多元积分方程自适应解法的最优化

In this paper it was considered that problem of optimization of adaptive direct algorithm of approximate solution of integral equations. For the Fredholm integral equations of second kind with kernels belonging to Besov classes there is determined the exact order of the error of an optimal adaptive direct algorithm and a algorithm for realizing it is indicated.     

多元Fredholm积分方程自适应直接方法的优化

考虑了以各向异性光滑函数为核及自由项的多元Fredholm积分方程类的自适应直接方法的优化,得到误差阶的精确估计及相应的最优算法。     

A meshless like method for the approximate solution of integral equations over polygons

In this paper, a numerical scheme based on a Gauss-like cubature formula from Sammariva and Vianello (2007) [1] is introduced for approximate solution of integral equations over a polygonal domain with a piecewise straight lines boundary in R²${\mathbb{R}}^2$. The proposed technique is a meshless like method with sufficient precision, which does not require any discretization of the polygon domain or any preprocessing such as mesh refinement. The error analysis of the method is provided and some numerical experiments are also presented to evaluate the performance of the proposed algorithm.     

Some algorithms for approximate solution of convolution integral equations

We study algorithms for approximate solution of convolution integral equations of first kind in which an approximate solution of the original equations can be constructed using the solution of equations having simpler structure or certain necessary properties, for example regularized equations. We obtain a number of algorithms for recursive refinement of the solution, where if the a priori information is given by linear algorithms, such algorithms are pseudo-recursive. The results may have application in problems of secondary processing of the data of indirect measurement. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Optimization of direct methods for the approximate solution of weakly singular integral equations

For a class of weakly singular integral equations with power and logarithmic singularities, we establish the optimal order of the error of direct methods and indicate the procedure which realizes this order.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1246–1254, September, 1994.     

An adaptive fast direct solver for boundary integral equations in two dimensions

We describe an algorithm for the rapid direct solution of linear algebraic systems arising from the discretization of boundary integral equations of potential theory in two dimensions. The algorithm is combined with a scheme that adaptively rearranges the parameterization of the boundary in order to minimize the ranks of the off-diagonal blocks in the discretized operator, thus obviating the need for the user to supply a parameterization r of the boundary for which the distance ‖r(s)−r(t)‖ between two points on the boundary is related to their corresponding distance |s−t| in the parameter space. The algorithm has an asymptotic complexity of , where N is the number of nodes in the discretization. The performance of the algorithm is illustrated with several numerical examples.     

Weighted Monte Carlo method for an approximate solution of the nonlinear coagulation equation

New weighted modifications of direct statistical simulation methods designed for the approximate solution of the nonlinear Smoluchowski equation are developed on the basis of stratification of the interaction distribution in a multiparticle system according to the index of a pair of interacting particles. The weighted algorithms are validated for a model problem with a known solution. It is shown that they effectively estimate variations in the functionals with varying parameters, in particular, with the initial number N ₀ of particles in the simulating ensemble. The computations performed for the problem with a known solution confirm the semiheuristic hypothesis that the model error is O(N ₀ ^?1 ). Estimates are derived for the derivatives of the approximate solution with respect to the coagulation coefficient.     

An approximate solution to the integral radiative transfer equation in an optically thick slab

We consider the problem of solving the integral form of the radiative transfer equation in an atmosphere with optical thickness τ₀?1. We propose two methods transforming this problem to a finite set of the independent problems of the same type set in an atmosphere with optical thickness much less then τ₀. The error estimates are derived. Copyright © 2007 John Wiley & Sons, Ltd.     

Fast adaptive algorithms in the non-standard form for multidimensional problems

We present a fast, adaptive multiresolution algorithm for applying integral operators with a wide class of radially symmetric kernels in dimensions one, two and three. This algorithm is made efficient by the use of separated representations of the kernel. We discuss operators of the class (−Δ+μ²I)^−α, where μ0 and 0<α<3/2, and illustrate the algorithm for the Poisson and Schrödinger equations in dimension three. The same algorithm may be used for all operators with radially symmetric kernels approximated as a weighted sum of Gaussians, making it applicable across multiple fields by reusing a single implementation. This fast algorithm provides controllable accuracy at a reasonable cost, comparable to that of the Fast Multipole Method (FMM). It differs from the FMM by the type of approximation used to represent kernels and has the advantage of being easily extendable to higher dimensions.     

Global well-posedness for the 2D critical dissipative quasi-geostrophic equation

We study the initial value problem for the 2D critical dissipative quasi-geostrophic equation. We prove the global existence for small data in the scale invariant Besov spaces Bp,12/p,1≤p≤∞. In particular, for p=∞, our result does not impose any regularity on the initial data. Our proofs are based on an exponential decay estimate of the semigroup e-tk(-Δ)αand the use of space-time Besov spaces.     

Projection solution methods for one nonlinear singular integral equation

In this paper we consider a general projection method for the solution of a nonlinear singular integral equation and its applications in the method of orthogonal polynomials, the subdomains method, and the collocation method.     

Global attractivity and positive almost periodic solution for delay logistic differential equation

In this paper, we establish some sufficient conditions for the global attractivity of positive solutions and the unique existence of positive almost periodic solutions for delay logistic differential equations of the form

x^′(t)=r(t)x(t)(a(t)−b(t)x(t)−L(_xt)).$x^{\prime }\left(t\right)=r\left(t\right)x\left(t\right)(a\left(t\right)-b\left(t\right)x\left(t\right)-L\left(x_t\right))\text{.}$

A free boundary problem of diffusion equation with integral condition

We consider a free boundary problem of heat equation with integral condition on the unknown free boundary. Results of solution regularity and problem well-posedness are presented.     

Finite time blow up for a Navier-Stokes like equation

We consider an equation similar to the Navier-Stokes equation. We show that there is initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space). The purpose is to show the limitations of the so-called semigroup method for the Navier-Stokes equation. We also consider the possibility of existence of solutions with initial data in the Besov space . We give initial data in this space for which there is no reasonable solution for the Navier-Stokes like equation.

     

Mesh adaptive direct search algorithms for mixed variable optimization

This paper introduces a new derivative-free class of mesh adaptive direct search (MADS) algorithms for solving constrained mixed variable optimization problems, in which the variables may be continuous or categorical. This new class of algorithms, called mixed variable MADS (MV-MADS), generalizes both mixed variable pattern search (MVPS) algorithms for linearly constrained mixed variable problems and MADS algorithms for general constrained problems with only continuous variables. The convergence analysis, which makes use of the Clarke nonsmooth calculus, similarly generalizes the existing theory for both MVPS and MADS algorithms, and reasonable conditions are established for ensuring convergence of a subsequence of iterates to a suitably defined stationary point in the nonsmooth and mixed variable sense.     

Symmetry of integral equation systems on bounded domains

In this paper, we investigate the symmetry of domains and solutions of integral equation systems on bounded domains. Under some natural integrability conditions, we prove that the domains are balls, all positive solutions of systems are radially symmetric and monotone decreasing with respect to the radius.     

Numerical solution of the abel integral equation

A numerical method for the solution of the Abel integral equation is presented. The known function is approximated by a sum of Chebyshev polynomials. The solution can then be expressed as a sum of generalized hypergeometric functions, which can easily be evaluated, using a simple recurrence relation.     

热方程的一些端点估计及其在Navier-Stokes方程中的应用——献给陆善镇教授75华诞

本文首先讨论热方程初值问题的解在Hardy、BMO（bounded mean oscillation）和Besov型空间中的估计.然后本文结合Coifmann-Lions-Meyer-Semmes在Hardy空间中的补偿紧性结果,给出Navier-Stokes方程整体弱解的二阶导数的一些端点估计.     

Exact and approximate resolution of integral multiflow and multicut problems: algorithms and complexity

This is a summary of the author’s PhD thesis supervised by Marie- Christine Costa and Frédéric Roupin and defended on 20 November 2006 at the Conservatoire National des Arts et Métiers in Paris (France). The thesis is written in French and is available upon request from the author. This work deals with two well-known optimization problems from graph theory: the maximum integral multiflow and the minimum multicut problems. The main contributions of this thesis concern the polynomial-time solvability and the approximation of these two problems (and of some of their variants) in classical classes of graphs: bounded tree-width graphs, planar graphs and grids, digraphs, etc.