On the method of moving planes and the sliding method

The method of moving planes and the sliding method are used in proving monotonicity or symmetry in, say, thex ₁ direction for solutions of nonlinear elliptic equationsF(x, u, Du, D ² u)=0 in a bounded domain in ⁿ which is convex in thex ₁ direction. Here we present a much simplified approach to these methods; at the same time it yields improved results. For example, for the Dirichlet problem, no regularity of the boundary is assumed. The new approach relies on improved forms of the Maximum Principle in narrow domains. Several results are also presented in cylindrical domains—under more general boundary conditions.dedicated to Shmuel Agmon     

One version of the projection-iterative method based on the method of chords

We consider the problem of application of one version of the projection-iterative method to nonlinear integral equations. Sufficient conditions for the convergence of this method are established. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 7, pp. 996–1000, July, 1999.     

A dual method to the collocation method

In this article we consider methods which are related to the collocation method by interchanging the test and the trial spaces. Error estimates are derived. As a by-product we obtain some extensions to the known convergence results for the collocation method.     

A nonlinear analog of the WKB method and the method of the averaged Lagrangian

Some connections between the methods indicated in the title are considered. Thus, for example, it is shown that the solvability in periodic functions of the equations of the zeroth approximation of the WKB method is equivalent (in the completely integrable case) to the vanishing of the variation with respect to the variables of the action of the averaged Lagrangian.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 162–171, 1981.In conclusion, the author would like to express his sincere thanks to V. M. Babich for his attention to this work and for useful discussions and, although this is not honored by tradition, to S. Yu. Dobrokhotov and V. P. Maslov whose work Finite-Zone Almost Periodic Solutions in the WKB Approximations served as the foundation and stimulus for writing the present note.     

Relation between the memory gradient method and the Fletcher-Reeves method

The minimization of a function of unconstrained variables is considered using the memory gradient method. It is shown that, for the particular case of a quadratic function, the memory gradient algorithm and the Fletcher-Reeves algorithm are identical.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-828-67. In more expanded form, it can be found in Ref. 1.     

A multicolour SOR method for the finite-element method

In this paper we present several algorithms to reorder unknowns in a finite-element mesh so that we can use the multicolour SOR method to solve the corresponding linear system on a pipelined computer or on a parallel computer. We also discuss the assembling process by reordering elements with our algorithms. Numerical tests on a pipelined computer indicate the efficiency of the multicolour SOR method.     

Foundation of the synthesis method

With the aid of the extension of the energy space, the synthesis method is interpreted in two ways: as the best approximation method and as the Galerkin-Petrov method in a new space. For elliptic problems one considers the problem of estimating the convergence rate of the mentioned method in norms which are generalizations of the Sobolev and Holder norms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 19–48, 1977.     

Analysis of the quasi-Laguerre method

The quasi-Laguerre's iteration formula, using first order logarithmic derivatives at two points, is derived for finding roots of polynomials. Three different derivations are presented, each revealing some different properties of the method. For polynomials with only real roots, the method is shown to be optimal, and the global and monotone convergence, as well as the non-overshooting property, of the method is justified. Different ways of forming quasi-Laguerre's iteration sequence are addressed. Local convergence of the method is proved for general polynomials that may have complex roots and the order of convergence is . Received June 30, 1996 / Revised version received August 12, 1996     

Convergence of the Nelder-Mead method

We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.

     

Analysis of the Kleiser-Schumann method

Summary The Kleiser-Schumann algorithm for the approximation of the Stokes problem by Fourier/Legendre polynomials is analized. Stability when the degree of the polynomials increases is established, whereas error estimates in Sobolev spaces are proven.The research of this author has been partially supported by the U.S. Army through its European Research Office under contract No. DAJA-84-C-0035     

A nonlinear analogue of the WKB method and the method of the averaged Lagrangian. II

This paper is a direct continuation of the author's previous paper. One considers questions concerning the uniqueness and the solvability of equations of the higher approximations of the WKB method and the relation between these questions and the properties of the averaged Lagrangian.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 95–101, 1983.     

A Tensor product generalized ADI method for the method of planes

We consider solving separable, second order, linear elliptic prtial differential equations in three independent variables. If the partial differential opertor separates into two terms, one depending on x and y, and one depending on z, then we use the method of planes to obtain a discrete problem, which we write in tensor product from as We apply a new interative method, the tensor product generalized alternating direction implicit method, to solve the discrete problem. We study a specific implementation that uses Hermite bicubic collocation in the xy direction and symmetric finite differences in the z direction. We demostrate that this method is a fast and accurate way to solve the large linear systems arising from three-dimensional elliptic problems.     

On the order-optimality of a method for evaluating unbounded operators and applications of the method

An order-optimal method is proposed for solving approximately the problem of evaluating unbounded operators in various spaces.     

Extensions of the method of quasilinearization

Recently, the method of quasilinearization has been generalized, extended, and refined (Refs. 1–2). In this paper, various results are obtained which offer monotone sequences that provide lower and upper bounds for the solution and converge quadratically, when the function involved admits a decomposition of the difference of two convex functions.     

On the convergence of the UOBYQA method

We analyze the convergence properties of Powell's UOBYQA method. A distinguished feature of the method is its use of two trust region radii. We first study the convergence of the method when the objective function is quadratic. We then prove that it is globally convergent for general objective functions when the second trust region radius ρ converges to zero. This gives a justification for the use of ρ as a stopping criterion. Finally, we show that a variant of this method is superlinearly convergent when the objective function is strictly convex at the solution.     

Stability of the method of lines

Summary It is well known that a necessary condition for the Lax-stability of the method of lines is that the eigenvalues of the spatial discretization operator, scaled by the time stepk, lie within a distanceO(k) of the stability region of the time integration formula ask0. In this paper we show that a necessary and sufficient condition for stability, except for an algebraic factor, is that the -pseudo-eigenvalues of the same operator lie within a distanceO()+O(k) of the stability region ask, 0. Our results generalize those of an earlier paper by considering: (a) Runge-Kutta and other one-step formulas, (b) implicit as well as explicit linear multistep formulas, (c) weighted norms, (d) algebraic stability, (e) finite and infinite time intervals, and (f) stability regions with cusps.In summary, the theory presented in this paper amounts to a transplantation of the Kreiss matrix theorem from the unit disk (for simple power iterations) to an arbitrary stability region (for method of lines calculations).Work supported by an NSF Presidential Young Investigator Award to L.N. Trefethen