Some Remarks on the Convex Feasibility Problem and Best Approximation Problem

In this paper we investigate several solution algorithms for the convex fea- sibility problem(CFP)and the best approximation problem(BAP)respectively.The algorithms analyzed are already known before,but by adequately reformulating the CFP or the BAP we naturally deduce the general projection method for the CFP from well-known steepest decent method for unconstrained optimization and we also give a natural strategy of updating weight parameters.In the linear case we show the connec- tion of the two projection algorithms for the CFP and the BAP respectively.In addition, we establish the convergence of a method for the BAP under milder assumptions in the linear case.We also show by examples a Bauschke's conjecture is only partially correct.

The Discrete Approximation Problem for a Special Case of Hermite-Type Polynomial Interpolation

Every univariate Hermite interpolation problem can be written as a pointwise limit of Lagrange interpolants.However,this property is not preserved for the multivariate case.In this paper,the authors first generalize the result of P.Gniadek.As an application,the authors consider the discrete approximation problem for a special case when the interpolation condition contains all partial derivatives of order less than n and one nth order differential polynomial.In addition,for the case of n≥3,the authors use the concept of Cartesian tensors to give a sufficient condition to find a sequence of discrete points,such that the Lagrange interpolation problems at these points converge to the given Hermite-type interpolant.

Optimal obstacle control problem

In the paper we discuss some properties of the state operators of the optimal obstacle control problem for elliptic variational inequality.Existence,uniqueness and regularity of the optimal control,problem are established.In addition,the approximation of the optimal obstacle problem is also studied.

Improved locality-sensitive hashing method for the approximate nearest neighbor problem

In recent years, the nearest neighbor search （NNS） problem has been widely used in various interesting applications. Locality-sensitive hashing （LSH）, a popular algorithm for the approximate nearest neighbor problem, is proved to be an efficient method to solve the NNS problem in the high-dimensional and large-scale databases. Based on the scheme of p-stable LSH, this paper introduces a novel improvement algorithm called randomness-based locality-sensitive hashing （RLSH） based on p-stable LSH. Our proposed algorithm modifies the query strategy that it randomly selects a certain hash table to project the query point instead of mapping the query point into all hash tables in the period of the nearest neighbor query and reconstructs the candidate points for finding the nearest neighbors. This improvement strategy ensures that RLSH spends less time searching for the nearest neighbors than the p-stable LSH algorithm to keep a high recall. Besides, this strategy is proved to promote the diversity of the candidate points even with fewer hash tables. Experiments are executed on the synthetic dataset and open dataset. The results show that our method can cost less time consumption and less space requirements than the p-stable LSH while balancing the same recall.

The{P,k+1}-reflexive Solution to System of Matrix Equations AX=C,XB=D

Let P∈C^(n×n)be a Hermitian and{k+1}-potent matrix,i.e.,P^(k+1)=P=P^(*),where(·)^(*)stands for the conjugate transpose of a matrix.A matrix X∈C^(n×n)is called{P,k+1}-reflexive(anti-reflexive)if PXP=X(P XP=-X).The system of matrix equations AX=C,XB=D subject to{P,k+1}-reflexive and anti-reflexive constraints are studied by converting into two simpler cases:k=1 and k=2,the least squares solution and the associated optimal approximation problem are also considered.

FINITE ELEMENT APPROXIMATION FOR A CLASS OF PARAMETER ESTIMATION PROBLEMS

This paper investigates the finite element approximation of a class of parameter estimation problems which is the form of performance as the optimal control problems governed by bilinear parabolic equations,where the state and co-state are discretized by piecewise linear functions and control is approximated by piecewise constant functions.The authors derive some a priori error estimates for both the control and state approximations.Finally,the numerical experiments verify the theoretical results.

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS,Ⅱ

In his series of three papers we study singularly perturbed (SP) boundary valueproblems for equations of elliptic and parabolic type. For small values of the pertur-bation parameter parabolic boundary and interior layers appear in these problems.If classical discretisation methods are used, the solution of the finite differencescheme and the approximation of the diffusive flux do not converge uniformly withrespect to this parameter. Using the method of special, adapted grids, we canconstruct difference schemes that allow approximation of the solution and the nor-malised diffusive flux uniformly with respect to the small parameter.We also consider sillgularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly We study what problems ap-pear, when classical schemes are used for the approximation of the spatial deriva-tives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Diriclilet, Neumann and RDbin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions-

FINITE ELEMENT APPROXIMATION OF EIGENVALUE PROBLEM FOR A COUPLED VIBRATION BETWEEN ACOUSTIC FIELD AND PLATE

We formulate a coupled vibration between plate and acoustic field in mathematically rigorous fashion. It leads to a non-standard eigenvalue problem. A finite element approximation is considered in an abstract way, and the approximate eigenvalue problem is written in an operator form by means of some Ritz projections. The order of convergence is proved based on the result of Babugka and Osborn. Some numerical example is shown for the problem for which the exact analytical solutions are calculated. The results shows that the convergence order is consistent with the one by the numerical analysis.

A BOUNDARY ELEMENT APPROXIMATION OF A SIGNORINI PROBLEM WITH FRICTION OBEYING COULOMB LAW

In this work, a Signorini problem with Coulomb friction in two dimensional elasticity is considered. Based on a new representation of the derivative of the double-layer potential, the original problem is reduced to a system of variational inequalities on the boundary of the given domain. The existence and uniqueess of this system are established for a small frictional coefficient. The boundary element approximation of this system is presented and an error estimate is given.

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS, III

In this series of three papers we study singularly perturbed (SP) boundary vaue problems for equations of elliptic and parabolic troe. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids,we can construct difference schemes that allow approkimation of the solution and the normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges E-uniformly We study what problems appear, when classical schemes are used for the approximation of the spatial deriva tives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic eqllation with discontinuous boundaxy conditions

BIFURCATION SOLUTION BRANCHESAND THEIR NUMERICAL APPROXIMATIONS OF A SEMI-LINEAR ELLIPTIC PROBLEM WITH TWO PARAMETERS

In this paper the bifurcation solution branches of a semi-linear elliptic problem are studied.

A STOCHASTIC MOVING BALLS APPROXIMATION METHOD OVER A SMOOTH INEQUALITY CONSTRAINT

We consider the problem of minimizing the average of a large number of smooth component functions over one smooth inequality constraint.We propose and analyze a stochastic Moving Balls Approximation(SMBA)method.Like stochastic gradient(SG)met hods,the SMBA method's iteration cost is independent of the number of component functions and by exploiting the smoothness of the constraint function,our method can be easily implemented.Theoretical and computational properties of SMBA are studied,and convergence results are established.Numerical experiments indicate that our algorithm dramatically outperforms the existing Moving Balls Approximation algorithm(MBA)for the structure of our problem.

The Equivalent Representation of the Breadth-One D-Invariant Polynomial Subspace and Its Discretization

This paper demonstrates the equivalence of two classes of D-invariant polynomial subspaces, i.e., these two classes of subspaces are different representations of the breadth-one D-invariant subspace. Moreover, the authors solve the discrete approximation problem in ideal interpolation for the breadth-one D-invariant subspace. Namely, the authors find the points, such that the limiting space of the evaluation functionals at these points is the functional space induced by the given D-invariant subspace, as the evaluation points all coalesce at one point.

On Scientific Data and Image Compression Based on Adaptive Higher-Order FEM

We present a novel compression algorithm for 2D scientific data and images based on exponentially-convergent adaptive higher-order finite element methods(FEM).So far,FEM has been used mainly for the solution of partial differential equations(PDE),but we show that it can be applied to data and image compression easily.The adaptive compression algorithm is trivial compared to adaptive FEM algorithms for PDE since the error estimation step is not present.The method attains extremely high compression rates and is able to compress a data set or an image with any prescribed error tolerance.Compressed data and images are stored in the standard FEM format,which makes it possible to analyze them using standard PDE visualization software.Numerical examples are shown.The method is presented in such a way that it can be understood by readers who may not be experts of the finite element method.

Simulations of Compressible Two-Medium Flow by Runge-Kutta Discontinuous Galerkin Methods with the Ghost Fluid Method

The original ghost fluid method (GFM) developed in [13] and the modifiedGFM (MGFM) in [26] have provided a simple and yet flexible way to treat twomediumflow problems. The original GFM and MGFM make the material interface"invisible" during computations and the calculations are carried out as for a singlemedium such that its extension to multi-dimensions becomes fairly straightforward.The Runge-Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservationlaws is a high order accurate finite element method employing the usefulfeatures from high resolution finite volume schemes, such as the exact or approximateRiemann solvers, TVD Runge-Kutta time discretizations, and limiters. In this paper,we investigate using RKDG finite element methods for two-medium flow simulationsin one and two dimensions in which the moving material interfaces is treated via nonconservativemethods based on the original GFM and MGFM. Numerical results forboth gas-gas and gas-water flows are provided to show the characteristic behaviors ofthese combinations.

Optimal Error Estimation of the Modified Ghost Fluid Method

The modified ghost fluid method(MGFM)has been shown to be robust and efficient when being applied to multi-medium compressible flows.In this paper,we rigorously analyze the optimal error estimation of the MGFM when it is applied to the multi-fluid Riemann problem.By analyzing the properties of the MGFM and the approximate Riemann problem solver(ARPS),we show that the interfacial status provided by the MGFM can achieve“third-order accuracy”in the sense of comparing to the exact solution of the Riemann problem,regardless of the solution type.In addition,our analysis further reveals that the ARPS based on a doubled shock structure in the MGFM is suitable for almost any conditions for predicting the interfacial status,and that the“natural”approach of“third-order accuracy”is practically less useful.Various examples are presented to validate the conclusions made.