Multistep methods for differential algebraic equations

Multistep methods for the differential/algebraic equations (DAEs) in the form of

Nonmonotone stabilization methods for nonlinear equations

We are concerned with defining new globalization criteria for solution methods of nonlinear equations. The current criteria used in these methods require a sufficient decrease of a particular merit function at each iteration of the algorithm. As was observed in the field of smooth unconstrained optimization, this descent requirement can considerably slow the rate of convergence of the sequence of points produced and, in some cases, can heavily deteriorate the performance of algorithms. The aim of this paper is to show that the global convergence of most methods proposed in the literature for solving systems of nonlinear equations can be obtained using less restrictive criteria that do not enforce a monotonic decrease of the chosen merit function. In particular, we show that a general stabilization scheme, recently proposed for the unconstrained minimization of continuously differentiable functions, can be extended to methods for the solution of nonlinear (nonsmooth) equations. This scheme includes different kinds of relaxation of the descent requirement and opens up the possibility of describing new classes of algorithms where the old monotone linesearch techniques are replace with more flexible nonmonotone stabilization procedures. As in the case of smooth unconstrained optimization, this should be the basis for defining more efficient algorithms with very good practical rates of convergence.This material is partially based on research supported by the Air Force Office of Scientific Research Grant AFOSR-89-0410, National Science Foundation Grant CCR-91-57632, and Istituto di Analisi dei Sistemi ed Informatica del CNR.     

An improved Menshov-Rademacher theorem

We study the a.e. convergence of orthogonal series defined over a general measure space. We give sufficient conditions which contain the Menshov-Rademacher theorem as an endpoint case. These conditions turn out to be necessary in the particular case where the measure space is the unit interval and the moduli of the coefficients form a nonincreasing sequence. We also prove a new version of the Menshov-Rademacher inequality.

     

Mosco approximation of integrands and integral functionals

It is shown that, given a lower semicontinuous convex integrandf satisfying a suitable integrability condition, there exists a sequence of Lipschitz simple integrands which Mosco converges tof and such that the sequence of conjugate integrands Mosco converges tof ^*. Moreover, this sequence can be chosen so that the sequence of associated integral functionals, respectively defined onL ¹(X) andL (X ^*), Mosco converges as well.We wish to thank Professor Erik J. Balder for interesting remarks on the first version of this work.     

Primal-dual interior point approach for computingl 1-solutions andl ∞-solutions of overdetermined linear systems

This paper concerns solving an overdetermined linear systemA ^T x=b in the leastl ₁-norm orl -norm sense, whereA ^m×n ,m<n. We show that the primal-dual interior point approach for linear programming can be applied, in an effective manner, to linear programming versions of thel ₁ andl -problems. The resulting algorithms are simple to implement and can attain quadratic or superlinear convergence rate. At each iteration, the algorithms must solve a linear system with anm×m positive-definite coefficient matrix of the formADA ^T , whereD is a positive diagonal matrix. The preliminary numerical results indicate that the proposed algorithms offer considerable promise.This research was supported in part by Grants NSF DMS-91-02761 and DOE DE-FG05-91-ER25100.     

Interior-point algorithm for quadratically constrained entropy minimization problems

In this paper, we develop an interior point algorithm for quadratically constrained entropy problems. The algorithm uses a variation of Newton's method to follow a central path trajectory in the interior of the feasible set. The primal-dual gap is made less than a given in at most steps, wheren is the dimension of the problem andm is the number of quadratic inequality constraints.     

A quadratically convergent O(\sqrt n L)-iteration algorithm for linear programming

Recently, Ye, Tapia and Zhang (1991) demonstrated that Mizuno—Todd—Ye's predictor—corrector interior-point algorithm for linear programming maintains the O( L)-iteration complexity while exhibiting superlinear convergence of the duality gap to zero under the assumption that the iteration sequence converges, and quadratic convergence of the duality gap to zero under the assumption of nondegeneracy. In this paper we establish the quadratic convergence result without any assumption concerning the convergence of the iteration sequence or nondegeneracy. This surprising result, to our knowledge, is the first instance of a demonstration of polynomiality and superlinear (or quadratic) convergence for an interior-point algorithm which does not assume the convergence of the iteration sequence or nondegeneracy.Supported in part by NSF Grant DDM-8922636 and NSF Coop. Agr. No. CCR-8809615, the Iowa Business School Summer Grant, and the Interdisciplinary Research Grant of the University of Iowa Center for Advanced Studies.Supported in part by NSF Coop. Agr. No. CCR-8809615, AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Supported in part by NSF Grant DMS-9102761 and DOE Grant DE-FG05-91ER25100.     

The mean spherical model in a random external field and the replica method

We provide a quick elementary solution of the mean spherical model in a random external field. This also allows an immediate proof of the self-averaging property of the free energy. We calculate the free energy by means of the replica method, i.e., for any (not necessarily integer) replica numbern, and show that when a phase transition occurs the limits andn 0 are not interchangeable.     

The convergence of the Rayleigh-Ritz Method in quantum chemistry

Two sufficient criteria for the convergence of the Rayleigh-Ritz Method (RRM) with respect to the eigenvalues (E-convergence) of non-relativistic electronic Hamiltonians of molecules are discussed and compared. Moreover, a necessary and sufficient criterion is given. By example (Sect. 9) it is shown that the L ²-completeness of the basis is not sufficient to guarantee E-convergence. The convergence of the wave functions in different norms (-convergence) is also investigated. In particular, sufficient conditions for the one-particle basis functions (orbitals) are given, such that a CI calculation in this basis is both E- and -convergent.     

The W algorithm and the D transformation for the numerical evaluation of three‐center nuclear attraction integrals

Three‐center nuclear attraction integrals over exponential‐type functions are required for ab initio molecular structure calculations and density functional theory (DFT). These integrals occur in many millions of terms, even for small molecules, and they require rapid and accurate numerical evaluation. The use of a basis set of B functions to represent atomic orbitals, combined with the Fourier transform method, led to the development of analytic expressions for these molecular integrals. Unfortunately, the numerical evaluation of the analytic expressions obtained turned out to be extremely difficult due to the presence of two‐dimensional integral representations, involving spherical Bessel integral functions. % The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of these spherical Bessel integrals. This algorithm, which is based on the nonlinear D transformation and the W algorithm of Sidi, can be computed recursively, allowing the control of the degree of accuracy. Numerical analysis tests were performed to further improve the efficiency of our algorithm. The numerical results section demonstrates the efficiency of this new algorithm for the numerical evaluation of three‐center nuclear attraction integrals. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007