Science News: Groups at Harvard and the University of Queensland in Brisbane, Australia, have designed and built a quantum computer to simulate and calculate the behavior of a molecular, quantum system.
http://blogs.physicstoday.org/newspicks/2010/01/an-quantum-simultion-of-realit.html
viernes, 29 de enero de 2010
jueves, 28 de enero de 2010
Multiscale investigation of shear bands in sand: Physical and numerical experiments
In many geotechnical systems, it is not uncommon to observe failure in zones of high localized strain called shear bands. The existing models predict the existence and the extent of these localizations, but provide little insight into the micromechanics within the shear bands. This research captures and compares the variation in microstructure both inside and outside of shear bands that formed in physical laboratory plane strain and companion numerical two-dimensional discrete element method (DEM) biaxial compression experiments. Unsheared and sheared laboratory specimens of Ottawa 20-30 sand of varying dilatancy were solidified using a two-stage resin impregnation procedure. The solidified specimens were sectioned and the resulting surfaces were prepared for microstructure observation using optical bright-field microscopy and stereological analysis. Statistical properties of microstructural parameters for sub-regions in a grid pattern and along predefined inclined zones were determined. Similar measurements were performed on 2D DEM simulation specimens at varying strain levels to characterize the evolution of microstructure with increasing strain. The results showed how differences evolved in the mean, standard deviation, and entropy of void distributions with increasing global strain levels. The results indicate how disorder increases and that the material within the shear band does not adhere to the classical concept of critical state, but reaches a terminal void ratio that is largely a function of initial void ratio. Furthermore, there appears to be a transition zone between the far field and the fully formed shear block, as opposed to an abrupt delineation as is traditionally inferred. Copyright © 2010 John Wiley & Sons, Ltd.
Molecular dynamics calculations suggest a conduction mechanism for the M2 proton channel from influenza A virus
The M2 protein of the influenza A virus is activated by low endosomal pH and performs the essential function of proton transfer into the viral interior. The resulting decrease in pH within the virion is essential for the uncoating and further replication of the viral genetic material. The x-ray crystal [Stouffer AL, et al. (2008) Nature 451:596–599] and solution NMR [Schnell JR, Chou JJ (2008) Nature 451:591–595] structures of the transmembrane region of the M2 homo-tetrameric bundle both revealed pores with narrow constrictions at one end, leaving a question as to how protons enter the channel. His-37, which is essential for proton-gating and selective conduction of protons, lies in the pore of the crystallographic and NMR structures. Here, we explore the different protonation states of the His-37 residues of the M2 bundle in a bilayer using molecular dynamics (MD) simulations. When the His-37 residues are neutral, the protein prefers an Openout-Closedin conformation in which the channel is open to the environment on the outside of the virus but closed to the interior environment of the virus. Diffusion of protons into the channel from the outside of the virus and protonation of His-37 residues in the tetramer stabilizes an oppositely gated Closedout-Openin conformation. Thus, protons might be conducted through a transporter-like mechanism, in which the protein alternates between Openout-Closedin and Closedout-Openin conformations, and His-37 is protonated/deprotonated during each turnover. The transporter-like mechanism is consistent with the known properties of the M2 bundle, including its relatively low rate of proton flux and its strong rectifying behavior.
martes, 26 de enero de 2010
Optimal tuning of the Hybrid Monte-Carlo Algorithm. (arXiv:1001.4460v1 [math.PR])
We investigate the properties of the Hybrid Monte-Carlo algorithm (HMC) in
high dimensions. HMC develops a Markov chain reversible w.r.t. a given target
distribution $Pi$ by using separable Hamiltonian dynamics with potential
$-logPi$. The additional momentum variables are chosen at random from the
Boltzmann distribution and the continuous-time Hamiltonian dynamics are then
discretised using the leapfrog scheme. The induced bias is removed via a
Metropolis-Hastings accept/reject rule. In the simplified scenario of
independent, identically distributed components, we prove that, to obtain an
$mathcal{O}(1)$ acceptance probability as the dimension $d$ of the state space
tends to $infty$, the leapfrog step-size $h$ should be scaled as $h= l imes
d^{-1/4}$. Therefore, in high dimensions, HMC requires $mathcal{O}(d^{1/4})$
steps to traverse the state space. We also identify analytically the
asymptotically optimal acceptance probability, which turns out to be 0.651 (to
three decimal places). This is the choice which optimally balances the cost of
generating a proposal, which {em decreases} as $l$ increases, against the cost
related to the average number of proposals required to obtain acceptance, which
{em increases} as $l$ increases.
Solving Schubert Problems with Littlewood-Richardson Homotopies. (arXiv:1001.4125v1 [math.NA])
We present a new numerical homotopy continuation algorithm for finding all
solutions to Schubert problems on Grassmannians. This Littlewood-Richardson
homotopy is based on Vakil's geometric proof of the Littlewood-Richardson rule.
Its start solutions are given by linear equations and they are tracked through
a sequence of homotopies encoded by certain checker configurations to find the
solutions to a given Schubert problem. For generic Schubert problems the number
of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is
implemented using the path trackers of the software package PHCpack.
Numerical Studies of Three-dimensional Stochastic Darcy’s Equation and Stochastic Advection-Diffusion-Dispersion Equation
Abstract Solute transport in randomly heterogeneous porous media is commonly described by stochastic flow and advection-dispersion
equations with a random hydraulic conductivity field. The statistical distribution of conductivity of engineered and naturally
occurring porous material can vary, depending on its origin. We describe solutions of a three-dimensional stochastic advection-dispersion
equation using a probabilistic collocation method (PCM) on sparse grids for several distributions of hydraulic conductivity.
Three random distributions of log hydraulic conductivity are considered: uniform, Gaussian, and truncated Gaussian (beta).
Log hydraulic conductivity is represented by a Karhunen-Loève (K-L) decomposition as a second-order random process with an
exponential covariance function. The convergence of PCM has been demonstrated. It appears that the accuracy in both the mean
and the standard deviation of PCM solutions can be improved by using the Jacobi-chaos representing the truncated Gaussian
distribution rather than the Hermite-chaos for the Gaussian distribution. The effect of type of distribution and parameters
such as the variance and correlation length of log hydraulic conductivity and dispersion coefficient on leading moments of
the advection velocity and solute concentration was investigated.
equations with a random hydraulic conductivity field. The statistical distribution of conductivity of engineered and naturally
occurring porous material can vary, depending on its origin. We describe solutions of a three-dimensional stochastic advection-dispersion
equation using a probabilistic collocation method (PCM) on sparse grids for several distributions of hydraulic conductivity.
Three random distributions of log hydraulic conductivity are considered: uniform, Gaussian, and truncated Gaussian (beta).
Log hydraulic conductivity is represented by a Karhunen-Loève (K-L) decomposition as a second-order random process with an
exponential covariance function. The convergence of PCM has been demonstrated. It appears that the accuracy in both the mean
and the standard deviation of PCM solutions can be improved by using the Jacobi-chaos representing the truncated Gaussian
distribution rather than the Hermite-chaos for the Gaussian distribution. The effect of type of distribution and parameters
such as the variance and correlation length of log hydraulic conductivity and dispersion coefficient on leading moments of
the advection velocity and solute concentration was investigated.
- Content Type Journal Article
- DOI 10.1007/s10915-010-9346-5
- Authors
- G. Lin, Pacific Northwest National Laboratory Computational Mathematics 902 Battelle Blvd. Box 999 Richland WA 99352 USA
- A. M. Tartakovsky, Pacific Northwest National Laboratory Computational Mathematics 902 Battelle Blvd. Box 999 Richland WA 99352 USA
- Journal Journal of Scientific Computing
- Online ISSN 1573-7691
- Print ISSN 0885-7474
A Compact Fourth Order Scheme for the Helmholtz Equation in Polar Coordinates
Abstract In many problems, one wishes to solve the Helmholtz equation in cylindrical or spherical coordinates which introduces variable
coefficients within the differentiated terms. Fourth order accurate methods are desirable to reduce pollution and dispersion
errors and so alleviate the points-per-wavelength constraint. However, the variable coefficients renders existing fourth order
finite difference methods inapplicable. We develop a new compact scheme that is provably fourth order accurate even for these
problems. The resulting system of finite difference equations is solved by a separation of variables technique based on the
FFT. Moreover, in the r direction the unbounded domain is replaced by a finite domain, and an exact artificial boundary condition is specified as
a closure. This global boundary condition fits naturally into the inversion of the linear system. We present numerical results
that corroborate the fourth order convergence rate for several scattering problems.
coefficients within the differentiated terms. Fourth order accurate methods are desirable to reduce pollution and dispersion
errors and so alleviate the points-per-wavelength constraint. However, the variable coefficients renders existing fourth order
finite difference methods inapplicable. We develop a new compact scheme that is provably fourth order accurate even for these
problems. The resulting system of finite difference equations is solved by a separation of variables technique based on the
FFT. Moreover, in the r direction the unbounded domain is replaced by a finite domain, and an exact artificial boundary condition is specified as
a closure. This global boundary condition fits naturally into the inversion of the linear system. We present numerical results
that corroborate the fourth order convergence rate for several scattering problems.
- Content Type Journal Article
- DOI 10.1007/s10915-010-9348-3
- Authors
- S. Britt, North Carolina State University Department of Mathematics Box 8205 Raleigh NC 27695 USA
- S. Tsynkov, North Carolina State University Department of Mathematics Box 8205 Raleigh NC 27695 USA
- E. Turkel, Tel Aviv University School of Mathematical Sciences Ramat Aviv Tel Aviv 69978 Israel
- Journal Journal of Scientific Computing
- Online ISSN 1573-7691
- Print ISSN 0885-7474
A note on the O(n)-storage implementation of the GKO algorithm and its adaptation to Trummer-like matrices
Abstract We propose a new O(n)-space implementation of the GKO-Cauchy algorithm for the solution of linear systems where the coefficient matrix is Cauchy-like.
Moreover, this new algorithm makes a more efficient use of the processor cache memory; for matrices of size larger than n ≈ 500–1,000, it outperforms the customary GKO algorithm. We present an applicative case of Cauchy-like matrices with non-reconstructible
main diagonal. In this special instance, the O(n) space algorithms can be adapted nicely to provide an efficient implementation of basic linear algebra operations in terms
of the low displacement-rank generators.
Moreover, this new algorithm makes a more efficient use of the processor cache memory; for matrices of size larger than n ≈ 500–1,000, it outperforms the customary GKO algorithm. We present an applicative case of Cauchy-like matrices with non-reconstructible
main diagonal. In this special instance, the O(n) space algorithms can be adapted nicely to provide an efficient implementation of basic linear algebra operations in terms
of the low displacement-rank generators.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-010-9361-5
- Authors
- Federico Poloni, Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa Italy
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
An O(h6) numerical solution of general nonlinear fifth-order two point boundary value problems
Abstract A sixth-order numerical scheme is developed for general nonlinear fifth-order two point boundary-value problems. The standard
sextic spline for the solution of fifth order two point boundary-value problems gives only O(h
2) accuracy and leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations
of the problem are generated and applied to construct the numerical algorithm. O(h
6) global error estimates obtained for these problems. The convergence properties of the method is studied. This scheme has
been applied to the system of nonlinear fifth order two-point boundary value problem too. Numerical results are given to illustrate
the efficiency of the proposed method computationally. Results from the numerical experiments, verify the theoretical behavior
of the orders of convergence.
sextic spline for the solution of fifth order two point boundary-value problems gives only O(h
2) accuracy and leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations
of the problem are generated and applied to construct the numerical algorithm. O(h
6) global error estimates obtained for these problems. The convergence properties of the method is studied. This scheme has
been applied to the system of nonlinear fifth order two-point boundary value problem too. Numerical results are given to illustrate
the efficiency of the proposed method computationally. Results from the numerical experiments, verify the theoretical behavior
of the orders of convergence.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-010-9362-4
- Authors
- Jalil Rashidinia, Iran University of Science & Technology Narmak School of Mathematics Tehran 16844-13114 Iran
- M. Ghasemi, Iran University of Science & Technology Narmak School of Mathematics Tehran 16844-13114 Iran
- R. Jalilian, Ilam University Department of Mathematics P.O. Box 69315-516 Ilam Iran
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
A variable step-size control algorithm for the weak approximation of stochastic differential equations
Abstract In this paper, we propose two local error estimates based on drift and diffusion terms of the stochastic differential equations
in order to determine the optimal step-size for the next stage in an adaptive variable step-size algorithm. These local error
estimates are based on the weak approximation solution of stochastic differential equations with one-dimensional and multi-dimensional
Wiener processes. Numerical experiments are presented to illustrate the effectiveness of this approach in the weak approximation
of several standard test problems including SDEs with small noise and scalar and multi-dimensional Wiener processes.
in order to determine the optimal step-size for the next stage in an adaptive variable step-size algorithm. These local error
estimates are based on the weak approximation solution of stochastic differential equations with one-dimensional and multi-dimensional
Wiener processes. Numerical experiments are presented to illustrate the effectiveness of this approach in the weak approximation
of several standard test problems including SDEs with small noise and scalar and multi-dimensional Wiener processes.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-010-9363-3
- Authors
- A. Valinejad, Tarbiat Modares University Department of Mathematics P.O. Box 14115-175 Tehran Iran
- S. Mohammad Hosseini, Tarbiat Modares University Department of Mathematics P.O. Box 14115-175 Tehran Iran
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
A binary powering Schur algorithm for computing primary matrix roots
Abstract An algorithm for computing primary roots of a nonsingular matrix A is presented. In particular, it computes the principal root of a real matrix having no nonpositive real eigenvalues, using
real arithmetic. The algorithm is based on the Schur decomposition of A and has an order of complexity lower than the customary Schur based algorithm, namely the Smith algorithm.
real arithmetic. The algorithm is based on the Schur decomposition of A and has an order of complexity lower than the customary Schur based algorithm, namely the Smith algorithm.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-009-9357-1
- Authors
- Federico Greco, Università di Perugia Dipartimento di Matematica e Informatica Via Vanvitelli 1 06123 Perugia Italy
- Bruno Iannazzo, Università di Perugia Dipartimento di Matematica e Informatica Via Vanvitelli 1 06123 Perugia Italy
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
A Kantorovich-type convergence analysis of the Newton–Josephy method for solving variational inequalities
Abstract We present a Kantorovich-type semilocal convergence analysis of the Newton–Josephy method for solving a certain class of variational
inequalities. By using a combination of Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions,
we provide an analysis with the following advantages over the earlier works (Wang 2009, Wang and Shen, Appl Math Mech 25:1291–1297, 2004) (under the same or less computational cost): weaker sufficient convergence conditions, larger convergence domain, finer
error bounds on the distances involved, and an at least as precise information on the location of the solution.
inequalities. By using a combination of Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions,
we provide an analysis with the following advantages over the earlier works (Wang 2009, Wang and Shen, Appl Math Mech 25:1291–1297, 2004) (under the same or less computational cost): weaker sufficient convergence conditions, larger convergence domain, finer
error bounds on the distances involved, and an at least as precise information on the location of the solution.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-010-9364-2
- Authors
- Ioannis K. Argyros, Cameron University Department of Mathematics Sciences Lawton OK 73505 USA
- Saïd Hilout, Poitiers University Laboratoire de Mathématiques et Applications Bd. Pierre et Marie Curie, Téléport 2 B.P. 30179 86962 Futuroscope Chasseneuil Cedex France
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
- Journal Volume Volume -1
- Journal Issue Volume -1, Online First / January, 2010
domingo, 24 de enero de 2010
Numerical analysis of a contact problem between two thermoviscoelastic beams
We propose and analyze in this article a finite element approximation, based on a penalty formulation, to a quasi-static unilateral contact problem between two thermoviscoelastic beams. An error bound is given and some numerical experiments discussed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010
Partition of Unity Refinement for local approximation
In this article, we propose a Partition of Unity Refinement (PUR) method to improve the local approximations of elliptic boundary value problems in regions of interest. The PUR method only needs to refine the local meshes and hanging nodes generate no difficulty. The mesh qualities such as uniformity or quasi-uniformity are kept. The advantages of the PUR include its effectiveness and relatively easy implementation. In this article, we present the basic ideas and implementation of the PUR method on triangular meshes. Numerical results for elliptic Dirichlet boundary value problem on an L-shaped domain are shown to demonstrate the effectiveness of the proposed method. The extensions of the PUR method to multilevel and higher dimension are straightforward. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010
A new approach for solving Stokes systems arising from a distributive relaxation method
The distributed relaxation method for the Stokes problem has been advertised as an adequate change of variables that leads to a lower triangular system with Laplace operators on the main diagonal for which multigrid methods are very efficient. We show that under high regularity of the Laplacian, the transformed system admits almost block-lower triangular form. We analyze the distributed relaxation method and compare it with other iterative methods for solving the Stokes system. We also present numerical experiments illustrating the effectiveness of the transformation which is well established for certain finite difference discretizations of Stokes problems. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010
Numerical simulation of generalized Newtonian blood flow past a couple of irregular arterial stenoses
This paper looked at the numerical investigations of the generalized Newtonian blood flow through a couple of irregular arterial stenoses. The flow is treated to be axisymmetric, with an outline of the stenoses obtained from a three dimensional casting of a mild stenosed artery, so that the flow effectively becomes two-dimensional. The Marker and Cell (MAC) method is developed for the governing unsteady generalized Newtonian equations in staggered grid for viscous incompressible flow in the cylindrical polar co-ordinates system. The derived pressure-Poisson equation was solved using Successive-Over-Relaxation (S.O.R.) method and the pressure-velocity correction formulae have been derived. Computations are performed for the pressure drop, the wall shear stress distribution and the separation region. The presented computations show that in comparison to the corresponding Newtonian model the generalized Newtonian fluid experiences higher pressure drop, lower peak wall shear stress and smaller separation region. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010
viernes, 22 de enero de 2010
A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy
Abstract While second order methods for computational simulations of fluid flow provide the basis of widely used commercial software,
there is a need for higher order methods for more accurate simulations of turbulent and vortex dominated flows. The discontinuous
Galerkin (DG) method is the subject of much current research toward this goal. The spectral difference (SD) method has recently
emerged as a promising alternative which can reduce the computational costs of higher order simulations. There remains some
questions, however, about the stability of the SD method. This paper presents a proof that for the case of one dimensional
linear advection the SD method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior
fluxes collocation points are placed at the zeros of the corresponding Legendre polynomial.
there is a need for higher order methods for more accurate simulations of turbulent and vortex dominated flows. The discontinuous
Galerkin (DG) method is the subject of much current research toward this goal. The spectral difference (SD) method has recently
emerged as a promising alternative which can reduce the computational costs of higher order simulations. There remains some
questions, however, about the stability of the SD method. This paper presents a proof that for the case of one dimensional
linear advection the SD method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior
fluxes collocation points are placed at the zeros of the corresponding Legendre polynomial.
- Content Type Journal Article
- DOI 10.1007/s10915-009-9339-4
- Authors
- Antony Jameson, Stanford University Department of Aeronautics and Astronautics Stanford CA 94305 USA
- Journal Journal of Scientific Computing
- Online ISSN 1573-7691
- Print ISSN 0885-7474
Mathematical analysis of the Navier-Stokes/Darcy coupling
We consider a differential system based on the coupling of the Navier-Stokes and Darcy equations for modeling the interaction between surface and porous media flows. We formulate the problem as an interface equation, we analyze the associated (nonlinear) Steklov-Poincare' operators, and we prove its wellposedness. We propose and analyze iterative methods to solve a conforming finite element approximation of the coupled problem.
Quantum Computing Resource Estimate of Molecular Energy Simulation. (arXiv:1001.3855v1 [quant-ph])
Over the last century, ingenious physical and mathematical insights paired
with rapidly advancing technology have allowed the field of quantum chemistry
to advance dramatically. However, efficient methods for the exact simulation of
quantum systems on classical computers do not exist. The present paper reports
an extension of one of the authors' previous work [Aspuru-Guzik et al., Science
{309} p. 1704, (2005)] where it was shown that the chemical Hamiltonian can be
efficiently simulated using a quantum computer. In particular, we report in
detail how a set of molecular integrals can be used to create a quantum circuit
that allows the energy of a molecular system with fixed nuclear geometry to be
extracted using the phase estimation algorithm proposed by Abrams and Lloyd
[Phys. Rev. Lett. {83} p. 5165, (1999)]. We extend several known results
related to this idea and present numerical examples of the state preparation
procedure required in the algorithm. With future quantum devices in mind, we
provide a complete example using the Hydrogen molecule, of how a chemical
Hamiltonian can be simulated using a quantum computer. Some of the results we
present here represent an extension of our recent collaboration appearing in B.
P. Lanyon et al. [Nature Chem., advance online publication, doi:
10.1038/nchem.483 (2010)].
Quantum Computing with Superqubits. (arXiv:1001.3753v1 [hep-th])
We analyze some aspects of quantum computing with super-qubits (squbits). We
propose the analogue of a superfield formalism, and give a physical
interpretation for the Grassmann coefficients in the squbit expansion as
fermionic creation operators of an auxiliary quantum system. In the simplest
case the squbit is a superposition of one Bose X Bose and one Fermi X Fermi
state, and its norm is invariant under a U(2) group realized with
Clifford-valued matrices. This case can be generalized to a superposition of
n_B bosonic and n_F fermionic states, with a norm invariant under U(n_B + n_F).
Entanglement between squbits, super quantum gates and teleportation are
discussed.
Effective polar potential in the central force Schrodinger equation. (arXiv:1001.3693v1 [quant-ph])
The angular part of the Schrodinger equation for a central potential is
brought to the one-dimensional 'Schrodinger form' where one has a kinetic
energy plus potential energy terms. The resulting polar potential is seen to be
a family of potentials characterized by the square of the magnetic quantum
number m. It is demonstrated that this potential can be viewed as a confining
potential that attempts to confine the particle to the xy-plane, with a
strength that increases with increasing m. Linking the solutions of the
equation to the conventional solutions of the angular equation, i.e. the
associated Legendre functions, we show that the variation in the spatial
distribution of the latter for different values of the orbital angular quantum
number l can be viewed as being a result of 'squeezing' with different
strengths by the introduced 'polar potential'.
The concept of minimized integrated exponential error for low dispersion and low dissipation schemes
We devise two novel techniques to optimize parameters which regulate dispersion and dissipation effects in numerical methods using the notion that dissipation neutralizes dispersion. These techniques are baptized as the minimized integrated error for low dispersion and low dissipation (MIELDLD) and the minimized integrated exponential error for low dispersion and low dissipation (MIEELDLD).These two techniques of optimization have an advantage over the concept of minimized integrated square difference error (MISDE), especially in the case when more than one optimal cfl is obtained, out of which only one of these values satisfy the shift condition. For instance, when MISDE is applied to the 1-D Fromm's scheme, we have obtained two optimal cfl numbers: 0.28 and 1.0. However, it is known that Fromm's scheme satisfies shift condition only at r=1.0. Using MIELDLD and MIEELDLD, the optimal cfl of Fromm's scheme is computed as 1.0.We show that like the MISDE concept, both the techniques MIELDLD and MIEELDLD are effective to control dissipation and dispersion. The condition [nu]2>4µ is satisfied for all these three techniques of optimization, where [nu] and µ are parameters present in the Korteweg-de-Vries-Burgers equation.The optimal cfl number for some numerical schemes namely Lax-Wendroff, Beam-Warming, Crowley and Upwind Leap-Frog when discretized by the 1-D linear advection equation is computed. The optimal cfl number obtained is in agreement with the shift condition. Some numerical experiments in 1-D have been performed which consist of discontinuities and shocks. The dissipation and dispersion errors at some different cfl numbers for these experiments are quantified. Copyright © 2010 John Wiley & Sons, Ltd.
Compact embeddings of broken Sobolev spaces and applications
In this paper, we present several extensions of theoretical tools for the analysis of discontinuous Galerkin (DG) method beyond the linear case. We define broken Sobolev spaces for Sobolev indices in [1, ), and we prove generalizations of many techniques of classical analysis in Sobolev spaces. Our targeted application is the convergence analysis for DG discretizations of energy minimization problems of the calculus of variations. Our main tool in this analysis is a theorem which permits the extraction of a ‘weakly’ converging subsequence of a family of discrete solutions and which shows that any ‘weak limit’ is a Sobolev function. As a second application, we compute the optimal embedding constants in broken Sobolev–Poincaré inequalities.
Improved contour integral methods for parabolic PDEs
One way of computing the matrix exponential that arises in semidiscrete parabolic partial differential equations is via the Dunford–Cauchy integral formula. The integral is approximated by the trapezoidal or midpoint rules on a Hankel contour defined by a suitable change of variables. In a recent paper by Weideman & Trefethen (2007, Math. Comput., 76, 1341–1356) two widely used contours were analysed. Estimates for the optimal parameters that define these contours were proposed. In this paper this analysis is extended in two directions. First, the effect of roundoff error is now included in the error model. Second, we extend the results to the case of a model convection–diffusion equation, where a large convective term causes the matrix to be highly non-normal.
Discretization error and modelling error in the context of the rapid inflation of hyperelastic membranes
The computational modelling of the rapid large inflation of hyperelastic circular sheets modelled as axisymmetric membranes is treated, with the aim of estimating engineering quantities of interest and their errors. Fine (involving inertia terms) and coarse (quasi-static) models of the inflation are considered and, using goal-oriented techniques, both modelling and discretization errors are estimated. Numerical results involving only discretization errors for the quasi-static problem and both modelling and discretization errors for the dynamic problem are presented.
On the convergence of a wide range of trust region methods for unconstrained optimization
We consider trust region methods for seeking the unconstrained minimum of an objective function F(
),
, when the gradient
(
),
, is available. The methods are iterative with
1 being given. The new vector of variables
k+1 is derived from a quadratic approximation to F that interpolates F(
k) and
, where k is the iteration number. The second derivative matrix of the quadratic approximation, Bk say, can be indefinite, because the approximation is employed only if the vector of variables
satisfies
, where k is a "trust region radius" that is adjusted automatically. Thus the approximation is useful if
is sufficiently large and if ||Bk|| and k are sufficiently small. It is proved under mild assumptions that the condition
is achieved after a finite number of iterations, where is any given positive constant, and then it is usual to end the calculation. The assumptions include a Lipschitz condition on
and also F has to be bounded below. The termination property is established in a single theorem that applies to a wide range of trust region methods that force the sequence F(
k), k = 1, 2, 3, ..., to decrease monotonically. Any choice of each symmetric matrix Bk is allowed, provided that ||Bk|| is bounded above by a constant multiple of k.
A survey of results on the q-Bernstein polynomials
It is now nearly a century since S. N. Bernstein introduced his well-known polynomials. This paper is concerned with generalizations of the Bernstein polynomials, mainly with the so called q-Bernstein polynomials. These are due to the author of this paper and are based on the q integers. They reduce to the Bernstein polynomials when we put q = 1 and share the shape-preserving properties of the Bernstein polynomials when q (0, 1). This paper also describes another earlier generalization of the Bernstein polynomials, a sequence of rational functions that are also based on the q-integers, proposed by A. Lupas, and two even earlier generalizations due to D. D. Stancu. The present author summarizes various results, due to a number of authors, that are concerned with the q-Bernstein polynomials and with Stancu's two generalizations.
ADI orthogonal spline collocation methods for parabolic partial integro-differential equations
Alternating direction implicit (ADI) orthogonal spline collocation schemes are formulated and analysed for a class of partial integro-differential equations of parabolic type. These techniques are based on the -method, for [1/2, 1], (where = 1 yields the backward Euler method and = 1/2 yields the Crank–Nicolson method) and the second-order backward differentiation formula (BDF) method. For each method, optimal estimates in various norms at each time step are derived and confirmed by results of numerical experiments.
Asymptotic behaviour in linear least squares problems
The asymptotic behaviour of a class of least squares problems when subjected to structured perturbations is considered. It is permitted that the number of rows (observations) in the design matrix can be unbounded while the number of degrees of freedom (variables) is fixed. It is shown that for certain classes of random data the solution sensitivity depends asymptotically on the condition number of the design matrix rather than on its square, which is the generic result for inconsistent systems when the norm of the residual is not small. Extension of these results to the case where the perturbations are due to rounding errors is considered.
The convection-diffusion Petrov-Galerkin story
The term ‘Petrov–Galerkin method’ is probably due to Ron Mitchell and his collaborators, and he was certainly very active in studying the application of finite-element methods to second-order partial differential equations ‘with significant first derivatives’. Our aim in the present paper is to trace links between such early methods and the more recent discontinuous Galerkin methods—not only the methods but also their analysis. Also as Mitchell, like the author, was initially steeped in finite-difference methods, we shall sometimes use their manipulation in our analysis.
Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation
In a previous paper, McLean & Thomée (2009, J. Integr. Equ. Appl. (to appear)), we studied three numerical methods for the discretization in time of a fractional-order evolution equation in a Banach space framework. Each of the methods applied a quadrature rule to a contour integral representation of the solution in the complex plane, where for each quadrature point an elliptic boundary-value problem had to be solved to determine the value of the integrand. The first two methods involved the Laplace transform of the forcing term, but the third did not. We analysed both the quadrature error and the error arising from a spatial discretization by finite elements, measured in the L2-norm. The present work extends our earlier results by proving error bounds in the technically more complicated case of the maximum norm. We also establish new regularity properties for the exact solution that are needed for our analysis.
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.
On the approximation and efficient evaluation of integral terms in PDE models of cell adhesion
Recently, a nonlocal term has been introduced in time-dependent partial differential equation (PDE) models of cell migration in tissue. This term is used to model adhesive effects between cells and also between cells and the extracellular matrix. We assume periodic boundary conditions for the model and that the PDE system is discretized following the method of lines and using a finite-volume scheme on a uniform grid in space. For high-resolution simulations of the PDE system an efficient evaluation of the approximation of the nonlocal term is crucial. For one and two spatial dimensions we develop suitable approximations of the nonlocal term and evaluate these using fast Fourier transform (FFT) techniques. Comprehensive numerical tests show the accuracy and efficiency of our approach. We also demonstrate the impact of the proposed scheme for the treatment of the nonlocal term on simulation times for a differential cell adhesion model. We discuss extensions and applicability of our work to systems with nonperiodic boundary conditions and for other nonlocal PDE models from mathematical biology.
Comparisons between pseudospectral and radial basis function derivative approximations
Fourier-based pseudospectral (PS) methods have been used since the 1970s for obtaining spectrally accurate solutions to partial differential equations (PDEs) in periodic geometries. Radial basis functions (RBFs) were introduced about the same time for interpolation on scattered nodes in irregular geometries. As was later recognized, they can also be used for accurate numerical solution of PDEs. Although the main strength of RBFs lies in their outstanding geometric flexibility, also offering possibilities of spectral accuracy over irregularly shaped finite domains, it is still of interest to compare them against Fourier-based PS methods in the extremely simple geometries (infinite or periodic domains) where the latter can also be used. Mostly by means of heuristic arguments and graphical illustrations based on Fourier analysis and numerical experiments, we show that there are notable differences (more pronounced in increasing numbers of dimensions) in how the two spectral approaches approximate derivatives.
Optimal stability for trapezoidal-backward difference split-steps
The marginal stability of the trapezoidal method makes it dangerous to use for highly non-linear oscillations. Damping is provided by backward differences. The split-step combination (t trapezoidal, (1 – )t for BDF2) retains second-order accuracy. The ‘magic choice’
allows the same Jacobian for both steps, when Newton's method solves these implicit difference equations. That choice is known to give the smallest error constant, and we prove that
also gives the largest region of linearized stability.
Trees, B-series and exponential integrators
Questions concerning the accuracy of numerical methods for differential equations are often analysed using B-series and other formulations based on rooted trees. The analysis of numerical methods, such as Rosenbrock and certain exponential methods, requires an additional algebraic structure to represent the direct use of Jacobian matrices in the computation. It is shown how this can be done within a context containing a broad review of the existing B-series formulation.
The spectral problem for a class of highly oscillatory Fredholm integral operators
Let
be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K0(x, y)ei|x–y|. We study the spectral problem for large , showing that the spectrum consists of infinitely many discrete (complex) eigenvalues and give a precise description of the way in which they converge to the origin. In addition, we investigate the asymptotic properties of the solutions f = f(x;) to the associated Fredholm integral equation f = µ
f + a as $$omega o mathrm{infty }$$, thus refining a classical result by Ursell. Possible extensions of these results to highly oscillatory Fredholm integral operators with more general highly oscillating kernels are also discussed.
A preconditioned Newton algorithm for the nearest correlation matrix
Various methods have been developed for computing the correlation matrix nearest in the Frobenius norm to a given matrix. We focus on a quadratically convergent Newton algorithm recently derived by Qi and Sun. Various improvements to the efficiency and reliability of the algorithm are introduced. Several of these relate to the linear algebra: the Newton equations are solved by minres instead of the conjugate gradient method, as it more quickly satisfies the inexact Newton condition; we apply a Jacobi preconditioner, which can be computed efficiently even though the coefficient matrix is not explicitly available; an efficient choice of eigensolver is identified; and a final scaling step is introduced to ensure that the returned matrix has unit diagonal. Potential difficulties caused by rounding errors in the Armijo line search are avoided by altering the step selection strategy. These and other improvements lead to a significant speed-up over the original algorithm and allow the solution of problems of dimension a few thousand in a few tens of minutes.
jueves, 21 de enero de 2010
Finite element approximations of harmonic map heat flows and wave maps into spheres of nonconstant radii
Abstract We prove the existence of weak solutions to the harmonic map heat flow, and wave maps into spheres of nonconstant radii. Weak
solutions are constructed as proper limits of iterates from a fully practical scheme based on lowest order conforming finite
elements, where discrete Lagrange multipliers are employed to exactly meet the sphere constraint at mesh-points. Computational
studies are included to motivate interesting dynamics in two and three spatial dimensions.
solutions are constructed as proper limits of iterates from a fully practical scheme based on lowest order conforming finite
elements, where discrete Lagrange multipliers are employed to exactly meet the sphere constraint at mesh-points. Computational
studies are included to motivate interesting dynamics in two and three spatial dimensions.
- Content Type Journal Article
- DOI 10.1007/s00211-009-0282-y
- Authors
- L’ubomír Baňas, Heriot-Watt University Department of Mathematics and the Maxwell Institute for Mathematical Sciences Edinburgh EH14 4AS UK
- Andreas Prohl, Mathematisches Institut der Universität Tübingen Auf der Morgenstelle 10 72076 Tübingen Germany
- Reiner Schätzle, Mathematisches Institut der Universität Tübingen Auf der Morgenstelle 10 72076 Tübingen Germany
- Journal Numerische Mathematik
- Online ISSN 0945-3245
- Print ISSN 0029-599X
MIT Researchers Simulate New Materials on Ranger to Improve Solar Photovoltaic Cells
To better understand the fundamentals of solar energy conversion and to identify potential new materials for cheaper and more efficient cells, researchers from MIT have been simulating the atomic behavior of solar cells using the Ranger supercomputer at TACC.
Curved Boundary Treatments for the Discontinuous Galerkin Method Applied to Aeroacoustic Propagation
AIAA Journal Feb. 2010, Vol. 48: 479-489.
Aerodynamic Optimization Algorithm with Integrated Geometry Parameterization and Mesh Movement
AIAA Journal Feb. 2010, Vol. 48: 400-413.
Finite Difference Lattice Boltzmann Method Applied to Acoustic-Scattering Problems
AIAA Journal Feb. 2010, Vol. 48: 354-371.
Further results on error estimators for local refinement with first-order system least squares (FOSLS)
Adaptive local refinement (ALR) can substantially improve the performance of simulations that involve numerical solution of partial differential equations. In fact, local refinement capabilities are one of the attributes of first-order system least squares (FOSLS) in that it provides an inexpensive but effective a posteriori local error bound that accurately identifies regions that require further refinement. Previous theory on FOSLS established the effectiveness of its local error estimator, but only under the assumption that the local region is not too 'thin'. This paper extends this theory to the case of a rectangular domain by showing that the estimator's effectiveness holds even for certain 'thin' local regions. Further, we prove that when the approximation satisfies a local saturation property, convergence of a FOSLS ALR scheme is guaranteed. Copyright © 2010 John Wiley & Sons, Ltd.
martes, 19 de enero de 2010
Entanglement of Periodic States, the Quantum Fourier Transform and Shor's Factoring Algorithm. (arXiv:1001.3145v1 [quant-ph])
The preprocessing stage of Shor's algorithm generates a class of quantum
states referred to as periodic states, on which the quantum Fourier transform
is applied. Such states also play an important role in other quantum algorithms
that rely on the quantum Fourier transform. Since entanglement is believed to
be a necessary resource for quantum computational speedup, we analyze the
entanglement of periodic states, and the way it is affected by the quantum
Fourier transform. To this end, we derive a formula that evaluates the
Groverian entanglement measure for periodic states. Using this formula, we
explain the surprising result that the Groverian entanglement of the periodic
states built up during the preprocessing stage is only slightly affected by the
quantum Fourier transform.
Gradual Variation Analysis for Groundwater Flow. (arXiv:1001.3190v1 [math.NA])
Groundwater flow in Washington DC greatly influences the surface water
quality in urban areas. The current methods of flow estimation, based on
Darcy's Law and the groundwater flow equation, can be described by the
diffusion equation (the transient flow) and the Laplace equation (the
steady-state flow). The Laplace equation is a simplification of the diffusion
equation under the condition that the aquifer has a recharging boundary. The
practical way of calculation is to use numerical methods to solve these
equations. The most popular system is called MODFLOW, which was developed by
USGS. MODFLOW is based on the finite-difference method in rectangular Cartesian
coordinates. MODFLOW can be viewed as a "quasi 3D" simulation since it only
deals with the vertical average (no z-direction derivative). Flow calculations
between the 2D horizontal layers use the concept of leakage. In this project,
we have established a mathematical model based on gradually varied functions
for groundwater data volume reconstruction. These functions do not rely on the
rectangular Cartesian coordinate system. A gradually varied function can be
defined in a general graph or network. Gradually varied functions are suitable
for arbitrarily shaped aquifers. Two types of models are designed and
implemented for real data processing: (1) the gradually varied model for
individual (time) groundwater flow data, (2) the gradually varied model for
sequential (time) groundwater flow data. In application, we also established a
MySQL database to support the related research. The advantage of the gradually
varied fitting and its related method does not need the strictly defined
boundary condition as it is required in MODFLOW.
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