Nonlinear function approximation: Computing smooth solutions with an adaptive greedy algorithm

In contrast to linear schemes, nonlinear approximation techniques allow for dimension independent rates of convergence. Unfortunately, typical algorithms (such as, e.g., backpropagation) are not only computationally demanding, but also unstable in the presence of data noise. While we can show stability for a weak relaxed greedy algorithm, the resulting method has the drawback that it requires in practise unavailable smoothness information about the data.In this work we propose an adaptive greedy algorithm which does not need this information but rather recovers it iteratively from the available data. We show that the generated approximations are always at least as smooth as the original function and that the algorithm also remains stable, when it is applied to noisy data. Finally, the applicability of this algorithm is demonstrated by numerical experiments.     

Time shifted aliasing error upper bounds for truncated sampling cardinal series

Time shifted aliasing error upper bound extremals for the sampling reconstruction procedure are fully characterized. Sharp upper bounds are found on the aliasing error of truncated cardinal series and the corresponding extremals are described for entire functions from certain specific Lp, p>1, classes. Analogous results are obtained in multidimensional regular sampling. Truncation error analysis is provided in all cases considered. Moreover, sharpness of bounding inequalities, convergence rates and various sufficient conditions are discussed.     

The spin-

General Galerkin (G2) is a new computational method for turbulent flow, where a stabilized Galerkin finite element method is used to compute approximate weak solutions to the Navier–Stokes equations directly, without any filtering of the equations as in a standard approach to turbulence simulation, such as large eddy simulation, and thus no Reynolds stresses are introduced, which need modelling. In this paper, G2 is used to compute the drag coefficient c_D for the flow past a circular cylinder at Reynolds number Re=3900, for which the flow is turbulent. It is found that it is possible to approximate c_D to an accuracy of a few percent, corresponding to the accuracy in experimental results for this problem, using less than 10⁵ mesh points, which makes the simulations possible using a standard PC. The mesh is adaptively refined until a stopping criterion is reached with respect to the error in a chosen output of interest, which in this paper is c_D. Both the stopping criterion and the mesh‐refinement strategy are based on a posteriori error estimates, in the form of a space–time integral of residuals times derivatives of the solution of a dual problem, linearized at the approximate solution, and with data coupling to the output of interest. Copyright © 2008 John Wiley & Sons, Ltd.     

Efficient computation of mean drag for the subcritical flow past a circular cylinder using general Galerkin G2

General Galerkin (G2) is a new computational method for turbulent flow, where a stabilized Galerkin finite element method is used to compute approximate weak solutions to the Navier–Stokes equations directly, without any filtering of the equations as in a standard approach to turbulence simulation, such as large eddy simulation, and thus no Reynolds stresses are introduced, which need modelling. In this paper, G2 is used to compute the drag coefficient c_D for the flow past a circular cylinder at Reynolds number Re=3900, for which the flow is turbulent. It is found that it is possible to approximate c_D to an accuracy of a few percent, corresponding to the accuracy in experimental results for this problem, using less than 10⁵ mesh points, which makes the simulations possible using a standard PC. The mesh is adaptively refined until a stopping criterion is reached with respect to the error in a chosen output of interest, which in this paper is c_D. Both the stopping criterion and the mesh‐refinement strategy are based on a posteriori error estimates, in the form of a space–time integral of residuals times derivatives of the solution of a dual problem, linearized at the approximate solution, and with data coupling to the output of interest. Copyright © 2008 John Wiley & Sons, Ltd.     

矩形校正公式的误差分析

This paper presents truncation errors among Corrector Formula for left Rectangular rule and Corrector Formula for middle Rectangular rule respectively.It also displays an analysis on convergence order of compound corrector formulas for rectangular rule.Examples of numerical calculation have validated theoretical analysis.     

THE GENERALIZED REFLEXIVE SOLUTION FOR A CLASS OF MATRIX EQUATIONS （AX- B, XC- D）

In this article, the generalized reflexive solution of matrix equations （AX = B, XC = D） is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is solved.     

Global existence of strong solutions to the Cauchy problem for a 1D radiative gas

We consider a one-dimensional radiation hydrodynamics model in the case of the equilibrium diffusion approximation which is described by the compressible Navier-Stokes system with the additional terms in the pressure and internal energy respectively, which embody the effect of radiation. Under the physical growth conditions on the heat conductivity, we establish the existence and uniqueness of strong solutions to the Cauchy problem with large initial data, where the initial density and velocity may have differing constant states at infinity. Moreover, we show that if there is no vacuum in the initial density, then, the vacuum and concentration of the density will never occur in any finite time.     

Asymptotic expansions of the Robbins-Monro process

Assume that the function values f(x) of an unknown regression function f: ℝ → ℝ can be observed with some random error V. To estimate the zero ϑ of f, Robbins and Monro suggested to run the recursion X _n+1 = X _n − a/n Y _n with Y _n = f(X _n ) − V _n . Under regularity assumptions, the normalized Robbins-Monro process, given by (X _n+1 − ϑ)/√Var(X _n+1, is asymptotically standard normal. In this paper Edgeworth expansions are presented which provide approximations of the distribution function up to an error of order o(1/√n) or even o(1/n). As corollaries asymptotic confidence intervals for the unknown parameter ϑ are obtained with coverage probability errors of order O(1/n). Further results concern Cornish-Fisher expansions of the quantile function, an Edgeworth correction of the distribution function and a stochastic expansion in terms of a bivariate polynomial in 1/√n and a standard normal random variable. The proofs of this paper heavily rely on recently published results on Edgeworth expansions for approximations of the Robbins-Monro process.      

On the estimation of optimal batch sizes in the analysis of simulation output

The estimation of the variance of point estimators is a classical problem of stochastic simulation. A more specific problem addresses the estimation of the variance of a sample mean from a steady-state autocorrelated process. Many proposed estimators of the variance of the sample mean are parameterized by batch size. A critical problem is to find an appropriate batch size that provides a good tradeoff between bias and variance. This paper proposes a procedure for determining the optimal batch size to minimize the mean squared error of estimators of the variance of the sample mean. This paper also presents the results of empirical studies of the procedure. The experiments involve symmetric two-state Markov chain models, first-order autoregressive processes, seasonal autoregressive processes, and queue-waiting times for several M/M/1 queueing models. The empirical results indicate that the estimation procedure works nearly as well as it would if the parameters of the processes were known.     

Comparison of GMRES and ORTHOMIN for the black oil model on unstructured grids

This paper addresses an application of ORTHOMIN and GMRES to petroleum reservoir simulation using the black oil model on unstructured grids. Comparisons between these two algorithms are presented in terms of storage and total flops per restart step. Numerical results indicate that GMRES is faster than ORTHOMIN for all tested petroleum reservoir problems, particularly for large scale problems. The control volume function approximation method is utilized in the discretization of the governing equations of the black oil model. This method can accurately approximate both the pressure and velocity in the simulation of multiphase flow in porous media, effectively reduce grid orientation effects, and be easily applied to arbitrarily shaped control volumes. It is particularly suitable for hybrid grid reservoir simulation. Copyright © 2006 John Wiley & Sons, Ltd.