scispace - formally typeset
Search or ask a question

Showing papers in "Siam Review in 1992"


Journal ArticleDOI
TL;DR: The main purpose of this paper is to advocate the use of the graph associated with Tikhonov regularization in the numerical treatment of discrete ill-posed problems, and to demonstrate several important relations between regularized solutions and the graph.
Abstract: When discrete ill-posed problems are analyzed and solved by various numerical regularization techniques, a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector. In particular, the graph associated with Tikhonov regularization plays a central role. The main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill-posed problems. The graph is characterized quantitatively, and several important relations between regularized solutions and the graph are derived. It is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic L-shaped “corner” of the graph.

3,585 citations


Journal ArticleDOI
TL;DR: A unified theory for a diverse group of iterative algorithms, such as Jacobi and Gauss–Seidel iterations, diagonal preconditioning, domain decomposition methods, multigrid methods,Multilevel nodal basis preconditionsers and hierarchical basis methods, is presented by using the notions of space decomposition and subspace correction.
Abstract: The main purpose of this paper is to give a systematic introduction to a number of iterative methods for symmetric positive definite problems. Based on results and ideas from various existing works...

1,176 citations


Journal ArticleDOI
TL;DR: This paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems, formulated in terms of frontal matrices, updateMatrices, and an assembly tree.
Abstract: This paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems. The method is formulated in terms of frontal matrices, update matrices, and an assembly tree. Formal definitions of these notions are given based on the sparse matrix structure. Various advances to the basic method are surveyed. They include the role of matrix reorderings, the use of supernodes, and other implementatjon techniques. The use of the method in different computational environments is also described.

434 citations


Journal ArticleDOI
TL;DR: First some well-known features of superconducting materials are reviewed and then various results concerning the model, the resultant differential equations, and their solution on bounded domains are derived.
Abstract: The authors consider the Ginzburg–Landau model for superconductivity. First some well-known features of superconducting materials are reviewed and then various results concerning the model, the resultant differential equations, and their solution on bounded domains are derived. Then, finite element approximations of the solutions of the Ginzburg–Landau equations are considered and error estimates of optimal order are derived.

398 citations


Journal ArticleDOI
TL;DR: A unified treatment of algorithms is described for linear programming methods based on the central path, which is a curve along which the cost decreases, and that stays always far from the centre.
Abstract: In this paper a unified treatment of algorithms is described for linear programming methods based on the central path. This path is a curve along which the cost decreases, and that stays always far...

347 citations


Journal ArticleDOI
TL;DR: The aim of this overview is to show that one of its roots is a basic paper of de la Vallee Poussin of 1908, and the historical development of sampling theory from 1908 to the present, especially the matter dealing with not necessarily band-limited functions.
Abstract: Shannon’s sampling theorem is one of the most powerful results in signal analysis. The aim of this overview is to show that one of its roots is a basic paper of de la Vallee Poussin of 1908. The historical development of sampling theory from 1908 to the present, especially the matter dealing with not necessarily band-limited functions (which includes the duration-limited case actually studied in 1908), is sketched. Emphasis is put on the study of error estimates, as well as on the delicate point-wise behavior of sampling sums at discontinuity points of the signal to be reconstructed.

233 citations


Journal ArticleDOI
TL;DR: The main objective of this survey is to study convergence properties of difference methods applied to differential inclusions to present a number of results scattered in the literature.
Abstract: The main objective of this survey is to study convergence properties of difference methods applied to differential inclusions. It presents, in a unified way, a number of results scattered in the li...

201 citations


Journal ArticleDOI
TL;DR: There are severe limitations on the range of validity of these analogies and outside this range many interesting open questions can be posed about the solutions to the macroscopic models.
Abstract: This paper reviews the derivation of some macroscopic models for superconductivity and also some of the mathematical challenges posed by these models. The paper begins by exploring certain analogies between phase changes in superconductors and those in solidification and melting. However, it is soon found that there are severe limitations on the range of validity of these analogies and outside this range many interesting open questions can be posed about the solutions to the macroscopic models.

175 citations


Journal ArticleDOI
TL;DR: The focus on the important issues of a more realistic discussion of electromagnetic wave propagation in themquite commonmpresence of corners, edges, and conical points is the most prominent feature of this monograph.
Abstract: The three parts into which this monograph is divided are loosely bound together by the fact that they have to do with singular behavior of time-harmonic or static electromagnetic fields. The first two parts are more closely connected. They discuss distributions and their application in formulating the presence and effects of sources concentrated on submanifolds (i.e., points, curves, surfaces). Here, in particular, multipole expansions and the Green’s tensor (dyadic) are discussed. Green’s tensors are presented for various physical situations and applied to representing solutions to the time-harmonic Maxwell equations (e.g., in free space, cavities, waveguides). The third part deals with the behavior of electromagnetic fields near corners, edges and conical points. It is well known that solutions with finite energy display interior regularity (if the data are regular) but may have singularities at singular boundary points (preventing for certain geometries, e.g., that all first derivatives are square-integrable). This type of singularity is studied by expansions in terms of function systems associated with separation of variables with respect to coordinates matching the local geometry of the boundary (cylinder, polar, spheroconal coordinates). The emphasis in this part \"is laid on the presentation of numerical data useful to the practising electromagneticst.\" The focus on the important issues of a more realistic discussion of electromagnetic wave propagation in themquite commonmpresence of corners, edges, and conical points is the most prominent feature of this text. The monograph is complemented by six short appendices to several issues and prerequisites referred to in the course of the monograph. This monograph is an engineering text in the strict sense of the word. Mathematics is used as a means of symbolic communication rather than as an instrument of establishing the validity of statements; in the author’s words, \"... the approach is unashamedly that of the ’applied mathematician.’\" The risk inherent in the lack of mathematical rigor will be gladly accepted by the intended audience, due to the admissibility of supporting evidence (measurements, numerical results, experiments, etc.). The audience addressed with this monograph consisting of \"graduate students of electrical engineering and physics, as well as to practising electromagneticists in industrial and academic laboratories\" will find this a valuable addition to the many more general books on electromagnetism. Last but not least, the fairly extensive references given at the end of each chapter should be very useful for graduate students as well as practitioners in the field.

156 citations


Journal ArticleDOI
TL;DR: An overview is given of the attempts to understand the behavior of scroll waves in three-dimensional excitable media using analytical methods, and these results are compared with recent numerical and experimental results.
Abstract: An overview is given of the attempts to understand the behavior of scroll waves in three-dimensional excitable media using analytical methods, and these results are compared with recent numerical and experimental results. The behavior of untwisted planar scroll rings, twisted rings and helices, and knotted scroll wave filaments is discussed.

128 citations


Journal ArticleDOI
TL;DR: The complexity of polynomial and matrix computations, as well as their various correlations to each other and some major techniques for the design of algebraic and numerical algorithms are reviewed.
Abstract: This paper reviews the complexity of polynomial and matrix computations, as well as their various correlations to each other and some major techniques for the design of algebraic and numerical algorithms.

Journal ArticleDOI
TL;DR: This book is a memorial not only to Fred but to the discipline of statistics, which owes much of its current healthy prospects to the seeds that he planted, and would be useful reading for any young person contemplating a career involving the diverse features of the discipline and profession of statistics.
Abstract: A StatisticalModel is an unusual book, written to honor the career of an unusual man and scientist, Frederick Mosteller. It is of great interest because it is a book about the history of statistics. I believe it would be useful reading for any young person contemplating a career involving the diverse features of the discipline and profession of statistics. Fred Mosteller was born in 1916, and received degrees in Statistics from Carnegie Mellon (Sc. M. in 1939) and Princeton (Ph.D. in 1946). As a faculty member at Harvard from 1946 until his retirement in 1986 (member of Departments of Social Relations, Statistics, Biostatistics, Health Policy, and Management), he has been involved in (and pioneered) many significant applications of statistics. The high esteem in which he is held was demonstrated by Harvard in 1991 by the honor of an honorary degree (Doctor of Laws). Fred Mosteller led an interesting and productive life, and stimulated the careers of many. This book records Fred’s life by the following chapters: biography (by John Tukey); bibliography; contributions as a scientific generalist (William Kruskal); contributions to mathematical statistics (Persi Diaconis and Erich Lehmann); contributions to method61ogy and applications (Stephen Fienberg); education (Judith Tanur); Harvard (Stephen Fienberg and David Hoaglin); new reviews of some of Mosteller’s books. The new reviews of 21 books are an original feature of this book. The reviewers (all ofwhom have been associates or students of Mosteller) place in perspective Fred’s contributions to statistics and also their own views of aspects of the past, present, and future state of statistics. Statistics is a field of mathematical sciences which, through the development of computationally intensive methods, is becoming an increasingly interdisciplinary field. These developments owe much to the career of Fred Mosteller. This book is a memorial not only to Fred but to the discipline of statistics, which owes much of its current healthy prospects to the seeds that he planted.


Journal ArticleDOI
TL;DR: This paper gives a short review of the mathematics involved in the study of asymptotic behavior of semigroups arising from structured models of population dynamics.
Abstract: This paper gives a short review of the mathematics involved in the study of asymptotic behavior of semigroups arising from structured models of population dynamics. First are considerations on the ...

Journal ArticleDOI
TL;DR: It is shown how refinement methods for smooth curve generation can be carried out efficiently through iterated function systems (IFS).
Abstract: It is shown how refinement methods for smooth curve generation can be carried out efficiently through iterated function systems (IFS). Affine transformations are constructed so that, when composed randomly, they generate the desired smooth curve. Underlying this random algorithm is the “tree traversal” property of IFS. Under a refinement method the points on the curve correspond to leaves on some N-tree. IFS theory enables one to generate all of these leaves through a single orbit of an appropriate Markov chain. Applications include Bezier curves, splines, wavelets and various interpolants.


Journal ArticleDOI
TL;DR: Selfadjoint operators, which lie in between the minimal and maximal operator, are easily described in terms of conditions on boundary coefficients, which look like, and indeed are, the same as those imposed on regular problems.
Abstract: Differential operators arising from the differential equation \[ - (py')' + qy = \lambda wy \] are put in the Hilbert space setting $L_2 (a,b;w)$. A new and simpler characterization of the minimal operator is exhibited.Selfadjoint operators, which lie in between the minimal and maximal operator, are easily described in terms of conditions on boundary coefficients, which look like, and indeed are, the same as those imposed on regular problems.Examples, drawn from mathematical physics, include the Legendre, Laguerre, Hermite, and Bessel problems.




Journal ArticleDOI
TL;DR: It is shown that if a satisfies some rather weak conditions then lim„_^xnh.n(X; f, N, da) exists and the limit is also evaluated and the corresponding limit will also be calculated.
Abstract: Let X = (*,, x2,..., xN), /: R -» C and let P„ be the class of polynomials of degree at most n. The generalized Christoffel function A„ corresponding to the measure da is defined by A„(X;f,N,da)= min f" \«(t)\2 da(t). ireP„_| •'-oo »(*,)-/(*,) 1-1,2.N It is shown that if a satisfies some rather weak conditions then lim„_^xnh.n(X; f, N, da) exists and the limit is also evaluated. The purpose of the present note is to investigate the asymptotic behavior of the functions A„: R^ -» R defined by An(X;f,N,da)= min f°° \m(t)\2 da(t). n-eP„_i ./-00 »(*,)-/(*,) /=1,2.N Here X = (xx, x2,. .., .%), f: R -» C is a fixed and almost everywhere finite function, P„ is the set of all polynomials m of a degree at most n and a is a weight function, that is a is nondecreasing on R, it has infinitely many points of increase and every polynomial m belongs to Lja. Therefore A„ is defined and finite for almost every X E RN. Estimates for A„ lead to several results in probability theory, statistics and in the theory of orthogonal polynomials. (See [1], [2] and [4].) In fact, it is not hard to explicitly compute A„ [3] but the formula for A„ is so complicated that it cannot be used to estimate A„ when a is not nice. It will be shown that \imn_>,xnAn(X;f, A, da) exists under rather weak assumptions on a and the corresponding limit will also be calculated. Let {pn(da))TM=0 be the system of polynomials which is orthogonal with respect to da, that ispn(da, x) = y„x" + . . . with yn > 0 and /oo Pn(da, t)Pm(da, t) da(t) = 8nm. -oo Let M denote the class of those weights a for which /oo tp2(da, t) da(t) = 0 Received by the editors February 14, 1977. AMS (MOS) subject classifications (1970). Primary 42A52.


Journal ArticleDOI
TL;DR: This volume is compulsory reading for all who are interested in mathematics in the real world; its contribution to the continuing development of a synergetic relationship between its readers and the IMA is Very welcome and would surely have met with approval of the late editor.
Abstract: mentary on each group, and if the open questions were not laid out, as at present, as \"student exercises.\" The use of grouping would help the hard-pressed reader to identify how areas are developing year by year and it would also enable major open questions to be targetted, with mutual benefit, at relevant specialists around the world. This volume is compulsory reading for all who are interested in mathematics in the real world; its contribution to the continuing development of a synergetic relationship between its readers and the IMA is Verywelcome and would surely have met with approval of our late editor.


Journal ArticleDOI
TL;DR: A consequence of the new measures is that power flux vanishes if the scalar and vector potentials are constant with respect to time, in contrast to the Poynting equation, another mathematical conservation law.
Abstract: In a vacuum the scalar field 4 and vector field A of electromagnetic radiation satisfy wave equations. Certain mathematically natural measures of energy density and power flux arise for any solutions of any wave equation, and, as components of a conservation law, these measures are utilized like Lyapunov functions in an associated PDE stability theory [Hahn, Stability of Motion, Springer-Verlag, New York, 1967, pp. 216’217], [Walker, Dynamical Systems and Evolution Equations, Plenum, New York, 1990, p. 106], [Zachmanoglov and Thoe, Introduction to Partial Differential Equations with Applications, Dover, New York, 1986, p. 283]. But energy conservation in electrodynamics is conventionally accounted for with the Poynting equation, another mathematical conservation law. The purpose of this paper is to compare the contending measures and conservation laws.A consequence of the new measures is that power flux vanishes if the scalar and vector potentials are constant with respect to time, in contrast to the Poyn...

Journal ArticleDOI
TL;DR: This paper purports to study Duffing’s equation by using the method of van der Pol, and provides an independent derivation of the response curves, which leads to a simplified discussion of the stability question.
Abstract: An analysis of the harmonic solutions of the Duffing equation can be accomplished by means of either the perturbation method or the method of iteration. However, this paper purports to study Duffing’s equation by using the method of van der Pol. In addition to providing an independent derivation of the response curves, this approach leads to a simplified discussion of the stability question. The variational equation takes the form of a Hill’s equation (in fact, the Mathieu equation). Hence, in order to investigate stability by the variational method one must study the boundedness properties of solutions of Mathieu’s equation. This need is obviated by using van der Pol’s method, since, in this case, the stability issue can be decided by applying the method of Andronow and Witt. That is, harmonic solutions are identified with the singular points of a first-order system of ordinary differential equations. Stability is then determined by utilizing Poincar~'s criteria for the classification of singularities of...


Journal ArticleDOI
TL;DR: The author did a good job of describing the field of computer systems performance analysis and of surveying the main techniques for quantitative analysis, and the book is certainly recommended to two audiences.
Abstract: and simulation to the most commonly used performance metrics. Part Two, \"Measurement Techniques and Tools,\" is more specific, and in my opinion is the most valuable of the book. Chapters 4, 5, and 6 contain a comprehensive treatment of computer systems workloads. The presentation is very good, punctuated by practical advice and realistic examples. My only suggestion would have been to either expand to a reasonable treatment or avoid altogether the superficial discussion on statistical techniques for workload characterization. The techniques mentioned by the author (principal component analysis, Markov chains, and cluster analysis) require familiarity with multivariate statistics beyond that of most computer professionals, and it is unlikely to dojustice to these topics in a few pages each. Chapters 7 and 8 are devoted to monitors and accounting logs, which are the main tools for collecting data about computer performance. Chapter 9 contains an overview of capacity planning and benchmarking, with brief discussions of numerous items, such as steps, problems, common mistakes, sources of bias, and limitations. These three chapters are also quite informative and reasonably well written. I have some mixed thoughts on the author’s extensive use of lists to present the material. On the positive side, this structured format makes the contents easily accessible to the casual reader. However, the material appears as sets of disjointed \"chunks,\" and the connections among them are lost. Chapter 10 is devoted to preparing graphics for data presentation. It offers good suggestions, in general, with a few inaccuracies, such as confusing Gantt charts, which by definition have the time dimension represented by the horizontal axis, with bar charts, which look similar but are not necessarily drawn on a time scale. Indeed, the examples on pages 150 and 153, mistakenly labelled as Gantt charts, show bar charts where the horizontal axis represents the percentage of time that the system is utilized by various activities on a 0-100% scale, rather than the actual periods of time during which each activity is executed by the system. This section concludes with Chapter 11, which is devoted to the manipulation of numbers. The chapter, appropriately titled \"Ratio Games,\" shows how results and conclusions can be distorted by the choice of numerator and denominator of the measures of performance. The remaining four parts of the book are devoted to Probability and Statistics; Design and Analysis of Experiments; Simulation; and Queueing Models. These are all weighty subjects that cannot be thoroughly covered in about 100 pages each. The treatment is adequate for a review for readers who are already familiar with the material, or as a synopsis for those readers who are not familiar with the subject and may need to decide whether or not to seek additional information or expertise. The book is designed as a textbook, and as such each chapter includes exercises. Most of the exercises in the first two parts are worded in an open-ended manner, and are suitable for discussions in graduate courses. The exercises in the last four parts are typical homework problems ofthe \"plug in the formula\" variety found in many texts on quantitative techniques, and as such are quite limited and not very exciting. Overall, the author did a good job of describing the field of computer systems performance analysis and of surveying the main techniques for quantitative analysis. Besides its original use as a textbook, this book would be very helpful to two audiences: those already working in performance analysis who need access to reference material on quantitative techniques, and those who are already trained in statistics and operations research but need to learn about performance analysis. In summary, although the presentation of the material is intuitive rather than rigorous, and could withstand some improvement, the book is certainly recommended to those two audiences.

Journal ArticleDOI
TL;DR: Several VCON networks motivated by observations by physiologists are presented, which make accessible the study of phase locking, an important physical phenomenon that makes possible stable frequency-encoded information processing even in the presence of noise.
Abstract: Voltage controlled oscillator model neurons (VCONs) are electronic circuits that are similar to phase locked loops, but designed to account for certain experimental observations of neurons. They are constructible electronic circuits, and they provide teaching tools that involve (relatively) simple mathematical models based on brilliant circuits designed by engineers. The model makes accessible the study of phase locking, an important physical phenomenon that makes possible stable frequency-encoded information processing even in the presence of noise. VCONs also enable the design of networks of circuits that might be useful as analog control devices in robotics, give interesting examples of rotation vectors in high-order dynamical systems, and can process, store, and recognize frequency-encoded information.Presented here are several VCON networks motivated by observations by physiologists. They fire bursting patterns similar to neural circuits in the thalamus and reticular complex of mammalian brains; they...