and is computed once and for all in a pre-processing stage using the detailed model equation of the system. The reduced system can then be advanced in time by successive applications of the mapping. The CPU cost of the method lies essentially in the mapping approximation which is performed offline, in a parallel fashion, and only once. Subsequent application of the mapping to perform a time-integration is carried out at a low cost thanks to its explicit character.Application of the method is considered for the 2-D flow around a circular cylinder. We investigate the effectiveness of the reduced model in rendering the dynamics for both asymptotic state and transient stages. It is shown that the method leads to a stable and accurate time-integration for only a fraction of the cost of a detailed simulation, provided that the mapping is properly approximated and the reduced basis remains relevant for the dynamics investigated. Copyright © 2009 John Wiley & Sons, Ltd.
Atomistic analysis of B clustering and mobility degradation in highly B-doped junctions
In this paper we discuss from an atomistic point of view some of the issues involved in the modeling of electrical characteristics evolution in silicon devices as a result of ion implantation and annealing processes in silicon. In particular, evolution of electrically active dose, sheet resistance and hole mobility has been investigated for high B concentration profiles in pre-amorphized Si. For this purpose, Hall measurements combined with atomistic kinetic Monte Carlo atomistic simulations have been performed. An apparent anomalous behavior has been observed for the evolution of the active dose and the sheet resistance, in contrast to opposite trend evolutions reported previously. Our results indicate that this anomalous behavior is due to large variations in hole mobility with active dopant concentration, much larger than that associated to the classical dependence of hole mobility with carrier concentration. Simulations suggest that hole mobility is significantly degraded by the presence of a large concentration of boron-interstitial clusters, indicating the existence of an additional scattering mechanism. Copyright © 2009 John Wiley & Sons, Ltd.
lunes, 28 de diciembre de 2009
Some Basic Difference Equations of Schrödinger Boundary Value Problems
We consider special basic difference equations which are related to discretizations of Schrödinger equations on time scales with special symmetry properties, namely, so-called basic discrete grids. These grids are of an adaptive grid type. Solving the boundary value problem of suitable Schrödinger equations on these grids leads to completely new and unexpected analytic properties of the underlying function spaces. Some of them are presented in this work.
A single-node characteristic collocation method for unsteady-state convection-diffusion equations in three-dimensional spaces
We develop a nonconventional single-node characteristic collocation method with piecewise-cubic Hermite polynomials for the numerical simulation to unsteady-state advection-diffusion transport partial differential equations. This method greatly reduces the number of unknowns in the conventional collocation method, and generates accurate numerical solutions even if very large time steps are taken. The reduction of number of nodes has great potential for problems defined on high space dimensions, which appears in such problems as quantification of uncertainties in subsurface porous media. The method developed here is easy to formulate. Numerical experiments are presented to show the strong potential of the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010
Explicit integration of bounding surface model for the analysis of earthquake soil liquefaction
This paper presents a new plasticity model developed for the simulation of monotonic and cyclic loading of non-cohesive soils and its implementation to the commercial finite-difference code FLAC, using its User-Defined-Model (UDM) capability. The new model incorporates the framework of Critical State Soil Mechanics, while it relies upon bounding surface plasticity with a vanished elastic region to simulate the non-linear soil response. Stress integration of constitutive relations is performed using a recently proposed explicit scheme with automatic error control and substepping, which so far has been employed in the literature only for constitutive models aiming at monotonic loading. The overall accuracy of this scheme is evaluated at element level by simulating cyclic loading along complex stress paths and by using iso-error maps for paths involving change of the Lode angle. The performance of the new constitutive model and its stress integration scheme in complex boundary value problems involving earthquake-induced liquefaction is evaluated, in terms of accuracy and computational cost, via a number of parametric analyses inspired by the successful simulation of the VELACS centrifuge Model Test No. 2 studying the lateral spreading response of a liquefied sand layer. Copyright © 2009 John Wiley & Sons, Ltd.
A partition of unity-based 'FE-Meshfree' QUAD4 element for geometric non-linear analysis
The recently published 'FE-Meshfree' QUAD4 element is extended to geometrical non-linear analysis. The shape functions for this element are obtained by combining meshfree and finite element shape functions. The concept of partition of unity (PU) is employed for the purpose. The new shape functions inherit their higher order completeness properties from the meshfree shape functions and the mesh-distortion tolerant compatibility properties from the finite element (FE) shape functions. Updated Lagrangian formulation is adopted for the non-linear solution. Several numerical example problems are solved and the performance of the element is compared with that of the well-known Q4, QM6 and Q8 elements. The results show that, for regular meshes, the performance of the element is comparable to that of QM6 and Q8 elements, and superior to that of Q4 element. For distorted meshes, the present element has better mesh-distortion tolerance than Q4, QM6 and Q8 elements. Copyright © 2009 John Wiley & Sons, Ltd.
Reduction in drag and vortex shedding frequency through porous sheath around a circular cylinder
A numerical study on the laminar vortex shedding and wake flow due to a porous-wrapped solid circular cylinder has been made in this paper. The cylinder is horizontally placed, and is subjected to a uniform cross flow. The aim is to control the vortex shedding and drag force through a thin porous wrapper around a solid cylinder. The flow field is investigated for a wide range of Reynolds number in the laminar regime. The flow in the porous zone is governed by the Darcy-Brinkman-Forchheimer extended model and the Navier-Stokes equations in the fluid region. A control volume approach is adopted for computation of the governing equations along with a second-order upwind scheme, which is used to discretize the convective terms inside the fluid region. The inclusion of a thin porous wrapper produces a significant reduction in drag and damps the oscillation compared with a solid cylinder. Dependence of Strouhal number and drag coefficient on porous layer thickness at different Reynolds number is analyzed. The dependence of Strouhal number and drag on the permeability of the medium is also examined. Copyright © 2009 John Wiley & Sons, Ltd.
Discretizing two-dimensional complex fractured fields for incompressible two-phase flow
A method is introduced to discretize irregular and complex two-dimensional fractured media. The geometry of the fractured media is first analysed by searching and treating the complex configurations. Based on that, the method generated a good mesh quality and allows for including finer grids. An incompressible two-phase flow problem is solved to compare the developed method and a public method based on the approximation of a 1D fracture by the edges of a 2D finite element grid of the porous media. The comparison showed that the developed method (i) represents better the fractured domain by maintaining the geometric integrity of input surfaces and geologic data, (ii) provides, for sample and complex fractured domains, excellent and more accurate results, and (iii) is much less sensitive to the grid sizes. Furthermore, the method has to be more efficient than the other methods for transport problems and has to provide better predictable results; this is mainly based on point (ii) and because the method produces optimal triangular grids. Copyright © 2009 John Wiley & Sons, Ltd.
The characteristic-based split (CBS) meshfree method for free surface flow problems in ALE formulation
A semi-implicit characteristic-based split (CBS) meshfree algorithm in the arbitrary Lagrangian Eulerian (ALE) framework is proposed for the numerical solution of incompressible free surface flow problem in the paper. The algorithm is the extension of general CBS method which was initially introduced in finite element framework, this is due to the fact that CBS method not only can enhance the stability, but also avoid LBB condition when equal order basis function is used to approximate velocity and pressure variables. Meanwhile, a simple way for node update and node speed calculation is developed which is used to capture the free surface exactly. The numerical solutions are compared with available analytical and numerical solutions, which shows that the proposed method has better ability to simulate the free surface incompressible flow problem. Copyright © 2009 John Wiley & Sons, Ltd.
Cell face velocity alternatives in a structured colocated grid for the unsteady Navier-Stokes equations
The use of a colocated variable arrangement for the numerical solution of fluid flow is becoming more and more popular due to its coding simplicity. The inherent decoupling of the pressure and velocity fields in this arrangement can be handled via a special interpolation procedure for the calculation of the cell face velocity named pressure-weighted interpolation method (PWIM) (AIAA J. 1983; 21(11):1525-1532). In this paper a discussion on the alternatives to extend PWIM to unsteady flows is presented along with a very simple criterion to ascertain if a given interpolation practice will produce steady results that are relaxation dependent or time step dependent. Following this criterion it will be shown that some prior schemes presented as time step independent are actually not, although by using special interpolations can be readily adapted to be. A systematic way of deriving different cell face velocity expressions will be presented and new formulae free of [Delta]t dependence will be derived. Several computational exercises will accompany the theoretical discussion to support our claims. Copyright © 2009 John Wiley & Sons, Ltd.
Numerical study of electromagnetic wave propagation through layered structures with chiral media
In this paper, the characteristics of layered structures (photonic or electromagnetic bandgaps), including chiral media, are studied by means of two different numerical methods, one in the time domain (finite differences in the time domain, FDTD) and the other in the frequency domain (coupled-mode method, CMM). The results (reflection and transmission coefficients for a plane wave normally incident over a layered structure) obtained by means of both well different techniques are practically identical. Copyright © 2009 John Wiley & Sons, Ltd.
A new algorithm for the general quadratic programming problems with box constraints
Abstract In this paper, we propose a new branch-and-bound algorithm for the general quadratic problems with box constraints. We, first,
transform the problem into a separable form by D. C. decomposition and Cholesky factorization of a positive definite matrix.
Then a lower bounding technique is derived and a branch-and-bound algorithm is presented based on the lower bounding and rectangular
bisection. Finally, preliminary computational results are reported.
transform the problem into a separable form by D. C. decomposition and Cholesky factorization of a positive definite matrix.
Then a lower bounding technique is derived and a branch-and-bound algorithm is presented based on the lower bounding and rectangular
bisection. Finally, preliminary computational results are reported.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-009-9358-0
- Authors
- Jianling Li, Guangxi University College of Mathematics and Information Science Nanning Guangxi 530004 China
- Peng Wang, Haikou College of Economics Department of Basic Course Wentan Road 2 Haikou Hainan 570203 China
- Lin Ma, Guangxi University College of Mathematics and Information Science Nanning Guangxi 530004 China
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
Compactly supported positive definite radial functions
Abstract We provide criteria for positive definiteness of radial functions with compact support. Based on these criteria we will produce
a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest
ones arecut-off polynomials, which consist of a single polynomial piece on [0, 1] and vanish on [1, ∞). More precisely, for any given dimensionn and prescribedC
k
smoothness, there is a function inC
k
(ℝ
n
), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean
distance. Another example is derived from odd-degreeB-splines.
a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest
ones arecut-off polynomials, which consist of a single polynomial piece on [0, 1] and vanish on [1, ∞). More precisely, for any given dimensionn and prescribedC
k
smoothness, there is a function inC
k
(ℝ
n
), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean
distance. Another example is derived from odd-degreeB-splines.
- Content Type Journal Article
- DOI 10.1007/BF03177517
- Authors
- Zongmin Wu, Fudan University Department of Mathematics 200433 Shanghai China
- Journal Advances in Computational Mathematics
- Online ISSN 1572-9044
- Print ISSN 1019-7168
- Journal Volume Volume 4
- Journal Issue Volume 4, Number 1 / December, 1995
On multivariate Hermite interpolation
Abstract We study the problem of Hermite interpolation by polynomials in several variables. A very general definition of Hermite interpolation
is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and
some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced
in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral
remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions.
is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and
some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced
in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral
remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions.
- Content Type Journal Article
- DOI 10.1007/BF03177515
- Authors
- Thomas Sauer, University Erlangen-Nuremberg Mathematical Institute D-91054 Erlangen Germany
- Yuan Xu, University of Oregon Department of Mathematics 97403 Eugene OR USA
- Journal Advances in Computational Mathematics
- Online ISSN 1572-9044
- Print ISSN 1019-7168
- Journal Volume Volume 4
- Journal Issue Volume 4, Number 1 / December, 1995
sábado, 26 de diciembre de 2009
Computed Chaos or Numerical Errors. (arXiv:math/0305212v2 [math.DS] UPDATED)
Discrete numerical methods with finite time-steps represent a practical
technique to solve initial-value problems involving nonlinear differential
equations. These methods seem particularly useful to the study of chaos since
no analytical chaotic solution is currently available. Using the well-known
Lorenz equations as an example, it is demonstrated that numerically computed
results and their associated statistical properties are time-step dependent.
There are two reasons for this behavior. First, chaotic differential equations
are unstable so that any small error is amplified exponentially near an
unstable manifold. The more serious and lesser-known reason is that stable and
unstable manifolds of singular points associated with differential equations
can form virtual separatrices. The existence of a virtual separatrix presents
the possibility of a computed trajectory actually jumping through it due to the
finite time-steps of discrete numerical methods. Such behavior violates the
uniqueness theory of differential equations and amplifies the numerical errors
explosively. These reasons imply that, even if computed results are bounded,
their independence on time-step should be established before accepting them as
useful numerical approximations to the true solution of the differential
equations. However, due to these exponential and explosive amplifications of
numerical errors, no computed chaotic solutions of differential equations
independent of integration-time step have been found. Thus, reports of computed
non-periodic solutions of chaotic differential equations are simply
consequences of unstably amplified truncation errors, and are not approximate
solutions of the associated differential equations.
jueves, 24 de diciembre de 2009
A Unitary Extension Principle for Shearlet Systems. (arXiv:0912.4529v1 [math.NA])
In this paper, we first introduce the concept of an adaptive MRA (AMRA)
structure which is a variant of the classical MRA structure suited to the main
goal of a fast flexible decomposition strategy adapted to the data at each
decomposition level. We then study this novel methodology for the general case
of affine-like systems, and derive a Unitary Extension Principle (UEP) for
filter design. Finally, we apply our results to the directional representation
system of shearlets. This leads to a comprehensive theory for fast
decomposition algorithms associated with shearlet systems which encompasses
tight shearlet frames with spatially compactly supported generators within such
an AMRA structure. Also shearlet-like systems associated with parabolic scaling
and unimodular matrices optimally close to rotation as well as 3D shearlet
systems are studied within this framework.
Fast Alternating Linearization Methods for Minimizing the Sum of Two Convex Functions. (arXiv:0912.4571v1 [math.OC])
Splitting and alternating direction methods have been widely used for solving
convex optimization problems. We present in this paper two first-order
alternating linearization algorithms based on variable splitting and
alternating linearization techniques for minimizing the sum of two convex
functions. We prove that the number of iterations needed by the first algorithm
is $O(1/epsilon)$ to obtain an $epsilon$-optimal solution. The second
algorithm is an accelerated version of the first one, where the complexity
result is improved to $O(1/sqrt{epsilon})$, while the computational effort
required at each iteration is almost unchanged. Our algorithms and complexity
results can also be extended to more general problems involving linear
operators. Algorithms in this paper are Gauss-Seidel type methods, so they are
different with the ones proposed by Goldfarb and Ma in [12] where the
algorithms are all Jacobi type methods. Preliminary numerical results are
reported to support our theoretical conclusions and demonstrate the practical
potential of our algorithms.
Fast Multiple Splitting Algorithms for Convex Optimization. (arXiv:0912.4570v1 [math.OC])
We present in this paper two different classes of general $K$-splitting
algorithms for solving convex optimization problems. We prove that the number
of iterations needed by the first class of algorithms to obtain an
$epsilon$-optimal solution is $O(1/epsilon)$. The algorithms in the second
class are accelerated versions of those in the first class, where the
complexity result is improved to $O(1/sqrt{epsilon})$ while the computational
effort required at each iteration is almost unchanged. The algorithms are also
extended to solving convex optimization problems involving linear operators
such as those that arise in variational formulations of partial differential
equations and in optimal control problems. To the best of our knowledge, the
complexity results presented in this paper are the first ones of this type that
have been given for splitting and alternating direction type methods. Moreover,
all algorithms proposed in this paper are parallelizable, which makes them
particularly attractive for solving large-scale problems.
miércoles, 23 de diciembre de 2009
On the Effectiveness of Projection Methods for Convex Feasibility Problems with Linear Inequality Constraints. (arXiv:0912.4367v1 [math.OC])
The effectiveness of projection methods for solving systems of linear
inequalities is investigated. It is shown that they have a computational
advantage over some alternatives and that this makes them successful in
real-world applications. This is supported by experimental evidence provided in
this paper on problems of various sizes (up to tens of thousands of unknowns
satisfying up to hundreds of thousands of constraints) and by a discussion of
the demonstrated efficacy of projection methods in numerous scientific
publications and commercial patents (dealing with problems that can have over a
billion unknowns and a similar number of constraints).
Analysis of a finite-volume-finite-element scheme for a nuclear transport model
We consider a problem of nuclear waste contamination, taking into account thermal effects. The temperature and the contaminant's concentration obey convection–diffusion reaction equations. The velocity and the pressure in the flow satisfy the Darcy equation, with a viscosity depending on both concentrations and temperature. The equations are nonlinear and strongly coupled. Using both finite-volume and nonconforming finite-element methods, we introduce a scheme adapted to this problem. We prove the convergence of this scheme and give error estimates.
Verification of the Clauser Technique in Predicting Wall Shear Stress
AIAA Journal Jan. 2010, Vol. 48: 252-256.
Laminar Near Wake of a Circular Cylinder at Hypersonic Speeds
AIAA Journal Jan. 2010, Vol. 48: 236-248.
Mixed-Variable Optimization Strategy Employing Multifidelity Simulation and Surrogate Models
AIAA Journal Jan. 2010, Vol. 48: 215-223.
Computations of Flapping Flow Propulsion for Unmanned Underwater Vehicle Design
AIAA Journal Jan. 2010, Vol. 48: 188-201.
Simulation of Supersonic Combustion Involving H2/Air and C2H4/Air
AIAA Journal Jan. 2010, Vol. 48: 166-173.
Piezoelectric Control of a Partially Propped Cantilever Subjected to a Follower Force
AIAA Journal Jan. 2010, Vol. 48: 144-157.
Optimization of Variable-Stiffness Panels for Maximum Buckling Load Using Lamination Parameters
AIAA Journal Jan. 2010, Vol. 48: 134-143.
Redundant Reactions of Indeterminate Beams by Principle of Quasi Work
AIAA Journal Jan. 2010, Vol. 48: 129-133.
Thermal Force and Moment Determination of an Integrated Thermal Protection System
AIAA Journal Jan. 2010, Vol. 48: 119-128.
Microramps Upstream of an Oblique-Shock/Boundary-Layer Interaction
AIAA Journal Jan. 2010, Vol. 48: 104-118.
Numerical Simulations of Effects of Micro Vortex Generators Using Immersed-Boundary Methods
AIAA Journal Jan. 2010, Vol. 48: 92-103.
Reliability Analysis for Multidisciplinary Systems with Random and Interval Variables
AIAA Journal Jan. 2010, Vol. 48: 82-91.
Finite Element Method Applied to Supersonic Flutter of Circular Cylindrical Shells
AIAA Journal Jan. 2010, Vol. 48: 73-81.
Fast algorithms for hierarchically semiseparable matrices
Semiseparable matrices and many other rank-structured matrices have been widely used in developing new fast matrix algorithms. In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. We represent HSS matrices in terms of general binary HSS trees and use simplified postordering notation for HSS forms. Fast HSS algorithms including new HSS structure generation and HSS form Cholesky factorization are developed. Moreover, we provide a new linear complexity explicit ULV factorization algorithm for symmetric positive definite HSS matrices with a low-rank property. The corresponding factors can be used to solve the HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the algorithms. All these algorithms have nice data locality. They are useful in developing fast-structured numerical methods for large discretized PDEs (such as elliptic equations), integral equations, eigenvalue problems, etc. Some applications are shown. Copyright © 2009 John Wiley & Sons, Ltd.
Factorized parallel preconditioner for the saddle point problem
The aim of this paper is to apply the factorized sparse approximate inverse (FSAI) preconditioner to the iterative solution of linear systems with indefinite symmetric matrices. Until now the FSAI technique has been applied mainly to positive definite systems and with a limited success for the indefinite case. Here, it is demonstrated that the sparsity pattern for the preconditioner can be chosen in such a way that it guarantees the existence of the factorization. The proposed scheme shows excellent parallel scalability, performance and robustness. It is applicable with short recurrence iterative methods such as MinRes and SymmLQ. The properties are demonstrated on linear systems arising from mixed finite element discretizations in linear elasticity. Copyright © 2009 John Wiley & Sons, Ltd.
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