Stability analysis of the cell centered finite-volume M<Emphasis Type="SmallCaps">uscl</Emphasis> method on unstructured grids

The goal of this study is to apply the Muscl scheme to the linear advection equation on general unstructured grids and to examine the eigenvalue stability of the resulting linear semi-discrete equation. Although this semi-discrete scheme is in general stable on cartesian grids, numerical calculations of spectra show that this can sometimes fail for generalizations of the Muscl method to unstructured three-dimensional grids. This motivates our investigation of the influence of the slope reconstruction method and stencil on the eigenvalue stability of the Muscl scheme. A theoretical stability analysis of the first order upwind scheme proves that this method is stable on arbitrary grids. In contrast, a general theoretical result is very difficult to obtain for the Muscl scheme. We are able to identify a local property of the slope reconstruction that is strongly related to the appearance of unstable eigenmodes. This property allows to identify the reconstruction methods that are best suited for stable discretizations. The explicit numerical computation of spectra for a large number of two- and three-dimensional test cases confirms and completes the theoretical results.     

On the carleman's model for the boltzmann equation and its generalizations

Summary T. Carleman [1] showed that a certain 2 x 2 hyperbolic, semilinear system of partial differential equations has many features in common with the integro-differential equation ofBoltzmann. We produce here a global existence theory for theCauchy problem for this system and its generalizations. The main result is derived by exploiting the fact that the solution of a certainRiccati equation depends isotonically (with respect to a suitable partial order) on the forcing term and the initial data. Research supported by the National Science Foundation, Grant NSF G-21053.     

Der senkrechte Verdichtungsstoss am gekrümmten Profil

Summary The present paper investigates a compression shock forming at a curved plane surface in stationary flow. The geometric form of this shock adjacent to the profile is determined, and the flow in the vicinity of the wall and of the shock is investigated. With convex profiles, an effect is confirmed which has been known for about 12 years through the measurements performed byAckeret,Feldmann andRott [1]. Our findings are of importance for both local supersonic flow fields and compressible flow through cascades.      

Zur Theorie der bedingten Tests

Summary The construction of a most powerful similar test with Neyman structure for composite hypotheses is frequently accomplished by the solution of a class of auxiliary problems asking for the determination of most powerful tests for simple hypotheses on the regions of constancy of a sufficient statistic. In detail, this class of so-called conditional tests has to be constructed so as to yield a test composed by them. This method due to Neyman is among others used by Fraser [3], Kendall-Stuart [6], Lehmann [7], Lehmann-Scheffé [8], Witting [11]. The present paper investigates the suppositions on which this construction is possible and the form of the tests obtained hereby. In particular, it is shown that these tests are easy to determine in the case of dominated hypotheses.

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Gekürzte Fassung einer von der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Münster i. Westf. angenommenen Dissertation.

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Der Autor ist am 20. 7. 1968 verstorben.

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Herrn Professor Dr. H. Witting, Münster, danke ich für zahlreiche Anregungen bei der Fertigstellung dieser Arbeit.     

A robust and efficient algorithm for planar competitive location problems

In this paper we empirically analyze several algorithms for solving a Huff-like competitive location and design model for profit maximization in the plane. In particular, an exact interval branch-and-bound method and a multistart heuristic already proposed in the literature are compared with uego (Universal Evolutionary Global Optimizer), a recent evolutionary algorithm. Both the multistart heuristic and uego use a Weiszfeld-like algorithm as local search procedure. The computational study shows that uego is superior to the multistart heuristic, and that by properly fine-tuning its parameters it usually (in the computational study, always) find the global optimal solution, and this in much less time than the interval branch-and-bound method. Furthermore, uego can solve much larger problems than the interval method.     

Ein quadratisch konvergentes Verfahren zur Eigenwertbestimmung bei unsymmetrischen Matrizen I

Summary J. C. F. Francis proposes for the determination of eigenvalues of an arbitrary matrix theQR-transformation which is a unitary analogue of theLR-transformation. In this process the current approximation matrix is always factorized into a unitary and a triangular matrix and the 2 factors are multiplied together in reverse order; this yields the next approximant. The present paper shows how this process may be extended to a process which transforms a given matrix with quadratic convergence orthogonally toWintner-Murnaghan's normal form. The process is formulated in the real domain.     

Eine wahrscheinlichkeitstheoretische Begründung der Integrationsformeln von Newton-Cotes

Summary A definite integral is approximated by a linear combination of values of the integrand at equidistant points. As a measure for the goodness of fit we use the mean square deviation; for this we take a wide sense stationary stochastic process as basis. We prove that there exist always a best integration formula in this sense. The Taylor expansion of its coefficients begins with the coefficients ofNewton-Cotes. It is shown that the error of the best integration formula is of the same order of magnitude as in the well-known formula ofNewton-Cotes. Therefore we get a justification for the formulae ofNewton-Cotes also if the integrand is not differentiable. Besides we obtain estimates of the error, which use higher differences instead of higher differential quotients of the integrand.     

Fourier multipliers on compact groups

In a recent paperIan Inglis gives some sufficient conditions for a function on a totally disconnected compact Abelian group to be anL ^p Fourier multiplier. His proof depends on an interpolation theorem ofE. M. Stein. In this note we prove a generalization ofInglis' theorem. Our result is deduced from a factorization theorem, the proof of which is elementary, and a standard multiplier theorem.This work was done while the second-named author held a visiting appointment at the University of Washington. He wishes to thank ProfessorsE. Hewitt andR. R. Phelps for making the visit possible.     

Machine learning for global optimization

In this paper we introduce the LeGO (Learning for Global Optimization) approach for global optimization in which machine learning is used to predict the outcome of a computationally expensive global optimization run, based upon a suitable training performed by standard runs of the same global optimization method. We propose to use a Support Vector Machine (although different machine learning tools might be employed) to learn the relationship between the starting point of an algorithm and the final outcome (which is usually related to the function value at the point returned by the procedure). Numerical experiments performed both on classical test functions and on difficult space trajectory planning problems show that the proposed approach can be very effective in identifying good starting points for global optimization.     

Ein “affiner” Beweis des Satzes von v. Staudt—Schleiermacher

The Theorem ofv. Staudt-Schleiermacher asserts that a projective incidence plane is pappian if every projectivity with 5 fixed points is the identity.Schleiermacher's original proof needs different treatments for a lot of cases. We give a simpler proof based on the fact that Desargues' and Pappos' theorems hold in a projective plane if this is true for some affine specializations.

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Mit 8 Abbildungen     

A simple proof of a general duality theorem of convex programming

Summary The first part of this note presents concisely and partially proves in logical terms the relations betweenUzawa's andKuhn andTucker's equivalence theorems of nonlinear programming. In the second part we give simple mathematical proofs of two lemmata linking theKuhn-Tucker conditions and dual solutions and use them to establish a nonlinear duality theorem of considerable generality.

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Zusammenfassung Im ersten Abschnitt dieses Beitrags demonstrieren wir in kurzer, symbolischlogischer Form die Relationen zwischenUzawas undKuhn undTuckers Äquivalenzsätzen des nichtlinearen Programmierens. Im zweiten Teil geben wir mathematische Beweise zweier Lemmata, die dieKuhn-Tucker-Bedingungen und duale Lösungen in Zusammenhang bringen, und benützen sie und die logische Struktur des ersten Teils, um ein nichtlineares Dualitätstheorem von besonderer Allgemeinheit zu beweisen.

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The author is indebted toP. Van Moeseke for teaching him programming theory some time ago, and toD. Bent for helpful comments and criticism.

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Vorgel. v.:G. Tintner.     

Fehlerabschätzung für die numerische Differentiation analytischer Funktionen durch Quadratur

Summary Numerical treatment of the integral in Cauchy's integral formula produces approximations for the derivatives of an analytic functionf; this fact has already been utilized byLyness andMoler [3, 4]. In the present paper this idea is investigated especially in view of the accuracy of these formulas regarded as quadrature formulas. Since the integration can be reduced to the integration of a periodic analytic function, it is possible to continue the considerations ofDavis [2] in order to find bounds for the error of the differentiation rules. For the application of these bounds one essentially needs estimations of the maximum off on a circle inside of its region of analyticity. Examples show the practical use of the bounds.

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Meinem verehrten LehrerH. Görtler zur Vollendung seines 60. Lebensjahres gewidmet     

An efficient model formulation for level of repair analysis

Given a product design and a repair network, a level of repair analysis (lora) determines for each component in the product (1) whether it should be discarded or repaired upon failure and (2) at which echelon in the repair network to do this. The objective of the lora is to minimize the total (variable and fixed) costs. We propose an ip model that generalizes the existing models, based on cases that we have seen in practice. Analysis of our model reveals that the integrality constraints on a large number of binary variables can be relaxed without yielding a fractional solution. As a result, we are able to solve problem instances of a realistic size in a couple of seconds on average. Furthermore, we suggest some improvements to the lora analysis in the current literature.     

Über die Konvergenz des Verfahrens vonGrammel und seine mögliche Erweiterung

Summary The paper deals with the convergence ofR. Grammel's method for the solution of boundary-eigenvalue-problems. It is further shown thatGrammel's method can be extended to the case where it is difficult to find the exactGreen function. In this case one may use a different, relatedGreen function if one adds certain supplemental terms to the properGrammel equations.     

Über den Satz von Weil-Cartier

It is shown that the theorem ofWeil-Cartier ([10, Th. 5], [4, Th. 3]) is connected with a homomorphism of groups of unitary operators. The existence proof for this homomorphism is based on simple results in harmonic analysis and on an extension property of the Schwartz-Bruhat functions. Some applications are given, including a result ofIgusa's [6, Th. 3] and the reciprocity formula ofKrazer-Siegel [9, Th. 2]. An outline of the proof has been given in [8].     

Eine Kuppel gleicher Festigkeit

Summary Thin shells of optimum strength are usually defined by the condition that the two principal stresses in any point are equal to the yield stress. This definition is too restrictive; it is sufficient that principal stresses satisfy the yield condition ofv. Mises or the one ofTresca.In this paper a proof is supplied for the existence of spherical cupolas of equal strength on the basis ofv. Mises' yield condition, and an exact solution is presented for the distribution of thickness.     

Essential spectra of spectral operators

The various essential spectra of a linear operator have been surveyed byB. Gramsch andD. Lay [4]. In this paper we characterize the essential spectra and the related quantities nullity, defect, ascent and descent of bounded spectral operators. It is shown that a number of these spectra coincide in the case of a spectral or a scalar type operator. Some results known for normal operators in Hilbert space are extended to spectral operators in Banach space.     

A parallel quasi-Newton method for partially separable large scale minimization

The parallel quasi-Newton method based on updating conjugate subspaces proposed in [4] can be very effective for large-scale sparse minimization because conjugate subspaces with respect to sparse Hessians are usually easy to obtain. We demonstrate this point in this paper for the partially separable case with matrices updated by a quasi-Newton scheme ofGriewank andToint [2,3]. The algorithm presented is suitable for parallel computation and economical in computer storage. Some testing results of the algorithm on an Alliant FX/8 minisupercomputer are reported.The material is based on work supported in part by the National Science Foundation under Grant No. DMS 8602419 and by the Center for Supercomputing Research and Development at the University of Illinois.     

Globalization strategies for Mesh Adaptive Direct Search

The class of Mesh Adaptive Direct Search (Mads) algorithms is designed for the optimization of constrained black-box problems. The purpose of this paper is to compare instantiations of Mads under different strategies to handle constraints. Intensive numerical tests are conducted from feasible and/or infeasible starting points on three real engineering applications.     

Das Taylor-Deansche Stabilitätsproblem für beliebige Spaltbreiten

Summary The stability of a viscous flow between two co-axial circular cylinders, produced by the simultaneous effect of rotation of the interior cylinder and by a constant pressure gradient in peripherical direction, has been examined. (The two single problems had been studied first byTaylor [1] resp.Dean [2].) The perturbations investigated are vortices of the Taylor-Görtler-type and are assumed to be small. It therefore becomes possible to operate on the basis of the linear differential equations for perturbations.The present stability problem had been treated under the same assumptions byDiPrima [5] for small gap-distances by using the series method ofTaylor. In this paper, however, arbitrary gap-distances are admitted, and the integral equation method ofGörtler [3] andHämmerlin [4] has been applied. The calculated dependences of the critical Reynolds number upon the pressure gradient are in good qualitative agreement with the results ofDiPrima. However, our computations performed by more exact methods yielded somewhat different results with regard to the dependence of the critical vortex number upon the pressure gradient.