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Intractability Results for Positive Quadrature Formulas and Extremal Problems for Trigonometric Polynomials

Erich Novak
1999 Journal of Complexity  
Lower bounds for the error of quadrature formulas with positive weights are proved. We get intractability results for quasi-Monte Carlo methods and, more generally, for positive formulas.  ...  problems for trigonometric polynomials.  ...  ACKNOWLEDGMENTS I thank Knut Petras, Klaus Ritter, and Henryk Woz niankowski for many valuable suggestions. The author was supported by a Heisenberg scholarship of the German Research Council (DFG).  ... 
doi:10.1006/jcom.1999.0507 fatcat:rdml7hwvgrg3feq7o7r4sf5lly

Page 1929 of Mathematical Reviews Vol. , Issue 2001C [page]

2001 Mathematical Reviews  
problems for trigonometric polynomials.  ...  {For the entire collection see MR 2000k:00042. } Gheorghe Coman (Cluj-Napoca) 2001¢:65031 65D32 41A55 Novak, Erich (D-ERL-MI; Erlangen) Intractability results for positive quadrature formulas and extremal  ... 

On positive positive-definite functions and Bochner's Theorem

Aicke Hinrichs, Jan Vybíral
2011 Journal of Complexity  
Jan Vybíral acknowledges the financial support provided by the FWF project Y 432-N15 START-Preis ''Sparse Approximation and Optimization in High Dimensions''.  ...  Acknowledgments We thank Erich Novak for useful discussions about the contents of the paper. The research of Aicke Hinrichs was supported by the DFG Heisenberg grant HI 584/3-2.  ...  It is relatively easy to prove an error formula for the best quadrature rules with positive weights which shows intractability of the tensor product problem for such rules.  ... 
doi:10.1016/j.jco.2011.01.002 fatcat:zv26l2embjdrjb6gaydf55dt5q

Random matrices in 2D, Laplacian growth and operator theory

Mark Mineev-Weinstein, Mihai Putinar, Razvan Teodorescu
2008 Journal of Physics A: Mathematical and Theoretical  
In this article, we review these recent developments and indicate some physical problems where the theory can be applied.  ...  The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density).  ...  Acknowledgments The authors wish to thank P Wiegmann, K Efetov and E Saff for useful discussions regarding parts of this project, and A Zabrodin for comments on the final manuscript.  ... 
doi:10.1088/1751-8113/41/26/263001 fatcat:ym43zdptdzhqfmlaiwlkt3ijee

Algorithm xxx --- ORTHPOL: A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules [article]

Walter Gautschi
1993 arXiv   pre-print
Once these are known, additional data can be generated, such as zeros of orthogonal polynomials and Gauss-type quadrature rules, for which routines are also provided.  ...  A collection of subroutines and examples of their uses, as well as the underlying numerical methods, are described for generating orthogonal polynomials relative to arbitrary weight functions.  ...  The author gratefully acknowledges a number of suggestions from two anonymous referees and from the associate editor, Dr. Ronald F. Boisvert, for improving the code of the package.  ... 
arXiv:math/9307212v1 fatcat:r52yd65x3je6zmsa56gsomqndi

Uniform Distribution Theory and Applications

Michael Gnewuch, Frances Kuo, Harald Niederreiter, Henryk Woźniakowski
2013 Oberwolfach Reports  
, computer science, numerics, pseudo-randomness and stochastic simulation.  ...  The topics of the workshop were recent progress in the theory of uniform distribution theory (also known as discrepancy theory) and new developments in its applications in analysis, approximation theory  ...  Discrepancy estimates for sequences: New results and open problems Gerhard LarcherIn my talk I presented two of my recent results on discrepancy theory and two open problems connected with these results  ... 
doi:10.4171/owr/2013/49 fatcat:wsm65bw7gfh7ndaepflzced3sm

Tomography on f-oscillators

I V Dudinets, V I Man'ko, G Marmo, F Zaccaria
2017 Physica Scripta  
The tomographic probability representations of oscillator coherent states and the problem of entanglement are then discussed.  ...  Symplectic tomographies of classical and quantum states are shortly reviewed. The concept of nonlinear f-oscillators and their properties are recalled.  ...  Explicit formulae for symplectic and photon number tomograms of the f -oscillator are given in Sec. 4.  ... 
doi:10.1088/1402-4896/aa8e22 fatcat:mi74th5vkvdczjctixsx5v2zly

How to solve dynamic stochastic models computing expectations just once

Kenneth L. Judd, Lilia Maliar, Serguei Maliar, Inna Tsener
2017 Quantitative Economics  
This technique is very general: it works for a broad class of approximating functions, including piecewise polynomials; it can be applied to both Bellman and Euler equations; and it is compatible with  ...  We are indebted to the editor and three anonymous referees for many thoughtful comments and suggestions. Errors are ours. Support from the Hoover  ...  (We had shown this result for a linear polynomial function in the Introduction, and we now generalize this result for polynomials of higher degrees.)  ... 
doi:10.3982/qe329 fatcat:53zqycav2rcpfaftoxmojyhw34

A class of deformational flow test cases for linear transport problems on the sphere

Ramachandran D. Nair, Peter H. Lauritzen
2010 Journal of Computational Physics  
The divergent flow is specifically aimed for conservative global transport schemes to test for conservation, consistency and monotonicity (or positivity) of limiters/filters in a challenging flow environment  ...  A class of new benchmark deformational flow test cases for the two-dimensional horizontal linear transport problems on the sphere is proposed.  ...  Christoph Erath, and Dr. Natasha Flyer for internal reviews of the manuscript. Discussions with Dr. Dale Durran, Dr. Mark A. Taylor, and Dr. Oksana Guba are gratefully acknowledged.  ... 
doi:10.1016/j.jcp.2010.08.014 fatcat:5mrye6xy5rhyneow4vvij4sibq

Gauß Cubature for the Surface of the Unit Sphere [article]

David De Wit
2000 arXiv   pre-print
In 1D, this system can be converted to a linear system, and a unique solution is obtained, for which the points lie within the region of integration, and the weights are all positive.  ...  The uncertainty of the existence of solutions, coupled with the size and degree of the system makes the problem daunting for numerical methods.  ...  Despite these problems, solutions to ( * ) have been computed for several R n , and various weighting functions. Primary results for R 2 and R 3 are contained in [17, 18, 20] .  ... 
arXiv:math/0003018v1 fatcat:yxebv5xivveerltkmhozymy5hq

Computing quantum dynamics in the semiclassical regime [article]

Caroline Lasser, Christian Lubich
2020 arXiv   pre-print
Making good use of classical mechanics is essential for all these approaches. The arising aspects of time integration and high-dimensional quadrature are also discussed.  ...  We present and analyse variationally evolving Gaussian wave packets, Hagedorn's semiclassical wave packets, continuous superpositions of both thawed and frozen Gaussians, and Wigner function approaches  ...  ℓ d > L and thus arrives at the quadrature formula S d L f = ℓ 1 +...  ... 
arXiv:2002.00624v2 fatcat:wtot6wuwvzfjrc7lxccyxroq4u

Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions

Dirk Nuyens, Gowri Suryanarayana, Markus Weimar
2015 Advances in Computational Mathematics  
In particular, we study the application of unshifted and randomly shifted rank-1 lattice rules in such a problem setting.  ...  Furthermore, we prove that those tractability results can be achieved with shifted lattice rules and that the shifts are indeed necessary.  ...  In contrast, problems are called polynomially intractable if (12) does not hold for any such choice of C, p, and q.  ... 
doi:10.1007/s10444-015-9411-6 fatcat:gtjnu3o6xngrpeg7ly3aagj4ga

Optimal Control of Inhomogeneous Ensembles

J. Ruths, Jr-Shin Li
2012 IEEE Transactions on Automatic Control  
This dissertation is concerned with formulating the problem and developing methods for the synthesis of optimal, open-loop inputs for large numbers of identically structured dynamical systems that exhibit  ...  We compose an optimal ensemble control problem and develop a computational method based on pseudospectral approximations to solve these complex problems.  ...  An optimal nonlinear control problem of this form is, in general, analytically intractable.  ... 
doi:10.1109/tac.2012.2195920 fatcat:n6epbr32j5gufjfpji4x5nj6pq

Page 1806 of Mathematical Reviews Vol. , Issue Subject Index [page]

Mathematical Reviews  
(English summary) 2001a:65032 Novak, Erich Intractability results for positive quadrature formulas and extremal problems for trigonometric polynomials.  ...  (English summary) 2001¢:65043 — Orthogonal polynomials and cubature formulae on balls, simplices, and spheres.  ... 

A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model

David J. Knezevic, Endre Süli
2009 Mathematical Modelling and Numerical Analysis  
The numerical methods we develop are fully-practical, and we present a range of numerical results demonstrating their accuracy and efficiency.  ...  Alternating-direction methods have been considered previously in the literature for this problem (e.g. in the work of Lozinski, Chauvière and collaborators [J. 607-625]), but this approach has not previously  ...  Alexei Lozinski of the Institut de Mathématiques de Toulouse, and Dr. David Kay of the University of Oxford for their helpful comments.  ... 
doi:10.1051/m2an/2009034 fatcat:ltjucfonqzh4fiv6azcwcfwawa
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