The adaptive operational Tau method for systems of ODEs

As the Tau method, like many other numerical methods, has the limitation of using a fixed step size with some high degree (order) of approximation for solving initial value problems over long intervals, we introduce here the adaptive operational Tau method. This limitation is very much problem dependent and in such case the fixed step size application of the Tau method loses the true track of the solution. But when we apply this new adaptive method the true solution is recovered with a reasonable number of steps. To illustrate the effectiveness of this method we apply it to some stiff systems of ordinary differential equations (ODEs). The numerical results confirm the efficiency of the method.     

Acceleration of the convergence of the Laguerre series in the problem of inverting the Laplace transform

When the Laplace transform is inverted numerically, the original function is sought in the form of a series in the Laguerre polynomials. To accelerate the convergence of this series, the Euler-Knopp method is used. The techniques for selecting the optimal value of the parameter of the transform on the real axis and in the complex plane are proposed.     

不确定非线性系统的周期信号自适应跟踪

考虑不确定非线性系统的周期信号的自适应跟踪问题. 系统的不确定性不能参数化,周期信号由一非线性系统产生.提出了跟踪周期信号的自适应控制律. 此控制律保证了闭环系统所有的信号有界和跟踪误差趋于零. 已有的有关的周期信号跟踪控制律只能保证跟踪误差的平方在一周期上的积分趋于零.     

带有固定步长的非单调自适应信赖域算法

提出了求解无约束优化问题带有固定步长的非单调自适应信赖域算法.信赖域半径的修正采用自适应技术,算法在试探步不被接受时,采用固定步长寻找下一迭代点.并在适当的条件下,证明算法具有全局收敛性和超线性收敛性.初步的数值试验表明算法对高维问题具有较好的效果.     

Non-linear sampling recovery based on quasi-interpolant wavelet representations

We investigate a problem of approximate non-linear sampling recovery of functions on the interval expressing the adaptive choice of n sampled values of a function to be recovered, and of n terms from a given family of functions Φ. More precisely, for each function f on , we choose a sequence of n points in , a sequence of n functions defined on and a sequence of n functions from a given family Φ. By this choice we define a (non-linear) sampling recovery method so that f is approximately recovered from the n sampled values f(ξ ¹), f(ξ ²),..., f(ξ ⁿ ), by the n-term linear combination

Rational approximations for the quotient of Gamma values

In this paper, we continue to study properties of rational approximations to Euler's constant and values of the Gamma function defined by linear recurrences, which were found recently by A.I. Aptekarev and T. Rivoal. Using multiple Jacobi-Laguerre orthogonal polynomials we present rational approximations to the quotient of values of the Gamma function at rational points. As a limit case of our result, we obtain new explicit formulas for numerators and denominators of the Aptekarev approximants to Euler's constant.     

ERROR REDUCTION IN ADAPTIVE FINITE ELEMENT APPROXIMATIONS OF ELLIPTIC OBSTACLE PROBLEMS

We consider an adaptive finite element method （AFEM） for obstacle problems associated with linear second order elliptic boundary value problems and prove a reduction in the energy norm of the discretization error which leads to R-linear convergence. This result is shown to hold up to a consistency error due to the extension of the discrete multipliers （point functionals） to H^-1 and a possible mismatch between the continuous and discrete coincidence and noncoincidence sets. The AFEM is based on a residual-type error estimator consisting of element and edge residuals. The a posteriori error analysis reveals that the significant difference to the unconstrained case lies in the fact that these residuals only have to be taken into account within the discrete noncoincidence set. The proof of the error reduction property uses the reliability and the discrete local efficiency of the estimator as well as a perturbed Galerkin orthogonality. Numerical results are given illustrating the performance of the AFEM.     

Adaptive mesh refinement and load balancing based on multi-level block-structured Cartesian mesh

We developed a framework for a distributed-memory parallel computer that enables dynamic data management for adaptive mesh refinement and load balancing. We employed simple data structure of the building cube method (BCM) where a computational domain is divided into multi-level cubic domains and each cube has the same number of grid points inside, realising a multi-level block-structured Cartesian mesh. Solution adaptive mesh refinement, which works efficiently with the help of the dynamic load balancing, was implemented by dividing cubes based on mesh refinement criteria. The framework was investigated with the Laplace equation in terms of adaptive mesh refinement, load balancing and the parallel efficiency. It was then applied to the incompressible Navier–Stokes equations to simulate a turbulent flow around a sphere. We considered wall-adaptive cube refinement where a non-dimensional wall distance y⁺ near the sphere is used for a criterion of mesh refinement. The result showed the load imbalance due to y⁺ adaptive mesh refinement was corrected by the present approach. To utilise the BCM framework more effectively, we also tested a cube-wise algorithm switching where an explicit and implicit time integration schemes are switched depending on the local Courant-Friedrichs-Lewy (CFL) condition in each cube.     

Adaptive Wavelet BEM for Boundary Integral Equations: Theory and Numerical Experiments

We are concerned with the numerical treatment of boundary integral equations by the adaptive wavelet boundary element method. In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in ?³. The corresponding operator equations are treated by adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions based on the Besov regularity of the exact solution.     

CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON'S EQUATION IN THE FEEC FRAMEWORK

Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained by solving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78 (2009) 35-53) established convergence and optimality of an adaptive mixed finite element method using Raviart-Thomas or Brezzi-Douglas-Marini elements for Poisson's equation on contractible domains in $\mathbb{R}^2$, which can be viewed as a boundary problem on the de Rham complex. Recently Demlow and Hirani (Found. Math. Comput. 14 (2014) 1337-1371) developed fundamental tools for a posteriori analysis on the de Rham complex. In this paper, we use tools in FEEC to construct convergence and complexity results on domains with general topology and spatial dimension. In particular, we construct a reliable and efficient error estimator and a sharper quasi-orthogonality result using a novel technique. Without marking for data oscillation, our adaptive method is a contraction with respect to a total error incorporating the error estimator and data oscillation.