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"A Discourse concerning the Methods of Approximation in the Extraction of Surd Roots. By John Wallis, S. T. D. and Savilian Professor of Geometry at Oxford" is an article from Philosophical Transactions (1683-1775), Volume 19 . View more articles from Philosophical Transactions (1683-1775) . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-102270

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

The research conducted contributes towards the development of a theory to analyze and design linear shift-invariant multivariable multidimensional discrete and continuous systems. Recursive schemes to compute rational approximants to a power series in two variables having constant matrices for coefficients are developed. Approximants to special matrix power series are investigated and the properties of these approximants are delineated in a strictly mathematical setting and their implications are interpreted via physical reasonings. Algebraic procedures to tet thee approximants for stability are provided, and criteria for guaranteeing the invariance of properties like positivity and stability under parameter changes or perturbation are advanced. Attention is directed throughout towards the reduction of algebraic computational complexity. In the important problem of filter stabilization without appreciable change in the magnitude of the frequency response, recent results on multiplicative computational complexity theory is exploited to demonstrate the feasibility of implementing efficiently a 2-D discrete Hilbert transform. Criteria for 2-D rational approximants to be maximally flat are obtained.

We study lacunary multiresolution methods from the point of view of their analogy to the use of near-degenerate elements in finite and boundary element methods. The main results are characterization of the best N-term approximation of solutions of nonlinear operator equations and best N-term approximation by near-degenerate normal approximating families in Hilbert spaces.

In this paper we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $C$. We assume that the set $C$ can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method the main idea of which is to project at each step onto a particular half-space constructed by using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima's method has so far been considered only in the Euclidean setting with different conditions on $F$. We provide several examples for the case where $C$ is the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results.

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

We investigate three different methods for systematically approximating the diffusion coefficient of a deterministic random walk on the line which contains dynamical correlations that change irregularly under parameter variation. Capturing these correlations by incorporating higher order terms, all schemes converge to the analytically exact result. Two of these methods are based on expanding the Taylor-Green-Kubo formula for diffusion, whilst the third method approximates Markov partitions and transition matrices by using the escape rate theory of chaotic diffusion. We check the practicability of the different methods by working them out analytically and numerically for a simple one-dimensional map, study their convergence and critically discuss their usefulness in identifying a possible fractal instability of parameter-dependent diffusion, in case of dynamics where exact results for the diffusion coefficient are not available.

Traditionally, stochastic approximation schemes for SVIs have relied on strong monotonicity and Lipschitzian properties of the underlying map. In contrast, we consider monotone stochastic variational inequality (SVI) problems where the strong monotonicity and Lipschitzian assumptions on the mappings are weakened. In the first part of the paper, to address such shortcomings, a regularized smoothed SA (RSSA) scheme is developed wherein the stepsize, smoothing, and regularization parameters are diminishing sequences updated after every iteration. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to a solution in an almost sure sense, extending the results in [16] to the non-Lipschitzian regime. Motivated by the need to develop non-asymptotic rate statements, in the second part of the paper, we develop a variant of the RSSA scheme, denoted by aRSSA$_r$, in which we employ a weighted iterate-averaging, parametrized by a scalar $r$ where $r = 1$ provides us with the standard averaging scheme. We make several contributions in this context: First, we show that the gap function associated with the sequences by the aRSSA$_r$ scheme tends to zero when the parameter sequences are chosen appropriately. Second, we show that the gap function associated with the averaged sequence diminishes to zero at the optimal rate $\cal{O}(1/\sqrt{K})$ after $K$ steps when smoothing and regularization are suppressed and $r < 1$, thus improving the rate statement for the standard averaging which admits a rate of $\cal{O}(\ln(K)/\sqrt{K})$. Third, we develop a window-based variant of this scheme that also displays the optimal rate for $r < 1$. Notably, we prove the superiority of the scheme with $r < 1$ with its counterpart with $r=1$ in terms of the constant factor of the error bound when the size of the averaging window is sufficiently large.

This is a reply to the comment by Gilbert and Sanders [arXiv:1111.6271 (2011)]. We point out that their comment is a follow-up of a previous discussion which we briefly summarize before we refute their new criticism.

Several different methods exist for efficient approximation of paths in multiscale stochastic chemical systems. Another approach is to use bursts of stochastic simulation to estimate the parameters of a stochastic differential equation approximation of the paths. In this paper, multiscale methods for approximating paths are used to formulate different strategies for estimating the dynamics by diffusion processes. We then analyse how efficient and accurate these methods are in a range of different scenarios, and compare their respective advantages and disadvantages to other methods proposed to analyse multiscale chemical networks.

Research during this period continued in the following topic areas: (1) Nonlinear filtering; (2) Large deviations problems; (3) Stochastic control of piecewise-deterministic processes; (4) Stochastic variational calculus; (5) Approximating multiple Ito integrals with band-limited processes; (6) Large deviations methods and asymptotic properties of recursive algorithms; and (7) Approximation and modelling for distributed stochastic systems. Progress is summarized and a list of publications supported by the grant is included in the report. (Author)

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

Many areas of stochastic control were studied in this period. Monte-Carlo algorithms of type appearing in adaptive systems were studied and stability, convergence and rate of convergence results obtained. The analytical techniques developed for this purpose have a wide applicability in the study of similar algorithms. Robust computationally oriented approximations to optimal non-linear filters were developed. A deep study of how to find the Markov system that best approximates a given non-linear system with a wide band-width input was undertaken and very good results obtained. The problem is important in control and communication theory because it is usually desired to use the many available methods for analyzing Markov systems. The results obtained seem to be the best available so far and application to control and communication theory are being pursued. Optimal stochastic control problems under the condition of partial information on the system state were studied and a new formulation, similar to that of the non-linear filtering problem, developed. The approach seems promising in providing qualitative information on this difficult class of problems. Part of the technique uses measure-valued processes in a way which has wider applicability to problems involving distributed states or information. (Author)

Many areas of stochastic control were studied in this period. Monte-Carlo algorithms of type appearing in adaptive systems were studied and stability, convergence and rate of convergence results obtained. The analytical techniques developed for this purpose have a wide applicability in the study of similar algorithms. Robust computationally oriented approximations to optimal non-linear filters were developed. A deep study of how to find the Markov system that best approximates a given non-linear system with a wide band-width input was undertaken and very good results obtained. The problem is important in control and communication theory because it is usually desired to use the many available methods for analyzing Markov systems. The results obtained seem to be the best available so far and application to control and communication theory are being pursued. Optimal stochastic control problems under the condition of partial information on the system state were studied and a new formulation, similar to that of the non-linear filtering problem, developed. The approach seems promising in providing qualitative information on this difficult class of problems. Part of the technique uses measure-valued processes in a way which has wider applicability to problems involving distributed states or information. (Author)

These are the lecture notes for the LMS/EPSRC short course on strong approximation methods in linear groups organized by Dan Segal in Oxford in September 2007.

Bio-image analysis is challenging due to inhomogeneous intensity distributions and high levels of noise in the images. Bayesian inference provides a principled way for regularizing the problem using prior knowledge. A fundamental choice is how one measures "distances" between shapes in an image. It has been shown that the straightforward geometric L2 distance is degenerate and leads to pathological situations. This is avoided when using Sobolev gradients, rendering the segmentation problem less ill-posed. The high computational cost and implementation overhead of Sobolev gradients, however, have hampered practical applications. We show how particle methods as applied to image segmentation allow for a simple and computationally efficient implementation of Sobolev gradients. We show that the evaluation of Sobolev gradients amounts to particle-particle interactions along the contour in an image. We extend an existing particle-based segmentation algorithm to using Sobolev gradients. Using synthetic and real-world images, we benchmark the results for both 2D and 3D images using piecewise smooth and piecewise constant region models. The present particle approximation of Sobolev gradients is 2.8 to 10 times faster than the previous reference implementation, but retains the known favorable properties of Sobolev gradients. This speedup is achieved by using local particle-particle interactions instead of solving a global Poisson equation at each iteration. The computational time per iteration is higher for Sobolev gradients than for L2 gradients. Since Sobolev gradients precondition the optimization problem, however, a smaller number of overall iterations may be necessary for the algorithm to converge, which can in some cases amortize the higher per-iteration cost.

Approximation concepts and dual method algorithms are combined to create a method for minimum weight design of structural systems. Approximation concepts convert the basic mathematical programming statement of the structural synthesis problem into a sequence of explicit primal problems of separable form. These problems are solved by constructing explicit dual functions, which are maximized subject to nonnegativity constraints on the dual variables. It is shown that the joining together of approximation concepts and dual methods can be viewed as a generalized optimality criteria approach. The dual method is successfully extended to deal with pure discrete and mixed continuous-discrete design variable problems. The power of the method presented is illustrated with numerical results for example problems, including a metallic swept wing and a thin delta wing with fiber composite skins.

In the spaces $S^p$ of functions of several variables, $2\pi$-periodic in each variable, we study the approximative properties of operators $A^\vartriangle_{\varrho,r}$ and $P^\vartriangle_{\varrho,s}$, which generate two summation methods of multiple Fourier series on triangular regions. In particular, in the terms of approximation estimates of these operators, we give a constructive description of classes of functions, whose generalized derivatives belong to the classes $S^pH_\omega$.

Bibliography: p. 106-109

Bibliography: p. 106-109

We propose a computational method that simplifies drastically the inclusion of spin-orbit interaction in density functional theory implemented on localised atomic orbital basis sets. Our method is based on a well-known procedure for obtaining pseudopotentials from atomic relativistic 'ab initio' calculations and on an on-site approximation for the spin-orbit matrix elements. We have implemented the technique in the SIESTA code, and we show that it provides accurate results for the overall band structure and splittings of group IV and III-IV semiconductors as well as for 5d metals.

We propose a computational method that simplifies drastically the inclusion of spin-orbit interaction in density functional theory implemented on localised atomic orbital basis sets. Our method is based on a well-known procedure for obtaining pseudopotentials from atomic relativistic 'ab initio' calculations and on an on-site approximation for the spin-orbit matrix elements. We have implemented the technique in the SIESTA code, and we show that it provides accurate results for the overall band structure and splittings of group IV and III-IV semiconductors as well as for 5d metals.

We present a systematic study on isogeometric approximations of second order elliptic eigenvalue problems. In the case of either Neumann boundaries or multi-patch geometries and a mortar approach spurious eigenvalues - typically referred to as "outliers" - occur. This pollution of the discrete spectrum can be avoided by a hybrid approach. In addition to the imposed weak consistency with respect to the Neumann boundary and to the interface mortar coupling condition, we apply higher-order penalty terms for the strong coupling. The mesh dependent weights are selected such that uniform continuity is guaranteed and optimal a priori estimates are preserved. Numerical results show the good behavior of the proposed method in the context of simple 1D and 2D Laplace problems, but also of more complex multi-patch mapped geometries and linear elasticity.

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This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis, their properties depend on combinatorics, topology, and geometry of a simplicial or polyhedral subdivision of a region in R^k, and are often quite subtle. We describe four algebraic techniques which are useful in the study of splines: homology, graded algebra, localization, and inverse systems. Our goal is to give a hands-on introduction to the methods, and illustrate them with concrete examples in the context of splines. We highlight progress made with these methods, such as a formula for the third coefficient of the polynomial giving the dimension of the spline space in high degree, much of which builds on pioneering work of Schumaker, Alfeld-Schumaker, and Billera. The objects appearing here may be computed using the Macaulay2 software system.

This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis, their properties depend on combinatorics, topology, and geometry of a simplicial or polyhedral subdivision of a region in R^k, and are often quite subtle. We describe four algebraic techniques which are useful in the study of splines: homology, graded algebra, localization, and inverse systems. Our goal is to give a hands-on introduction to the methods, and illustrate them with concrete examples in the context of splines. We highlight progress made with these methods, such as a formula for the third coefficient of the polynomial giving the dimension of the spline space in high degree, much of which builds on pioneering work of Schumaker, Alfeld-Schumaker, and Billera. The objects appearing here may be computed using the Macaulay2 software system.

We derive a three-term recurrence relation for computing the polynomial of best approximation in the uniform norm to $x^{-1}$ on a finite interval with positive endpoints. As application, we consider two-level methods for scalar elliptic partial differential equation (PDE), where the relaxation on the fine grid uses the aforementioned polynomial of best approximation. Based on a new smoothing property of this polynomial smoother that we prove, combined with a proper choice of the coarse space, we obtain as a corollary, that the convergence rate of the resulting two-level method is uniform with respect to the mesh parameters, coarsening ratio and PDE coefficient variation.

We derive a three-term recurrence relation for computing the polynomial of best approximation in the uniform norm to $x^{-1}$ on a finite interval with positive endpoints. As application, we consider two-level methods for scalar elliptic partial differential equation (PDE), where the relaxation on the fine grid uses the aforementioned polynomial of best approximation. Based on a new smoothing property of this polynomial smoother that we prove, combined with a proper choice of the coarse space, we obtain as a corollary, that the convergence rate of the resulting two-level method is uniform with respect to the mesh parameters, coarsening ratio and PDE coefficient variation.

This is a comment on [G. Knight and R. Klages, Phys. Rev. E 84, 041135 (2011); also available at arXiv:1107.5293v2 [math-ph]].

Journal of Research of the National Bureau of Standards

The WKB method is a powerful tool to obtain solutions for many physical problems. It is generally applicable to problems of wave propagation in which the frequency of the wave is very high or, equivalently, the wavelength of the wave is very short. The WKB solutions are approximate solutions, but sometimes they are surprisingly accurate. In this chapter we�ll discuss this method, which is applicable to linear equations only.

The WKB method is a powerful tool to obtain solutions for many physical problems. It is generally applicable to problems of wave propagation in which the frequency of the wave is very high or, equivalently, the wavelength of the wave is very short. The WKB solutions are approximate solutions, but sometimes they are surprisingly accurate. In this chapter we�ll discuss this method, which is applicable to linear equations only.

Belief networks (also known as Bayesian networks, causal networks, or probabilistic networks) represent dependencies between variables and give a concise specification of a joint probability distribution. They enable a general-purpose inference method that can answer a broad class of queries given information that is uncertain or incomplete. In this research project, we have investigated methods and implemented algorithms for efficiently making certain classes of inference in belief networks, and for automatically learning certain classes of belief networks to make more accurate inferences. The progress on this project falls into two related areas: inference and learning. In each case, progress has been both on understanding and unifying existing approaches and the development of new methods.

Belief networks (also known as Bayesian networks, causal networks, or probabilistic networks) represent dependencies between variables and give a concise specification of a joint probability distribution. They enable a general-purpose inference method that can answer a broad class of queries given information that is uncertain or incomplete. In this research project, we have investigated methods and implemented algorithms for efficiently making certain classes of inference in belief networks, and for automatically learning certain classes of belief networks to make more accurate inferences. The progress on this project falls into two related areas: inference and learning. In each case, progress has been both on understanding and unifying existing approaches and the development of new methods.

We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map. We propose and analyze a generalized version of Smolyak's algorithm, which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments. We apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel-based approximation. The theoretical results are supplemented by numerical experiments.

Discretization methods for ordinary differential equations based on the use of matrix exponentials have been known for decades. This set of ideas has come off age and acquired greater urgency recently, within the context of geometric integration and discretization methods on manifolds based on the use of Lie-group actions. In the present paper we study the approximation of the matrix exponential in a particular context: given a Lie group $G$ and its Lie algebra $g$, we seek approximants $F(tB)$ of $\exp(tB)$ such that $F(tB)\in G$ if $B\in g$. Having fixed a basis of the Lie algebra, we write $F(tB)$ as a composition of exponentials of the basis elements pre-multiplied by suitable scalar functions.

We compare three different methods to obtain solutions of Sturm-Liouville problems: a successive approximation method and two other iterative methods. We look for solutions with periodic or anti periodic boundary conditions. With some numerical test over the Mathieu equation, we compare the efficiency of these three methods. As an application, we make a numerical analysis on a model for carbon nanotubes.

Operations research applications often pose multicriteria problems. Mathematical research on multicriteria problems predominantly revolves around the set of Pareto optimal solutions, while in practice, methods that output a single solution are more widespread. In real-world multicriteria optimization, reference point methods are widely used and successful examples of such methods. A reference point solution is the solution closest to a given reference point in the objective space. We study the approximation of reference point solutions. In particular, we establish that approximating reference point solutions is polynomially equivalent to approximating the Pareto set. Complementing these results, we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization. In particular, we lift the link between dynamic programming and FPTAS, as well as oblivious LP-rounding techniques. The latter applies, e.g., to Set Cover and several machine scheduling problems.

A Discourse concerning the Methods of Approximation in the Extraction of Surd Roots. By John Wallis, S. T. D. and Savilian Professor of Geometry at Oxford. Wallis, J Philosophical Transactions (1683-1775). 1753-01-01. 19:2–11

This report summarizes the results of research produced under this grant, and lists titles of papers produced. (Author)

This is a summary of research completed during the period of this award, 1 October 1985-30 September 1988 by Fleming, Kushner together with associated postdoctoral and graduate student personnel. The research covers a number of problems in many areas of stochastic control, recursive stochastic algorithms, and related areas of analysis. It is part of a continuing research program pursued successfully for a number of years. The program has been motivated both by traditional applications in control and filtering and by newer areas of application arising in queueing/communication networks and production systems. Other research issues addressed include numerical methods for stochastic control and recursive algorithms for distributed and parallel processing and/or control. The work of Fleming and Kushner will be summarized in turn, with references to research publications supported under this award. (kr)

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

First, new stability tests were developed for multidimensional recursive digital filters and any double-ended n-dimensional noncausal linear processor which is said to be stable if its impulse response decreases exponentially in all 2-n directions. It was than shown that the impulse response operator for a 2-D discrete Hilbert transformer, though not by itself sum-separable, becomes so after appropriate classification. Subsequently it was proved that the multiplicative complexity of computation of a 2-D DHT is not greater than twice the sum of multiplicative complexities of two 1-D DHT's. Subsequently, the 1-D matrix Pade approximation problem via a three-term recursive computation scheme was tackled as a prelude to the solution of 2-D and n-D cases. Specifically, given a 1-D matrix power series, it was shown that a recurrence relation relates the ((L+1)/(M+1)), (L/M), ((L-1)/(M-1)) order Pade approximants, which are guaranteed to exist provided a certain rank condition is satisfied by characterizing matrices possessing block-Hankel structure. Attention to stability, algebraic computational complexity and approximation were necessary because efficient implementation of stable recursion is desired. (Author)

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

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A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

Refined estimates for the errors in eigenvalue and eigenvector approximation by finite element, or, more generally, Galerkin methods, as they apply to self-adjoint problems, are presented. Particular attention is given to the case of multiple eigenvalues. The results are new in this case. The proof is based on a novel approach which yields the known results for simple eigenvalues in a simple way. Numerical computations are presented and analyzed in light of the theoretical results.

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

This work addresses problems of sensor resource management (SRM) in which one or more sensors obtain measurements of the state of one or more targets. For example, an airborne radar may be attempting to track several ground targets, which are sometimes stationary (requiring a synthetic aperture radar mode) and sometimes moving (requiring a ground moving target indication radar mode). The challenge is to schedule the radar modes as the scenario evolves. Such problems can generally be formulated as partially observable Markov decision processes (POMDPs), which can express essential characteristics of the SRM problem such as uncertainty and dynamics. This work emphasizes a farsighted approach; the highest long-term payoff may not be generated by the action providing the highest immediate payoff. Accomplishments of this effort include the establishment of a boundary on optimal SRM performance, analysis of farsighted SRM strategies for controlling a multimode sensor, and the derivation of a novel set of sufficient conditions for optimality in Markov decision processes.