Showing papers in "Siam Review in 1980"

Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations

Abstract: This paper presents a digest of recently developed simplicial and continuation methods for approximating fixed-points or zero-points of nonlinear finite-dimensional mappings. Underlying the methods are algorithms for following curves which are implicitly defined, as for example, in the case of homotopies. The details of several algorithms are outlined sufficiently that they should be easily implemented. Applications of simplicial and continuation methods to nonlinear complementarily, location of critical points, location of multiple solutions and bifurcation are presented.

Algorithms for separable nonlinear least squares problems

Abstract: Iterative algorithms of Gauss–Newton type for the solution of nonlinear least squares problems are considered. They separate the variables into two sets in such a way that in each iteration, optimization with respect to the first set is performed first, and corrections to those of the second after that. The linear-nonlinear case, where the first set consists of variables that occur linearly, is given special attention, and a new algorithm is derived which is simpler to apply than the variable projection algorithm as described by Golub and Pereyra, and can be performed with no more arithmetical operations than the unseparated Gauss–Newton algorithm. A detailed analysis of the asymptotical convergence properties of both separated and unseparated algorithms is performed. It is found that they have comparable rates of convergence, and all converge almost quadratically for almost compatible problems. Simpler separation schemes, on the other hand, converge only linearly. An efficient and simple computer impleme...

Recent Developments in Synovial Joint Biomechanics

Abstract: This review is concerned with the modern development of animal joint biomechanics. The understanding of the biomechanical function of synovial joints depends upon our ability to quantitatively measure and model the biorheological behavior of the major tissues of these joints (synovial fluid and articular cartilage), and their dynamic interactions during motion. Since synovial fluid is a macromolecular solution of the polyanionic hyaluronic acid and a dialysate of blood plasma, its rheological behavior is highly nonlinear and therefore most difficult to characterize quantitatively. And, since articular cartilage is composed of a permeable fiber-reinforced composite solid matrix swollen with water, its deformational behavior is also very difficult to characterize. As such, the dynamic interactions of these two biorheologically complex materials offer what are indeed challenging biomechanics problems. In this survey, we will discuss both the evolution, over the past 50 years or so, of our understanding of jo...

Geometric Programming: Methods, Computations and Applications

Abstract: In this paper, we give an applications-oriented survey of geometric programming. This important class of nonlinear programming problems has been intensively studied over the past decade and has played a crucial role in an extremely broad range of applications. The focus of the paper will be on posynomial programs and the slightly more general class of signomial programs. The approach used in studying posynomial programs has been generalized to a much broader class of optimization problems, and an excellent survey of that generalized theory recently appeared in a paper by E. L. Peterson [SIAM Rev., 18 (1976), pp. 1–51]. We do not consider that generalized theory here. It is our intent to provide the reader with an understanding of the basic results in posynomial and signomial programming that have played a crucial role in applications. After reviewing these basic results, we consider computational methods and applications. Three specific applications are considered in detail, and a bibliography indicating ...

Singular Perturbation Methods in Stochastic Differential Equations of Mathematical Physics

Abstract: Stochastic differential equations are used as models for various physical phenomena, such as chemical reactions, atomic migration in crystals, thermal fluctuations in electrical networks, noisy sig...

Discrete and Continuous Bang-Bang and Facial Spaces Or: Look for the Extreme Points

Abstract: A simple geometrical argument is used to establish seemingly different continuous and discrete hang-hang type results. Among other applications we discuss the bang-bang principle for linear continu...

Relations Between Several Path Following Algorithms and Local and Global Newton Methods

Abstract: This paper discusses certain common features of several algorithms for finding approximations to a zero of a mapping $F:R^n \to R^n $. In particular, a recent simplicial pivot algorithm of the authors, Merrill’s algorithm, Newton’s method, and a “global Newton method” presented by Smale can be described in terms of the differential equations (or their discretizations) which determine the paths “followed by the algorithms.” This paper surveys the relations between these differential equations and presents simple proofs of several new results.

Some Recent Approaches to Solving Large Residual Nonlinear Least Squares Problems

Abstract: Gauss–Newton based algorithms are widely used for solving nonlinear least squares problems. However, for reasons we shall discuss, they can be expected to perform poorly in certain circumstances. This leads to two other classes of algorithms corresponding to methods which implicitly take account of the second term in the Hessian of the function and methods which explicitly take account of the second term. We discuss the more promising approaches in these two areas and illustrate our discussion with a set of test results.

Error Bounds for Asymptotic Expansions of Integrals

Abstract: The purpose of this paper is to give an up-to-date account of the error analysis of asymptotic expansions of integral transforms. Particular attention is paid to two new asymptotic tools that have been developed recently, one based on summability methods, the other based on the use of distributions. After a brief discussion of integrals with exponentially decaying kernels, we first turn to the consideration of integral transforms whose kernels are oscillatory. This includes the Fourier, Hankel and mufti-dimensional Fourier transforms. Next, we consider the class of integral transforms whose kernels are algebraic functions, e.g., Stieltjes, Hilbert and fractional integral transforms. Also, a quite thorough treatment is given to a Mellin-type convolution integral in which both the function and the kernel are algebraically dominated at $t = 0^ + $ and $t = + \infty $. Finally, we construct error bounds for the remainders associated with two uniform asymptotic expansions, one for the coalescence of a stationa...

Projection Methods for Two-Point Boundary Value Problems

Abstract: A survey of the many different projection methods for the numerical solution of two-point boundary value problems is given along with an introduction to the techniques by which their convergence is...

The linear-Quadratic Control Problem

Abstract: The classical linear-quadratic-Gaussian (LOG) problem from optimal control theory is surveyed with respect to recent developments in the following areas: singular control, state/control constraints...

Plate Failure under Pressure

Abstract: Let us consider a flat plate occupying a plane domain P and rigidly supported along its boundary aP. Let a uniform pressure p act on one of its faces. We suppose that the plate fails or yields or breaks along a curve if the shear force per unit length along that curve exceeds a certain constant o'. The shear force is just the force normal to the plate exerted by one part of the plate upon an adjacent part across their common boundary. What is the largest value of p which can be applied before the plate fails? This question was considered and solved by Gilbert Strang of MIT [1], [2]. His method, which differs from that given here, is described at the end of this note. To answer this question, we consider the balance of forces on any subdomain D of P. The total force normal to the plate exerted on D by the pressure p is pA, where A is the area of D. If the plate has not failed, the total opposing force exerted on D by the surrounding portion of the plate is at most oL, where L is the length of the boundary of D. Thus if failure has not occurred, we must have

Conservation Laws of Fluid Dynamics–A Survey

Abstract: The conservation law problem in fluid dynamics and a.generalization to Riemannian manifolds is surveyed in this paper. In particular the fluid dynamic equations in coordinate free notation and also...

Asymptotic Approximations and Error Bounds

Abstract: The purpose of this paper is to demonstrate that well-constructed error bounds for asymptotic approximations can provide useful analytical insight into the nature and reliability of the approximati...

Chebyshev Sets of Polynomials which Satisfy an Ordinary Differential Equation

Abstract: It has been known for a long time (S. Bochner, Math. Z., (1929)) that the only polynomial sets $\{ {y_n (x)} \}_{n = 0}^\infty $, where $y_n (x)$ is of degree exactly n in x, satisfying a second-order ordinary differential equation \[ Py''_n + Qy'_n + Ry_n = \lambda _n wy_n ,\] where P, Q, R, W are functions of x, and $\lambda _n $ is an appropriate constant, are the Jacobi, Laguerre, Hermite, Bessel polynomials and powers of x. Considerable recent work has substantially enlarged the class of weight functionals available, however. Further those polynomial sets satisfying similar fourth-order differential equations can also be completely classified.

A Practical Guide to Splines (Carl de Boor)

Abstract: This book is based on the authors experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shapepreserving properties of the B-spline series.

Von Neumann’s Hilbert Space Theory and Partial Differential Equations

Abstract: In this von Neumann Lecture I shall relate my early encounters with von Neumann, describe the part of my work that was strongly influenced by his work, and briefly discuss subsequent work I performed, some years ago, that is related to the earlier work. I first met John von Neumann in Gottingen in 1927 or 1926 in a seminar run by Max Born which was concerned with the new quantum theory. After a seminar session most participants walked up a hill to a garden restaurant. On one of these occasions Born asked von Neumann about his interest in science. Von Neumann then told us that his father wanted him to become a mathematician since he had shown very early a gift for mathematics. But he himself wanted to become a banker, as his father was. Finally, they settled on chemistry. Von Neumann studied chemistry, not in his native Budapest, Hungary, but in Zurich, Switzerland. There he was influenced by the mathematician Hermann Weyl, who at that time was interested in mathematical logic. So von Neumann wrote a paper in this field which turned out to be rather important. At the same time Hilbert was also concerned with mathematical logic, and so von Neumann went to Gottingen to work with him. In Gottingen he learned about the new quantum theory and participated in the seminar on that field. Von Neumann's interest in quantum theory led him to write a book about the mathematical foundations of this field. The book appeared in 1932; I reviewed it for the Zentralblatt. This book contained introductory mathematical sections, and sections involving quantum theory. At present I think that the mathematical sections, which were based on papers that had appeared in 1929, were the more significant ones. They were rather abstract. We, in the group around Courant, were quite suspicious about the significance of such abstract work. Nevertheless, when in late 1930 I studied these abstract papers, I was dumbfounded. In fact, I had just handed to Courant for publication a manuscript on spectral theory. I asked him to return the manuscript. I then rewrote the paper in von Neumann's abstract language. That was the origin of a substantial part of my later work. Before describing this work I should like to go back some years. When the basic formula of the new quantum theory, pq qp = -ih, was set up by Heisenberg, Born, and Jordan, some mathematicians in Gottingen claimed, somewhat sneeringly, that such a formula could not be valid. They claimed that they could prove this by using Hilbert's theory of infinite matrices, since p and q were such matrices. An infinite matrix is given by a system of real numbers Ma,f3 with a and f3 infinite sequences a = 1, 2, 3, . , f3 = 1, 2, 3, * . . . Such a matrix is called bounded if for any sequence of numbers a,, bs the finiteness of the sums a, a, I2 and E,, I b,s I2 implies the finiteness of the sum

Some Bubble and Contact Problems

Abstract: A number of problems involving bubbles in a fluid, or contact of surfaces, or both are considered. In each case the size and/or the shape of the bubble, or the location of the points of contact, are unknown in advance and must be found. When the problems are formulated mathematically, boundary conditions must be satisfied at the bubble surface and at the contact points, which are thus “free” boundaries. The problems are: postbuckling behavior of an elastic tube, contact problems involving a buckled elastica, steep capillary waves with trapped bubbles, deformation of a bubble in a uniform flow, distortion of a bubble in a straining flow, free oscillation of an underwater explosion bubble, and forced oscillation of a bubble in a sound field.

Reliability of Multichannel Systems

Abstract: The results of simple probability theory are applied to the analysis of the reliability of multichannel redundant systems. The significant reductions in instantaneous probability of system failure ...