Approximation by linear methods of summation of orthogonal series

We give an estimate of the rapidity of convergence of certain linear means of the orthogonal series _k =0 c_{k k} (x) to the functionf (x) L₂ (a, b) defined by this series according to the Riesz-Fisher theorem for almost all x (a, b). The results obtained are, in a certain sense, final.Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 345–356, March, 1968.     

Hybrid BDF methods for the numerical solutions of ordinary differential equations

In this article, we have presented the details of hybrid methods which are based on backward differentiation formula (BDF) for the numerical solutions of ordinary differential equations (ODEs). In these hybrid BDF, one additional stage point (or off-step point) has been used in the first derivative of the solution to improve the absolute stability regions. Stability domains of our presented methods have been obtained showing that all these new methods, we say HBDF, of order p, p = 2,4,..., 12, are A(α)-stable whereas they have wide stability regions comparing with those of some known methods like BDF, extended BDF (EBDF), modified EBDF (MEBDF), adaptive EBDF (A-EBDF), and second derivtive Enright methods. Numerical results are also given for five test problems.     

Approximation of integrable functions by linear methods almost everywhere

It is shown that 2π periodic functions whose (r-1)-th derivatives have bounded variation (r > 0) can be approximated by de La Vallée-Poussin sums σ _n,m (an ?m =m (n) ?An,0 <a<A<1) at almost all points with a rate o(n^?r). For functions belonging to the class Lip (α, L) (0 <α < 1), any natural N, and a positive ?, we have almost everywhere $$|f(x) - \sigma _{n,m} (f;x)| \leqslant c(f,x)n^{ - \alpha } lnn \ldots ln_N^{1 + \varepsilon } n,$$ where \(ln_k x = \underbrace {ln \ldots ln x}_k(k = 1, 2, \ldots )\) . For any triangular method of summation T with bounded coefficients we construct functions belonging to Lip (α, L) (0 < α < 1) and such that almost everywhere, $$\mathop {\overline {\lim } }\limits_{n \to \infty } |f(x) - \tau _n (f;x)|n^a (ln n \ldots ln_N n)^{ - a} = \infty $$ where the τ_n(f; x) are the means of the method T.     

Exact first and second derivatives by automatic differentiation

Automatic differentiation (AD) is a method for computing derivatives of outputs of numerical programs with respect to its inputs both accurately (with machine precision) and efficiently. The two basic modes of AD – forward and reverse – and combinations thereof yield products of a vector with the Jacobian, its transposed, or the Hessian, respectively. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Improving the accuracy of BDF methods for index 3 differential-algebraic equations

Methods for solving index 3 DAEs based on BDFs suffer a loss of accuracy when there is a change of step size or a change of order of the method. A layer of nonuniform convergence is observed in these cases, andO(1) errors may appear in the algebraic variables. From the viewpoint of error control, it is beneficial to allow smooth changes of step size, and since most codes based on BDFs are of variable order, it is also of interest to avoid the inaccuracies caused by a change of order of the method. In the case of BDFs applied to index 3 DAEs in semi-explicit form, we present algorithms that correct toO(h) the inaccurate approximations to the algebraic variables when there are changes of step size in the backward Euler method. These algorithms can be included in an existing code at a very small cost. We have also described how to obtain formulas that correct theO(1) errors in the algebraic variables appearing after a change of order.This author thanks the Centro de Estadística y Software Matemático de la Universidad Simón Bolívar (CESMa) for permitting her free use of its research facilities.     

Conservative set valued fields,automatic differentiation,stochastic gradient methods and deep learning

Mathematical Programming - Modern problems in AI or in numerical analysis require nonsmooth approaches with a flexible calculus. We introduce generalized derivatives called conservative fields for...     

Numerical solution of the Burgers’ equation by automatic differentiation

We compute the solution of the one-dimensional Burgers’ equation by marching the solution in time using a Taylor series expansion. Our approach does not require symbolic manipulation and does not involve the solution of a system of linear or non-linear algebraic equations. Instead, we use recursive formulas obtained from the differential equation to calculate exact values of the derivatives needed in the Taylor series. We illustrate the effectiveness of our method by solving four test problems with known exact solutions. The numerical solutions we obtain are in excellent agreement with the exact solutions, while being superior to other previously reported numerical solutions.     

Approximation of weak sense stationary stochastic processes from local averages

We show that a weak sense stationary stochastic process can be approximated by local averages. Explicit error bounds are given. Our result improves an early one from Splettstosser.     

Limits of nonlinear weak asymptotic methods

We propose a mathematical limit of $L^1$-stable weak asymptotic methods. A family of $L^1$-stable approximate solutions is transformed into a normal family of holomorphic functions defined in a complex domain having the real space on its boundary. This provides a holomorphic function which is the same mathematical object as the solutions from explicit calculations. The weak limit of the approximate solutions from weak asymptotic methods in the space of bounded Radon measures is recovered as a boundary value of this holomorphic function.     

New class of hybrid BDF methods for the computation of numerical solutions of IVPs

A new class of hybrid BDF-like methods is presented for solving systems of ordinary differential equations (ODEs) by using the second derivative of the solution in the stage equation of class 2 + 1hybrid BDF-like methods to improve the order and stability regions of these methods. An off-step point, together with two step points, has been used in the first derivative of the solution, and the stability domains of the new methods have been obtained by showing that these methods are A-stable for order p, p =?3,4,5,6,7and A(α)-stable for order p, 8 ≤ p ≤?14. The numerical results are also given for four test problems by using variable and fixed step-size implementations.     

Approximation with best order by Toeplitz-irregular methods of summability of Fourier series

Translated from Matematicheskie Zametki, Vol. 56, No. 6, pp. 117–121, December, 1994.     

Approximation of classes of convolutions by linear methods of summation of Fourier series

We consider a family of special linear methods of summation of Fourier series and establish exact equalities for the approximation of classes of convolutions with even and odd kernels by polynomials generated by these methods.     

Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods

In this paper, we analyze the biharmonic eigenvalue problem by two nonconforming finite elements, Q and E Q. We obtain full order convergence rate of the eigenvalue approximations for the biharmonic eigenvalue problem based on asymptotic error expansions for these two nonconforming finite elements. Using the technique of eigenvalue error expansion, the technique of integral identities, and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008