We present a reformulation of stochastic global optimization as a filtering
problem. The motivation behind this reformulation comes from the fact that for
many optimization problems we cannot evaluate exactly the objective function to
be optimized. Similarly, we may not be able to evaluate exactly the functions
involved in iterative optimization algorithms. For example, we may only have
access to noisy measurements of the functions or statistical estimates provided
through Monte Carlo sampling. This makes iterative optimization algorithms
behave like stochastic maps. Naive global optimization amounts to evolving a
collection of realizations of this stochastic map and picking the realization
with the best properties. This motivates the use of filtering techniques to
allow focusing on realizations that are more promising than others. In
particular, we present a filtering reformulation of global optimization in
terms of a special case of sequential importance sampling methods called
particle filters. The increasing popularity of particle filters is based on the
simplicity of their implementation and their flexibility. For parametric
exponential density estimation problems, we utilize the flexibility of particle
filters to construct a stochastic global optimization algorithm which converges
to the optimal solution appreciably faster than naive global optimization.
Several examples are provided to demonstrate the efficiency of the approach.
lunes, 21 de diciembre de 2009
Stochastic global optimization as a filtering problem. (arXiv:0912.4072v1 [math.NA])
How to increase convergence order of Newton's method to $2 imes m$?. (arXiv:0912.4140v1 [math.NA])
We present a simple yet powerful technique for forming iterative methods of
various convergence orders. Methods of various convergence orders (four, six,
eight and ten) are formed through a modest modification of the classical Newton
method. The technique can be easily implemented in existing software packages
as suggested by the presented C$^{++}$ algorithm. Finally some problems are
solved through the proposed algorithm.
A global method for coupling transport with chemistry in heterogeneous porous media. (arXiv:0912.3867v1 [math.NA])
Modeling reactive transport in porous media, using a local chemical
equilibrium assumption, leads to a system of advection-diffusion PDE's coupled
with algebraic equations. When solving this coupled system, the algebraic
equations have to be solved at each grid point for each chemical species and at
each time step. This leads to a coupled non-linear system. In this paper a
global solution approach that enables to keep the software codes for transport
and chemistry distinct is proposed. The method applies the Newton-Krylov
framework to the formulation for reactive transport used in operator splitting.
The method is formulated in terms of total mobile and total fixed
concentrations and uses the chemical solver as a black box, as it only requires
that on be able to solve chemical equilibrium problems (and compute
derivatives), without having to know the solution method. An additional
advantage of the Newton-Krylov method is that the Jacobian is only needed as an
operator in a Jacobian matrix times vector product. The proposed method is
tested on the MoMaS reactive transport benchmark.
Finite element method by using quartic B-splines
In this article, we discuss a kind of finite element method by using quartic B-splines to solve Dirichlet problem for elliptic equations. Bivariate spline proper subspace of S42,3([Delta]mn(2)) satisfying homogeneous boundary conditions on Type-2 triangulations and quadratic B-spline interpolating boundary functions are primarily constructed. Linear and nonlinear elliptic equations are solved by Galerkin quartic B-spline finite element method. Numerical examples are provided to illustrate the proposed method is flexible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010
A new parallel iterative algorithm for solving 2D Poisson equation
In this article, a finite difference parallel iterative (FDPI) algorithm for solving 2D Poisson equation was presented. Based on the domain decomposition, the domain was divided into four sub-domains and the four iterative schemes were constructed from the classical five-point difference scheme to implement the algorithm differently with the number of iterations of odd or even. Although the iterative schemes are semi-implicit, they can be computed explicitly and in parallel in combining with the boundary conditions. The convergence of the presented algorithm was proved. Particularly, a relaxation factor [omega] was added into the iterative schemes to improve the convergence rate and decrease the number of iterations. Finally, several numerical experiments were presented to examine the efficiency and accuracy of the iterative algorithm. Also, the comparison between the numerical results that were derived from Jacobi, Gauss-Seidel iterative algorithms, Mathematics Stencil (Fen et al., China Sci 35 (2005), 901-909) method, and the presented algorithm with the optimal relaxation factor [omega]opt demonstrated that the presented algorithm has smaller number of iterations, shorter computational time, and faster convergence rate. Furthermore, the presented algorithm is also applicable to 2D variable coefficient elliptic problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010
Parallel characteristic finite difference method for convection-diffusion equations
Based on the overlapping domain decomposition, an efficient parallel characteristic finite difference scheme is proposed for solving convection-diffusion equations numerically. We give the optimal convergence order in error estimate analysis, which shows that we just need to iterate once or twice at each time level to reach the optimal convergence order. Numerical experiments also confirm the theoretical analysis. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010
Computer identifies authentic Van Gogh
(PhysOrg.com) -- Dutch researcher Igor Berezhnoy has developed computer algorithms to support art historians and other art experts in their visual assessment of paintings. His digital technology is capable of distinguishing a forgery from an authentic Van Gogh based on the painter's characteristic brush work.
When does two-grid optimality carry over to the V-cycle?
We investigate additional condition(s) that confirm that a V-cycle multigrid method is satisfactory (say, optimal) when it is based on a two-grid cycle with satisfactory (say, level-independent) convergence properties. The main tool is McCormick's bound on the convergence factor (SIAM J. Numer. Anal. 1985; 22:634-643), which we showed in previous work to be the best bound for V-cycle multigrid among those that are characterized by a constant that is the maximum (or minimum) over all levels of an expression involving only two consecutive levels; that is, that can be assessed considering only two levels at a time. We show that, given a satisfactorily converging two-grid method, McCormick's bound allows us to prove satisfactory convergence for the V-cycle if and only if the norm of a given projector is bounded at each level. Moreover, this projector norm is simple to estimate within the framework of Fourier analysis, making it easy to supplement a standard two-grid analysis with an assessment of the V-cycle potentialities. The theory is illustrated with a few examples that also show that the provided bounds may give a satisfactory sharp prediction of the actual multigrid convergence. Copyright © 2009 John Wiley & Sons, Ltd.
Theoretical analyses of nonaqueous phase liquid dissolution-induced instability in two-dimensional fluid-saturated porous media
This paper deals with the theoretical aspects of nonaqueous phase liquid (NAPL)-dissolution-induced instability in two-dimensional fluid-saturated porous media including solute dispersion effects.After some weaknesses associated with the previous work are analyzed and overcome, a comprehensive dimensionless number, known as the Zhao number, is proposed to represent the main driving force and three controlling mechanisms of an NAPL-dissolution system that has a finite domain. The linear stability analysis is carried out to derive the critical value of the comprehensive dimensionless number of the NAPL-dissolution system in a limit case as the ratio of the equilibrium concentration to the density of the NAPL approaches zero. As a result, a theoretical criterion that can be used to assess the instability of planar NAPL-dissolution fronts in two-dimensional fluid-saturated porous media of finite domains has been established. Not only can the present theoretical results be used for the theoretical understanding of the effect of solute dispersion on the instability of an NAPL-dissolution front in the fluid-saturated porous medium of either a finite domain or an infinite domain, but also they can be used as benchmark solutions for verifying numerical methods employed to simulate detailed morphological evolution processes of NAPL-dissolution fronts in two-dimensional fluid-saturated porous media. Copyright © 2009 John Wiley & Sons, Ltd.
Shape of hexagonal hydrostatic menisci
A numerical method is implemented for computing the shape of an infinite, hexagonal, doubly periodic, hydrostatic meniscus originating from contact lines whose projections in the horizontal plane are circles. The contact lines themselves may lie on vertical cylinders or spherical objects. The Laplace-Young equation determining the meniscus shape is solved by a finite-difference method in orthogonal curvilinear coordinates generated by conformal mapping. The elevation of the contact lines is either prescribed or computed as part of the solution to ensure a specified contact angle. The results illustrate that the contact line spacing has an important effect on the contact angle or contact line distribution. Meniscus shapes exist only for a limited range of pressure differences between the upper and lower fluid that depend on the capillary length. Copyright © 2009 John Wiley & Sons, Ltd.
A posteriori error of a discontinuous Galerkin scheme for compressible miscible displacement problems with molecular diffusion and dispersion
A kind of compressible miscible displacement problems including molecular diffusion and dispersion in porous media is studied. A symmetric interior penalty discontinuous Galerkin method is applied to the coupled system of flow and transport. Explicit a posteriori error estimates are obtained based on the duality argument, approximation properties and residual notations. The resulting error bound is shown to be reliable and worthwhile for guiding dynamic mesh adaptivity. Copyright © 2009 John Wiley & Sons, Ltd.
Lattice Boltzmann study of bubble entrapment during droplet impact
The entrapment of bubble during droplet impact on solid surfaces was studied by using a phase-field lattice Boltzmann model (LBM). Numerous three-dimensional simulations were performed and several types of entrapment were found under specific impact conditions. Some cases are similar to previous studies, whereas others are reported for the first time. Dissections of the entrapment processes were carried out, and detailed analyses of the flow fields and interface motions were provided at selected moments for certain cases. The effects of the Reynolds number, the Weber number and the surface wettability were analyzed based on the observation. Besides, some computational issues of phase-field simulations were discussed for the present problem. Finally, the impact condition to prevent entrapment was preliminarily estimated in terms of the Ohnesorge number upon examination of numerous simulations. Copyright © 2009 John Wiley & Sons, Ltd.
Parallel Factorizations in Numerical Analysis. (arXiv:0912.3678v1 [math.NA])
In this paper we review the parallel solution of sparse linear systems,
usually deriving by the discretization of ODE-IVPs or ODE-BVPs. The approach is
based on the concept of parallel factorization of a (block) tridiagonal matrix.
This allows to obtain efficient parallel extensions of many known matrix
factorizations, and to derive, as a by-product, a unifying approach to the
parallel solution of ODEs.
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