Approximation Schemes for Functional Optimization Problems

Approximation schemes for functional optimization problems with admissible solutions dependent on a large number d of variables are investigated. Suboptimal solutions are considered, expressed as linear combinations of n-tuples from a basis set of simple computational units with adjustable parameters. Different choices of basis sets are compared, which allow one to obtain suboptimal solutions using a number n of basis functions that does not grow “fast” with the number d of variables in the admissible decision functions for a fixed desired accuracy. In these cases, one mitigates the “curse of dimensionality,” which often makes unfeasible traditional linear approximation techniques for functional optimization problems, when admissible solutions depend on a large number d of variables. Marcello Sanguineti was partially supported by a PRIN grant from the Italian Ministry for University and Research (project “Models and Algorithms for Robust Network Optimization”).     

Efficient Collocation Schemes for Singular Boundary Value Problems

We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully designed modification of this error estimate which not only results in less computational work but also appears to perform satisfactorily for singular problems. We give a full analytical justification for the asymptotical correctness of the error estimate when it is applied to a general nonlinear regular problem. For the singular case, we are presently only able to provide computational evidence for the full convergence order, the related analysis is still work in progress. This global estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. This procedure yields meshes suitable for an efficient numerical solution. Most importantly, we observe that the grid is refined in a way reflecting only the behavior of the solution and remains unaffected by the unsmooth direction field close to the singular point.     

NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs

We present NC-approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance trade-off as the best known approximation schemes for planar graphs. We also define the concept of λ-precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance trade-off than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for λ-precision unit disk graphs many more graph problems have efficient approximation schemes.Our NC-approximation schemes can also be extended to obtain efficient NC-approximation schemes for several PSPACE-hard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann, and Widmayer. The approximation schemes for hierarchically specified unit disk graphs presented in this paper are among the first approximation schemes in the literature for natural PSPACE-hard optimization problems.     

高维抛物方程迎风区域分裂差分方法

结合迎风方法和区域分裂思想,采用一阶迎风、二阶修正迎风法逼近高维抛物方程的对流项.内边界处和子区域分别对应区域分裂显隐格式;并运用极值原理和嵌入定理给出了收敛性分析,最后给出数值试验,说明其实际意义.     

Augmented Macro-Hybrid Mixed Finite Element Schemes for Elastic Contact Problems

On a setting of subdifferential models, variational augmented macro-hybrid mixed finite element schemes are formulated and analyzed for elastic unilateral contact problems with prescribed friction. Composition duality principles determine primal and dual mixed solvability, adopting coupling surjectivity for dualization. Macro-hybridization corresponds to nonoverlapping decompositions of elastic solid body systems, with displacement continuity and traction equilibrium transmission conditions dualized. In general, traction and displacement multipliers synchronize sub-bodies through nonmatching finite element interfaces. Three-field formulations give the basis for variational augmentation, in a sense of exact penalization, allowing speed-up of rates of convergence as well as proximation procedures of parallel numerical resolution algorithms.     

Additive Difference Schemes and Iteration Methods for Problems of Mathematical Physics

We construct additive difference schemes for first-order differential–operator equations. The exposition is based on the general theory of stability for operator–difference schemes in lattice Hilbert spaces. The main focus is on the case of additive decomposition with an arbitrary number of mutually noncommuting operator terms. Additive schemes for second-order evolution equations are considered in the same way.     

Multisplitting Iteration Schemes for Solving a Class of Nonlinear Complementarity Problems

We consider several synchronous and asynchronous multisplitting iteration schemes for solving aclass of nonlinear complementarity problems with the system matrix being an H-matrix.We establish theconvergence theorems for the schemes.The numerical experiments show that the schemes are efficient forsolving the class of nonlinear complementarity problems.     

Approximate Gradient Projection Method with Runge-Kutta Schemes for Optimal Control Problems

We consider an optimal control problem for systems governed by ordinary differential equations with control constraints. The state equation is discretized by the explicit fourth order Runge-Kutta scheme and the controls are approximated by discontinuous piecewise affine ones. We then propose an approximate gradient projection method that generates sequences of discrete controls and progressively refines the discretization during the iterations. Instead of using the exact discrete directional derivative, which is difficult to calculate, we use an approximate derivative of the cost functional defined by discretizing the continuous adjoint equation by the same Runge-Kutta scheme and the integral involved by Simpson's integration rule, both involving intermediate approximations. The main result is that accumulation points, if they exist, of sequences constructed by this method satisfy the weak necessary conditions for optimality for the continuous problem. Finally, numerical examples are given.     

A Hybrid Method for Nonlinear Least Squares Problems

A negative curvature method is applied to nonlinear least squares problems with indefinite Hessian approximation matrices. With the special structure of the method, a new switch is proposed to form a hybrid method. Numerical experiments show that this method is feasible and effective for zero-residual, small-residual and large-residual problems.     

Novel Hybrid Methods for Pseudomonotone Equilibrium Problems and Common Fixed Point Problems

The purpose of this article is to introduce some hybrid algorithms for finding a common element of the solution sets of pseudomonotone equilibrium problems and the fixed point sets of nonexpansive mappings in real Hilbert spaces. Our algorithms combine Mann’s iterative methods and Armijo line-search with parallel splitting-up and hybrid techniques. The strong convergence of the proposed algorithms are established without the assumption on the Lipschitz-type condition for the bifunctions involved.     

Some Uniformly Convergent Spline Difference Schemes for Singularly Parturbed Boundary Value Problems

In this paper some difference schemes for singularly perturbedtwo-point boundary value problems are derived using cubic splinesv(x) C¹[0,1]. One of them is the well-known Allen-Southwell-Il'inscheme. These schemes are first-order uniformly convergent.Numerical examples support the theoretical results.     

相对非扩展映射不动点的修正杂交迭代算法及其应用

在实一致凸、一致光滑Banach空间中,提出了新的修正杂交迭代算法,用以逼近相对非扩展映射的不动点.证明了一些强收敛定理,并讨论了迭代算法在逼近极大单调算子零点上的应用,推进了以往的研究成果.     

An Improved Hybrid Method for Inverse Obstacle Scattering Problems

An improved hybrid method is introduced in this paper as a numerical method to reconstruct the scatterer by far-field pattern for just one incident direction with unknown physical properties of the scatterer. The improved hybrid method inherits the idea of the hybrid method by Kress and Serranho which is a combination of Newton and decomposition method, and it improves the hybrid method by introducing a general boundary condition. The numerical experiments show the feasibility of this method.     

An Adaptive Hybrid Spectral Method for Stochastic Helmholtz Problems

The implementation of an adaptive hybrid spectral method for Helmholtz equations with random parameters is addressed in this work. New error indicators for generalized polynomial chaos for stochastic approximations and spectral element methods for physical approximations are developed, and systematic adaptive strategies are proposed associated with these error indicators. Numerical results show that these error indicators provide effective estimates for the approximation errors, and the overall adaptive procedure results in efficient approximation method for the stochastic Helmholtz equations.