×

A singular value homotopy for finding critical parameter values. (English) Zbl 1470.65106

Summary: Various applications in science and engineering depend upon computing real solutions to systems of analytic equations which depend upon real parameters. Locally in the parameter space, the qualitative behavior of the solutions remains the same except at critical parameter values. This article develops a singular value homotopy that aims to compute critical parameter values. Several examples are presented including computing critical parameter values for nonlinear boundary value problems, turning points for a steady-state system connected to learning and memory, and computing the maximum Gaussian curvature of a surface.

MSC:

65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations

Software:

AUTO; Bertini; MATCONT; Maple
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allgower, E. L.; Bates, D. J.; Sommese, A. J.; Wampler, C. W., Solution of polynomial systems derived from differential equations, Computing, 76, 1-2, 1-10 (2006) · Zbl 1086.65075
[2] Ascher, U. M.; Petzold, L. R., Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, vol. 61 (1998), SIAM · Zbl 0908.65055
[3] Baginski, F. E.; Whitaker, N., Numerical solutions of boundary value problems for \(\mathcal{K} \)-surfaces in \(\mathbb{R}^3\), Numer. Methods Partial Differ. Equ., 12, 4, 525-546 (1996) · Zbl 0856.65078
[4] Bates, D. J.; Hauenstein, J. D.; Peterson, C.; Sommese, A. J., Numerical decomposition of the rank-deficiency set of a matrix of multivariate polynomials, (Robbiano, L.; Abbott, J., Approximate Commutative Algebra (2010), Springer Vienna: Springer Vienna Vienna), 55-77 · Zbl 1191.14071
[5] Bates, D. J.; Hauenstein, J. D.; Sommese, A. J.; Wampler, C. W., Numerically Solving Polynomial Systems with Bertini, Software, Environments, and Tools, vol. 25 (2013), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 1295.65057
[6] Dhooge, A.; Govaerts, W.; Kuznetsov, Y. A., MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs, SIGSAM Bull., 38, 1, 21-22 (2004)
[7] Doedel, E., AUTO: a program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30, 265-284 (1981)
[8] Eckart, C.; Young, G., The approximation of one matrix by another of lower rank, Psychometrika, 1, 211-218 (1936) · JFM 62.1075.02
[9] Farrell, P. E.; Birkisson, Á.; Funke, S. W., Deflation techniques for finding distinct solutions of nonlinear partial differential equations, SIAM J. Sci. Comput., 37, 4, A2026-A2045 (2015) · Zbl 1327.65237
[10] Gautschi, W., Numerical Analysis (1997), Springer Science & Business Media
[11] Graef, J. R.; Kong, L.; Wang, H., Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem, J. Differ. Equ., 245, 5, 1185-1197 (2008) · Zbl 1203.34028
[12] Hao, W.; Hauenstein, J. D.; Hu, B.; Sommese, A. J., A bootstrapping approach for computing multiple solutions of differential equations, J. Comput. Appl. Math., 258, 181-190 (2014) · Zbl 1294.65085
[13] Hao, W.; Hauenstein, J. D.; Hu, B.; Sommese, A. J., A three-dimensional steady-state tumor system, Appl. Math. Comput., 218, 6, 2661-2669 (2011) · Zbl 1238.92019
[14] Harrington, H. A.; Mehta, D.; Byrne, H. M.; Hauenstein, J. D., Decomposing the parameter space of biological networks via a numerical discriminant approach, (Gerhard, J.; Kotsireas, I., Maple in Mathematics Education and Research (2020), Springer International Publishing: Springer International Publishing Cham), 114-131
[15] Hartley, R.; Zisserman, A., Multiple View Geometry in Computer Vision (2003), Cambridge University Press: Cambridge University Press Cambridge
[16] Hauenstein, J. D.; Regan, M. H., Adaptive strategies for solving parameterized systems using homotopy continuation, Appl. Math. Comput., 332, 19-34 (2018) · Zbl 1427.65086
[17] Hauenstein, J. D.; Sommese, A. J.; Wampler, C. W., Regeneration homotopies for solving systems of polynomials, Math. Comput., 80, 273, 345-377 (2011) · Zbl 1221.65121
[18] Laetsch, T., On the number of solutions of boundary value problems with convex nonlinearities, J. Math. Anal. Appl., 35, 389-404 (1971) · Zbl 0191.40102
[19] LeVeque, R. J., Finite Difference Methods for Ordinary and Partial Differential EquationsSteady-State and Time-Dependent Problems, vol. 98 (2007), SIAM
[20] Lin, S. S., Positive radial solutions and nonradial bifurcation for semilinear elliptic equations in annular domains, J. Differ. Equ., 86, 2, 367-391 (1990) · Zbl 0734.35073
[21] Mirsky, L., Symmetric gauge functions and unitarily invariant norms, Q. J. Math., 11, 1, 50-59 (1960) · Zbl 0105.01101
[22] Papageorgiou, N. S.; Smyrlis, G., A bifurcation-type theorem for singular nonlinear elliptic equations, Methods Appl. Anal., 22, 2, 147-170 (2015) · Zbl 1323.35042
[23] Pettigrew, D. B.; Smolen, P.; Baxter, D. A.; Byrne, J. H., Dynamic properties of regulatory motifs associated with induction of three temporal domains of memory in Aplysia, J. Comput. Neurosci., 18, 2, 163-181 (2005)
[24] Piret, K.; Verschelde, J., Sweeping algebraic curves for singular solutions, J. Comput. Appl. Math., 234, 4, 1228-1237 (2010) · Zbl 1189.65101
[25] Shi, J.; Yao, M., On a singular nonlinear semilinear elliptic problem, Proc. R. Soc. Edinb., Sect. A, 128, 6, 1389-1401 (1998) · Zbl 0919.35044
[26] Sommese, A. J.; Wampler, C. W., The Numerical Solution of Systems of Polynomials Arising in Engineering and Science (2005), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ · Zbl 1091.65049
[27] Song, H.; Smolen, P.; Av-Ron, E.; Baxter, D. A.; Byrne, J. H., Bifurcation and singularity analysis of a molecular network for the induction of long-term memory, Biophys. J., 90, 7, 2309-2325 (2006)
[28] Strogatz, S. H., Nonlinear Dynamics and Chaos (2015), Westview Press: Westview Press Boulder, CO, With applications to physics, biology, chemistry, and engineering
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.