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Computing sharp and scalable bounds on errors in approximate zeros of univariate polynomials [article]

P. H. D. Ramakrishna, Sudebkumar Prasant Pal, Samir Bhalla, Hironmay Basu, Sudhir Kumar Singh
2003 arXiv   pre-print
In this paper, we develop a simple and novel method for determining sharp upper bounds on errors in approximate zeros of a given polynomial using Rouche's theorem from complex analysis.  ...  We compute the error bounds using non-linear optimization. Our bounds are scalable in the sense that we compute sharper error bounds for better approximations of zeros.  ...  ÁÒ ×Ô Ø Ó Ø × Ò Ó ×ÓÐÙØ ÓÒ ÓÖÑÙÐ ¸Ø f undamental theorem of algebra ×Ø Ø × Ø Ø ÕÙ Ø ÓÒ ½ ÐÛ Ý× × Ò Ø ÖÑ Ò Ò Þ ÖÓ× ÔÖ × ÐÝ ÓÖ Ö ÙÒ Ú Ö Ø ÔÓÐÝÒÓÑ Ð׺ ÁÒ ÓÑÔÙØ Ö Ð Ö ÔÔÐ Ø ÓÒ׸ÓÒ Ù×Ù ÐÐÝ  ... 
arXiv:cs/0306015v1 fatcat:aa6krwjuyresbfienmjr5nrbzq

Fitting subdivision surfaces

N. Litke, A. Levin, P. Schroder
2001 Proceedings Visualization, 2001. VIS '01.  
We introduce a new algorithm for fitting a Catmull-Clark subdivision surface to a given shape within a prescribed tolerance, based on the method of quasi-interpolation.  ...  We demonstrate the power and versatility of our method with examples from interactive modeling, surface fitting, and scientific visualization.  ...  [9] focused on Catmull-Clark surfaces and (semi-)sharp crease rules to achieve a number of modeling effects.  ... 
doi:10.1109/visual.2001.964527 dblp:conf/visualization/LitkeLS01 fatcat:nckrladv5vcapjb6xwpprjqcn4

DSOS and SDSOS Optimization: More Tractable Alternatives to Sum of Squares and Semidefinite Optimization [article]

Amir Ali Ahmadi, Anirudha Majumdar
2018 arXiv   pre-print
In this paper, we introduce DSOS and SDSOS optimization as linear programming and second-order cone programming-based alternatives to sum of squares optimization that allow one to trade off computation  ...  These are optimization problems over certain subsets of sum of squares polynomials (or equivalently subsets of positive semidefinite matrices), which can be of interest in general applications of semidefinite  ...  We would like to thank Pablo Parrilo for acquainting us with scaled diagonally dominant matrices, and Georgina Hall for many corrections and simplifications on the first draft of this work.  ... 
arXiv:1706.02586v3 fatcat:a7kmmfs435ejbb5mtufd7jitli

Learning Activation Functions: A new paradigm for understanding Neural Networks [article]

Mohit Goyal, Rajan Goyal, Brejesh Lall
2020 arXiv   pre-print
We prove that SLNNs can approximate any neural network with lipschitz continuous activations, to any arbitrary error highlighting their capacity and possible equivalence with standard NNs.  ...  Also, SLNNs can be completely represented as a collections of finite degree polynomial upto the very last layer obviating several hyper parameters like width and depth.  ...  Taylor polynomials have easier analytic form and its basis elements can be computed in polynomial time.  ... 
arXiv:1906.09529v3 fatcat:r3ukudsb4ja2xl3wu2pgvrkokm

Performance Evaluation of Block Diagonal Preconditioners for Divergence-Conforming B-Spline Discretization of Stokes Equations [article]

Adriano M. A. Cortes, Alvaro L. G. A. Coutinho
2013 arXiv   pre-print
One of such methods, for symmetric systems, is the Minimum Residual Method (MINRES).  ...  In this paper, we compare the performance of block-diagonal preconditioners for several block choices.  ...  Univariate B-Splines To define a univariate B-spline basis one needs to specify the number n of basis functions wanted, the polynomial degree p of the basis and a knot vector Ξ.  ... 
arXiv:1309.5048v1 fatcat:i4vrinbomjeqvhkpk6prwuhaim

Performance evaluation of block-diagonal preconditioners for the divergence-conforming B-spline discretization of the Stokes system

A.M.A. Côrtes, A.L.G.A. Coutinho, L. Dalcin, V.M. Calo
2015 Journal of Computational Science  
One of such methods, for symmetric systems, is the Minimum Residual Method (MINRES).  ...  We also compare the number of iterations and wall-clock timings.  ...  Also, the bounds for (44) are sharp, as can be seen by computing its values with β 2 0 ≈ 0.1852 and C 2 b ≈ 1.0367, and comparing with the eigenvalues of Table 16 .  ... 
doi:10.1016/j.jocs.2015.01.005 fatcat:x3fwemzy75hyrpwfj6oyrirpvq

Padé approximants and the modal connection: Towards increased robustness for fast parametric sweeps

Romain Rumpler
2017 International Journal for Numerical Methods in Engineering  
In particular, this enables for the intervals of approximation to be chosen a priori in direct connection with a given choice of Padé approximants.  ...  Padé approximants and their associated interval of convergence for an effective and more robust approximation.  ...  The author would like to thank Mr Lorenzo Lavorini (Hitachi Rail Italy) for providing the exterior geometry after which the model of the last example is created and for the permission to include it in  ... 
doi:10.1002/nme.5603 fatcat:xyzf2nb6kjanlpidy4utk6hx34

The multi-dimensional Hermite-discontinuous Galerkin method for the Vlasov-Maxwell equations [article]

Oleksandr Koshkarov and Gianmarco Manzini and Gian Luca Delzanno and Cecilia Pagliantini and Vadim Roytershteyn
2020 arXiv   pre-print
The resulting system of time-dependent nonlinear equations is discretized by the discontinuous Galerkin (DG) method in space and by the method of lines for the time integration using explicit Runge-Kutta  ...  The method is based on a spectral expansion of the velocity space with the asymmetrically weighted Hermite functions.  ...  Acknowledgments The work of GLD, OK, GM was supported by the Laboratory Directed Research and Development -Exploratory and Research (LDRD-ER) Program of Los Alamos National Laboratory under project number  ... 
arXiv:2004.12010v1 fatcat:7vvs4dktk5bb3jujxibgtazdbe

Biharmonic diffusion curve images from boundary elements

Peter Ilbery, Luke Kendall, Cyril Concolato, Michael McCosker
2013 ACM Transactions on Graphics  
From left to right: toys image; sharp-profile (solid) and smooth-profile (dotted) curves of the toys image, with thumbnail images below showing examples of the 4 building block segment fields; car image  ...  Abstract There is currently significant interest in freeform, curve-based authoring of graphic images.  ...  Orzan and co-authors, and S. Boyé and M. Finch for permission to use images from their papers. We also thank N. Andronikos and A. Chang for assistance in preparing images.  ... 
doi:10.1145/2508363.2508426 fatcat:bndkhb6ry5dyhdq5enfb7gqcy4

Additive Models with Trend Filtering [article]

Veeranjaneyulu Sadhanala, Ryan J. Tibshirani
2018 arXiv   pre-print
Analogous to its advantages in the univariate case, additive trend filtering has favorable theoretical and computational properties, thanks in large part to the localized nature of the (discrete) total  ...  derivatives are of bounded variation.  ...  Acknowledgements We are very thankful to Garvesh Raskutti for his generous help and insight on various issues, and Martin Wainwright for generously sharing his unpublished book with us.  ... 
arXiv:1702.05037v4 fatcat:6jzeel2arvcjxlncgcxe354s5e

Rigorous bounds on the stationary distributions of the chemical master equation via mathematical programming [article]

Juan Kuntz, Philipp Thomas, Guy-Bart Stan, Mauricio Barahona
2019 arXiv   pre-print
In the unique case, the bounds form an approximation of the stationary distribution with a computable bound on its error.  ...  Here, we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy.  ...  ACKNOWLEDGEMENTS We thank Justine Dattani and Diego Oyarzún for stimulating discussions, and Michela Ottobre and Jure Vogrinc for insights on the stability and long-term behaviour of continuous-time chains  ... 
arXiv:1702.05468v3 fatcat:d7q3vnfrsnhsxa62adbbnyzsti

Abstracts of Presentations

2001 Brain and Cognition  
PREFACE This volume contains abstracts of the talks presented at the eleventh conference on Applications of Computer Algebra, held at Nara Women's University, Nara, Japan.  ...  This meeting had 14 special sessions and 100 talks. Accordingly, the abstracts are organized by special session.  ...  In our talk, we will explain the details of AsirPad, and report the results of the lecture.  ... 
doi:10.1006/brcg.2000.1285 fatcat:k6c2qcfpubdxzjdkbgzktkigz4

Bounding the stationary distributions of the chemical master equation via mathematical programming

Juan Kuntz, Philipp Thomas, Guy-Bart Stan, Mauricio Barahona
2019 Journal of Chemical Physics  
(c) Approximation errorε l r of the lower boundl r (red) and boundε u r on error of û r (black). In total, 3900 bounds were computed (solver time = 48.8 min, average of 0.75 s per bound).  ...  In Figs. 3(c) and 3(d), we show that the approximation error tends to zero as the size of the truncation |Sr| = ⌈r 1/α ⌉ is increased.  ... 
doi:10.1063/1.5100670 fatcat:dvobnq5ezrhslft7m4epw7b3pm

Polynomial-time algorithms for Multimarginal Optimal Transport problems with structure [article]

Jason M. Altschuler, Enric Boix-Adsera
2021 arXiv   pre-print
Indeed, MOT in general requires exponential time in the number of marginals k and their support sizes n.  ...  For structures (2)-(3), we give the first poly(n,k) time algorithms, even for approximate computation. Together, these three structures encompass many -- if not most -- current applications of MOT.  ...  Jayanti who were involved in the brainstorming stages and provided helpful references.  ... 
arXiv:2008.03006v3 fatcat:i3iz4krphremxiihr4beiw5nue

Chance-Constrained AC Optimal Power Flow -- A Polynomial Chaos Approach [article]

T. Mühlpfordt and L. Roald and V. Hagenmeyer and T. Faulwasser and S. Misra
2019 arXiv   pre-print
Two case studies highlight the efficacy of the method, with a focus on satisfaction of the AC power flow equations and on the accurate computation of moments of all random variables.  ...  As the share of renewables in the grid increases, the operation of power systems becomes more challenging.  ...  There is a sharp decrease in the power flow inaccuracies from N d = 1, with errors between 1 E-3 and 2 E-3, to N d = 2 with errors approximately at 4 E-4.  ... 
arXiv:1903.11337v2 fatcat:dyrw63zjuzceldytqbl44krywq
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