American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A general framework for validated continuation of periodic orbits in systems of polynomial ODEs
  • JCD Home
  • This Issue
  • Next Article
    A self-consistent dynamical system with multiple absolutely continuous invariant measures
January  2021, 8(1): 33-58. doi: 10.3934/jcd.2021003

The geometry of convergence in numerical analysis

George W. Patrick

Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK S7N5E6, Canada

Received  September 2019 Revised  June 2020 Published  August 2020

The domains of mesh functions are strict subsets of the underlying space of continuous independent variables. Spaces of partial maps between topological spaces admit topologies which do not depend on any metric. Such topologies geometrically generalize the usual numerical analysis definitions of convergence.

Keywords: Numerical method, convergence, covariance, geometric integration, hypertopology.
Mathematics Subject Classification: Primary: 65LXX; Secondary: 54B20.
Citation: George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, Addison-Wesley, 1978.

[2]

U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-algebraic Equations, SIAM, 1998. doi: 10.1137/1.9781611971392.

[3]

K. Back, Concepts of similarity for utility functions, J. Math. Econom., 15 (1986), 129-142.  doi: 10.1016/0304-4068(86)90004-2.

[4]

G. Beer, On the Fell topology, Set-valued Analysis, 1 (1993), 69-80.  doi: 10.1007/BF01039292.

[5]

G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8149-3.

[6]

G. Beer, A. Caserta, G. D. Maio and R. Lucchetti, Convergence of partial maps, J. Math. Anal. Appl., 419 (2014), 1274-1289.  doi: 10.1016/j.jmaa.2014.05.040.

[7]

R. D. Canary, D. B. A. Epstein and P. L. Green, Notes on notes of Thurston, in Fundamentals of Hyperbolic Geometry: Selected Expositions, London Math. Soc. Lecture Note Ser., 328, Cambridge Univ. Press, Cambridge, 2006.

[8]

C. Chabauty, Limite dénsembles et géométrie des nombres, Bull. Soc. Math. France, 78 (1950), 143-151. 

[9]

C. Cuell and G. W. Patrick, Geometric discrete analogues of tangent bundles and constrained Lagrangian systems, J. Geom. Phys., 59 (2009), 976-997.  doi: 10.1016/j.geomphys.2009.04.005.

[10]

P. de la Harpe, Spaces of closed subgroups of locally compact groups, preprint, arXiv: 0807.2030.

[11]

J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 13 (1962), 472-476.  doi: 10.1090/S0002-9939-1962-0139135-6.

[12]

A. C. Hansen, A theoretical framework for backward error analysis on manifolds, J. Geom. Mech., 3 (2011), 81-111.  doi: 10.3934/jgm.2011.3.81.

[13]

A. Illanes and S. B. Nadler, Hyperspaces. Fundamentals and Recent Advances., Marcel Dekker, Inc., New York, 1999.

[14]

A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009.

[15]

A. Iserles, H. Z. Munthe-Kass, S. P. Norsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365.  doi: 10.1017/S0962492900002154.

[16]

K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. (4), 40 (1955), 61–67. doi: 10.1007/BF02416522.

[17]

K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966.

[18]

K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Pańtwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968.

[19]

R. Lucchetti and A. Pasquale, A new approach to hyperspace theory, J. Convex Anal., 1 (1994), 173-193. 

[20]

J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395.  doi: 10.1007/s002200050505.

[21]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.

[22]

K. Matsuzaki, The Chabauty and the Thurston topologies on the hyperspace of closed subsets, J. Math. Soc. Japan, 69 (2017), 263-292.  doi: 10.2969/jmsj/06910263.

[23]

E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152-182.  doi: 10.1090/S0002-9947-1951-0042109-4.

[24]

J. R. Munkres, Topology, Prentice Hall, 2000.

[25]

S. B. Nadler, Hyperspaces of Sets. A Text with Research Questions, Marcel Dekker, Inc., New York-Basel, 1978.

[26]

R. S. Palais, When proper maps are closed, Proc. Amer. Math. Soc., 24 (1970), 835-836.  doi: 10.2307/2037337.

[27]

G. W. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained Lagrangian systems, Numerische Mathematik, 113 (2009), 243-264.  doi: 10.1007/s00211-009-0245-3.

[28]

W. Rudin, Functional Analysis, McGraw-Hill, 1973.

[29]

V. Runde, A Taste of Topology, Universitext, Springer, New York, 2005.

[30] M. Schatzman, Numerical Analysis: A Mathematical Introduction, Clarendon Press, Oxford, 2002. 
[31] E. Süli and D. F. Mayer, An Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511801181.
[32]

S. Willard, General Topology, Addison-Wesley, 1970.

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, Addison-Wesley, 1978.

[2]

U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-algebraic Equations, SIAM, 1998. doi: 10.1137/1.9781611971392.

[3]

K. Back, Concepts of similarity for utility functions, J. Math. Econom., 15 (1986), 129-142.  doi: 10.1016/0304-4068(86)90004-2.

[4]

G. Beer, On the Fell topology, Set-valued Analysis, 1 (1993), 69-80.  doi: 10.1007/BF01039292.

[5]

G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8149-3.

[6]

G. Beer, A. Caserta, G. D. Maio and R. Lucchetti, Convergence of partial maps, J. Math. Anal. Appl., 419 (2014), 1274-1289.  doi: 10.1016/j.jmaa.2014.05.040.

[7]

R. D. Canary, D. B. A. Epstein and P. L. Green, Notes on notes of Thurston, in Fundamentals of Hyperbolic Geometry: Selected Expositions, London Math. Soc. Lecture Note Ser., 328, Cambridge Univ. Press, Cambridge, 2006.

[8]

C. Chabauty, Limite dénsembles et géométrie des nombres, Bull. Soc. Math. France, 78 (1950), 143-151. 

[9]

C. Cuell and G. W. Patrick, Geometric discrete analogues of tangent bundles and constrained Lagrangian systems, J. Geom. Phys., 59 (2009), 976-997.  doi: 10.1016/j.geomphys.2009.04.005.

[10]

P. de la Harpe, Spaces of closed subgroups of locally compact groups, preprint, arXiv: 0807.2030.

[11]

J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 13 (1962), 472-476.  doi: 10.1090/S0002-9939-1962-0139135-6.

[12]

A. C. Hansen, A theoretical framework for backward error analysis on manifolds, J. Geom. Mech., 3 (2011), 81-111.  doi: 10.3934/jgm.2011.3.81.

[13]

A. Illanes and S. B. Nadler, Hyperspaces. Fundamentals and Recent Advances., Marcel Dekker, Inc., New York, 1999.

[14]

A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009.

[15]

A. Iserles, H. Z. Munthe-Kass, S. P. Norsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365.  doi: 10.1017/S0962492900002154.

[16]

K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. (4), 40 (1955), 61–67. doi: 10.1007/BF02416522.

[17]

K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966.

[18]

K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Pańtwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968.

[19]

R. Lucchetti and A. Pasquale, A new approach to hyperspace theory, J. Convex Anal., 1 (1994), 173-193. 

[20]

J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395.  doi: 10.1007/s002200050505.

[21]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.

[22]

K. Matsuzaki, The Chabauty and the Thurston topologies on the hyperspace of closed subsets, J. Math. Soc. Japan, 69 (2017), 263-292.  doi: 10.2969/jmsj/06910263.

[23]

E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152-182.  doi: 10.1090/S0002-9947-1951-0042109-4.

[24]

J. R. Munkres, Topology, Prentice Hall, 2000.

[25]

S. B. Nadler, Hyperspaces of Sets. A Text with Research Questions, Marcel Dekker, Inc., New York-Basel, 1978.

[26]

R. S. Palais, When proper maps are closed, Proc. Amer. Math. Soc., 24 (1970), 835-836.  doi: 10.2307/2037337.

[27]

G. W. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained Lagrangian systems, Numerische Mathematik, 113 (2009), 243-264.  doi: 10.1007/s00211-009-0245-3.

[28]

W. Rudin, Functional Analysis, McGraw-Hill, 1973.

[29]

V. Runde, A Taste of Topology, Universitext, Springer, New York, 2005.

[30] M. Schatzman, Numerical Analysis: A Mathematical Introduction, Clarendon Press, Oxford, 2002. 
[31] E. Süli and D. F. Mayer, An Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511801181.
[32]

S. Willard, General Topology, Addison-Wesley, 1970.

Figure 1.  Illustrating convergence in the lower and upper Vietoris topologies. Right: a subbasic neighbourhood of a subset $ A $, in the upper Vietoris topology, is defined by an open set $ U $. $ A $ is contained in $ U $ and the green sets are in the subbasic neighbourhood are contained in $ U $. As $ U $ shrinks, every point in the green sets in drawn to some point in $ A $—everything approximable is in $ {\rm cl}A $. At left, in the lower Vietoris topology, the green sets only have to intersect $ U $, and shrinking $ U $ around a single point of $ A $ generates approximations when the green sets also meet $ U $—everything in $ A $ is approximable
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Left: a neighbourhood of the geometric topology is defined by an open set $ U $, a compact set $ K $, and open sets $ V_i $. Containment within $ U $ has to occur only inside the compact set $ K $, with effect that a convergent sequence of subsets $ \langle A_i\rangle $ has $ K\cap A_i $ finally contained in $ U $, say for some $ i\ge N $, but larger $ K $ require larger $ N $. As shown the set $ A $ is inside the neighbourhood because its intersection with $ K $ is contained in $ U $ and contacts each $ V_i $. Smaller $ U $, and larger $ K $, and more and smaller $ V_i $, correspond to a smaller more restrictive neighbourhood. Right: a basic neighbourhood of a compact set $ B $ in the geometric topology. A subset of $ X $ is inside such a neighbourhood if it is contained in $ U $ and contacts each $ V_i $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Left: the discrete approximations $ y_i $, $ i = 1, 2, 3\ldots $ (circles) are limiting to a continuous $ y $. Shown (squares on red curves) are three selections of subsequences from the graphs of $ y_i $. Every such subsequence converges to the graph of $ y $, and that graph is the limit of such subsequences. Right: an open neighbourhood of the red graph is defined by a compact set $ K $, an open set $ U $, and open sets $ V_i $. $ K $, which may be restricted to a product of compact sets, may be thought of as a frame within which proximity to the graph is controlled by $ U $. The other black curves are in the neighbourhood because they also contact the $ V_i $. Larger $ K $, smaller $ U $, and more and smaller $ V_i $, correspond to smaller neighbourhoods
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[2]

Enrico Gerlach, Charlampos Skokos. Comparing the efficiency of numerical techniques for the integration of variational equations. Conference Publications, 2011, 2011 (Special) : 475-484. doi: 10.3934/proc.2011.2011.475
[3]

Cheng Wang. Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1143-1172. doi: 10.3934/dcdsb.2004.4.1143
[4]

Hongguang Xiao, Wen Tan, Dehua Xiang, Lifu Chen, Ning Li. A study of numerical integration based on Legendre polynomial and RLS algorithm. Numerical Algebra, Control and Optimization, 2017, 7 (4) : 457-464. doi: 10.3934/naco.2017028
[5]

Sanjit Kumar Mohanty, Rajani Ballav Dash. A quadrature rule of Lobatto-Gaussian for numerical integration of analytic functions. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021031
[6]

Sergio Blanes, Fernando Casas, Alejandro Escorihuela-Tomàs. Applying splitting methods with complex coefficients to the numerical integration of unitary problems. Journal of Computational Dynamics, 2022, 9 (2) : 85-101. doi: 10.3934/jcd.2021022
[7]

Sebastián J. Ferraro, David Iglesias-Ponte, D. Martín de Diego. Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods. Conference Publications, 2009, 2009 (Special) : 220-229. doi: 10.3934/proc.2009.2009.220
[8]

Ruijun Zhao, Yong-Tao Zhang, Shanqin Chen. Krylov implicit integration factor WENO method for SIR model with directed diffusion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4983-5001. doi: 10.3934/dcdsb.2019041
[9]

Hao-Chun Lu. An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, 2012, 8 (2) : 429-455. doi: 10.3934/jimo.2012.8.429
[10]

Zhanyuan Hou. Geometric method for global stability of discrete population models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3305-3334. doi: 10.3934/dcdsb.2020063
[11]

Wen Li, Song Wang, Volker Rehbock. A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 273-287. doi: 10.3934/naco.2017018
[12]

Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79.

[13]

Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223
[14]

Annalisa Cesaroni, Valerio Pagliari. Convergence of nonlocal geometric flows to anisotropic mean curvature motion. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4987-5008. doi: 10.3934/dcds.2021065
[15]

Alberto Zingaro, Ivan Fumagalli, Luca Dede, Marco Fedele, Pasquale C. Africa, Antonio F. Corno, Alfio Quarteroni. A geometric multiscale model for the numerical simulation of blood flow in the human left heart. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2391-2427. doi: 10.3934/dcdss.2022052
[16]

Shuang Liu, Xinfeng Liu. Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 141-159. doi: 10.3934/dcdsb.2019176
[17]

Antonia Katzouraki, Tania Stathaki. Intelligent traffic control on internet-like topologies - integration of graph principles to the classic Runge--Kutta method. Conference Publications, 2009, 2009 (Special) : 404-415. doi: 10.3934/proc.2009.2009.404
[18]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 1-21. doi: 10.3934/dcdss.2021006
[19]

Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial and Management Optimization, 2011, 7 (1) : 199-210. doi: 10.3934/jimo.2011.7.199
[20]

Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems and Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033

 Impact Factor: 

Tools

Metrics

  • PDF downloads (314)
  • HTML views (502)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy