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A Homotopy Method with Adaptive Basis Selection for Computing Multiple Solutions of Differential Equations

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Abstract

The homotopy continuation method has been widely used to compute multiple solutions of nonlinear differential equations, but the computational cost grows exponentially based on the traditional finite difference and finite element discretizations. In this work, we presented a new method by constructing a spectral approximation space adaptively based on a greedy algorithm for nonlinear differential equations. Then multiple solutions were computed by the homotopy continuation method on this low-dimensional approximation space. Various numerical examples were given to illustrate the feasibility and the efficiency of this new approach.

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References

  1. Allgower, E., Bates, D., Sommese, A., Wampler, C.: Solution of polynomial systems derived from differential equations. Computing 76(1–2), 1–10 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allgower, E., Cruceanu, S., Tavener, S.: Application of numerical continuation to compute all solutions of semilinear elliptic equations. Adv. Geom. 9(3), 371–400 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bates, D., Hauenstein, J., Sommese, A., Wampler, C.: Adaptive multiprecision path tracking. SIAM J. Numer. Anal. 46(2), 722–746 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bates, D., Hauenstein, J., Sommese, A., Wampler, C.: Stepsize control for path tracking. Contemp. Math. 496(3), 21–31 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bates, D., Hauenstein, J., Sommese, A., Wampler, C.: Numerically Solving Polynomial Systems with Bertini, vol. 25. SIAM, Philadelphia (2013)

    Book  MATH  Google Scholar 

  6. Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Adaptive multiprecision path tracking. SIAM J. Nume. Anal. 46(2), 722–746 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Numerically Solving Polynomial Systems with Bertini, vol. 25. SIAM, Philadelphia (2013)

    Book  MATH  Google Scholar 

  8. Bauer, L., Keller, H.B., Reiss, E.: Multiple eigenvalues lead to secondary bifurcation. SIAM Rev. 17(1), 101–122 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bernardi, C., Maday, Y.: Spectral methods. Handb. Numer. Anal. 5, 209–485 (1997)

    MathSciNet  Google Scholar 

  10. Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., Wojtaszczyk, P.: Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43, 1457–1472 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Braun, M., Golubitsky, M.: Differential Equations and Their Applications, vol. 4. Springer, New York (1983)

    Book  Google Scholar 

  12. Breuer, B., McKenna, P., Plum, M.: Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof. J. Differ. Equ. 195(1), 243–269 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, C., Xie, Z.: Structure of multiple solutions for nonlinear differential equations. Sci. China Ser. A Math. 47(1), 172–180 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, F., Shen, J., Yu, H.: A new spectral element method for pricing european options under the black-scholes and merton jump diffusion models. J. Sci. Comput. 52(3), 499–518 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, Y., Hesthaven, J., Maday, Y., Rodrguez, J.: Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell’s problem. ESAIM Math. Model. Numer. Anal. 43(6), 1099–1116 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, Y., Hesthaven, J.S., Maday, Y., Rodríguez, J.: Certified reduced basis methods and output bounds for the harmonic Maxwell’s equations. SIAM J. Sci. Comput. 32(2), 970–996 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chossat, P., Golubitsky, M.: Symmetry-increasing bifurcation of chaotic attractors. Physica D 32(3), 423–436 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cristini, V., Li, X., Lowengrub, J., Wise, S.: Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching. J. Math. Biol. 58(4), 723–763 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cristini, V., Lowengrub, J., Nie, Q.: Nonlinear simulation of tumor growth. J. Math. Biol. 46(3), 191–224 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fontelos, M., Friedman, A.: Symmetry-breaking bifurcations of free boundary problems in three dimensions. Asymptot. Anal. 35(3), 187–206 (2003)

    MathSciNet  MATH  Google Scholar 

  21. Friedman, A.: Free boundary problems in biology. Philos. Trans. R. Soc. A 373(2050), 20140368 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Friedman, A., Hao, W.: A mathematical model of atherosclerosis with reverse cholesterol transport and associated risk factors. Bull. Math. Biol. 77(5), 758–781 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Friedman, A., Hu, B.: Bifurcation from stability to instability for a free boundary problem arising in a tumor model. Arch. Ration. Mech. Anal. 180(2), 293–330 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Friedman, A., Hu, B.: Bifurcation for a free boundary problem modeling tumor growth by Stokes equation. SIAM J. Math. Anal. 39(1), 174–194 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Friedman, A., Reitich, F.: Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth. Trans. Am. Math. Soc. 353(4), 1587–1634 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Golubitsky, M., Stewart, I.: The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, vol. 200. Springer, New York (2003)

    MATH  Google Scholar 

  27. Golubitsky, M., Stewart, I.: Singularities and Groups in Bifurcation Theory, vol. 2. Springer, New York (2012)

    Google Scholar 

  28. Grepl, M., Patera, A.: A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM Math. Model. Numer. Anal. 39(1), 157–181 (2005)

    Article  MATH  Google Scholar 

  29. Haber, R., Unbehauen, H.: Structure identification of nonlinear dynamic systems survey on input/output approaches. Automatica 26(4), 651–677 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hao, W., Crouser, E., Friedman, A.: Mathematical model of sarcoidosis. Proc. Nat. Acad. Sci. 111(45), 16065–16070 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hao, W., Friedman, A.: The ldl-hdl profile determines the risk of atherosclerosis: a mathematical model. PLoS ONE 9(3), e90497 (2014)

    Article  Google Scholar 

  32. Hao, W., Hauenstein, J., Hu, B., Liu, Y., Sommese, A., Zhang, Y.-T.: Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core. Nonlinear Anal. Real World Appl. 13(2), 694–709 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hao, W., Hauenstein, J., Hu, B., Sommese, A.: A bootstrapping approach for computing multiple solutions of differential equations. J. Comput. Appl. Math. 258, 181–190 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hao, W., Hauenstein, J., Shu, C.-W., Sommese, A., Xu, Z., Zhang, Y.-T.: A homotopy method based on weno schemes for solving steady state problems of hyperbolic conservation laws. J. Comput. Phys. 250, 332–346 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hao, W., Nepomechie, R., Sommese, A.: Completeness of solutions of bethe’s equations. Phys. Rev. E 88(5), 052113 (2013)

    Article  Google Scholar 

  36. Hao, W., Nepomechie, R., Sommese, A.: Singular solutions, repeated roots and completeness for higher-spin chains. J. Stat. Mech Theory Exp. 2014(3), P03024 (2014)

    Article  MathSciNet  Google Scholar 

  37. Hao, W., Sommese, A., Zeng, Z.: Algorithm 931: an algorithm and software for computing multiplicity structures at zeros of nonlinear systems. ACM Trans. Math. Softw. TOMS 40(1), 5–20 (2013)

    MathSciNet  MATH  Google Scholar 

  38. Hou, T., Lowengrub, J., Shelley, M.: Boundary integral methods for multicomponent fluids and multiphase materials. J. Comput. Phys. 169(2), 302–362 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Li, T.-Y., Sauer, T., Yorke, J.: The Cheater’s homotopy: an efficient procedure for solving systems of polynomial equations. SIAM J. Numer. Anal. 26(5), 1241–1251 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  40. Li, T.-Y., Zeng, Z.: Homotopy-determinant algorithm for solving nonsymmetric eigenvalue problems. Math. Comput. 59(200), 483–502 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  41. Morgan, A., Sommese, A.: Computing all solutions to polynomial systems using homotopy continuation. Appl. Math. Comput. 24(2), 115–138 (1987)

    MathSciNet  MATH  Google Scholar 

  42. Morgan, A., Sommese, A.: A homotopy for solving general polynomial systems that respects m-homogeneous structures. Appl. Math. Comput. 24(2), 101–113 (1987)

    MathSciNet  MATH  Google Scholar 

  43. Peaceman, D., Rachford, H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  44. Pearson, J.: Complex patterns in a simple system. Science 261(3), 189–189 (1993)

    Article  Google Scholar 

  45. Pousin, J., Rappaz, J.: Consistency, stability, a priori and a posteriori errors for petrov-galerkin methods applied to nonlinear problems. Numer. Math. 69(2), 213–231 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  46. Rabier, P., Rheinboldt, W.: On a computational method for the second fundamental tensor and its application to bifurcation problems. Numer. Math. 57(1), 681–694 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rheinboldt, W.: Numerical methods for a class of finite dimensional bifurcation problems. SIAM J. Numer. Anal. 15(1), 1–11 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  48. Rheinboldt, W.: Numerical analysis of continuation methods for nonlinear structural problems. Comput. Struct. 13(1), 103–113 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  49. Rheinboldt, W., Burkardt, J.: A locally parameterized continuation process. ACM Trans. Math. Softw. TOMS 9(2), 215–235 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  50. Rozza, G., Huynh, D., Patera, A.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 1 (2007)

    Article  Google Scholar 

  51. Rozza, G., Huynh, D., Patera, A.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  52. Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications, vol. 41. Springer, New York (2011)

    Book  MATH  Google Scholar 

  53. Smith, G.: Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press, Oxford (1985)

    Google Scholar 

  54. Sommese, A., Verschelde, J., Wampler, C.: Homotopies for intersecting solution components of polynomial systems. SIAM J. Numer. Anal. 42(4), 1552–1571 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  55. Sommese, A., Verschelde, J., Wampler, C.: An intrinsic homotopy for intersecting algebraic varieties. J. Complex. 21(4), 593–608 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  56. Sommese, A., Wampler, C.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, vol. 99. World Scientific, Singapore (2005)

    Book  MATH  Google Scholar 

  57. Strogatz, S.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview press, Boulder (2014)

    MATH  Google Scholar 

  58. Turton, R., Bailie, R., Whiting, W., Shaeiwitz, J.: Analysis, Synthesis and Design of Chemical Processes. Pearson Education, London (2008)

    Google Scholar 

  59. Veroy, K., Prud’Homme, C., Rovas, D., Patera, A.: A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: 16th AIAA Computational Fluid Dynamics Conference, p. 3847 (2003)

  60. Xu, J.: Two-grid discretization techniques for linear and nonlinear pdes. SIAM J. Numer. Anal. 33(5), 1759–1777 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zhang, X., Zhang, J., Yu, B.: Eigenfunction expansion method for multiple solutions of semilinear elliptic equations with polynomial nonlinearity. SIAM J. Numer. Anal. 51(5), 2680–2699 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  62. Zhou, J.: Solving multiple solution problems: computational methods and theory revisited. Commun. Appl. Math. 3(1), 1–31 (2017)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

W. Hao’s research was supported by the American Heart Association (Grant 17SDG33660722) and National Science Foundation (Grant DMS-1818769). GL would like to gratefully acknowledge the support from National Science Foundation (DMS-1555072, DMS-1736364 and DMS-1821233). B. Zheng would like to acknowledge the support by Beijing Institute for Scientific and Engineering Computing and by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials and a Laboratory Directed Research and Development (LDRD) Program from Pacific Northwest National Laboratory. The PNNL is operated by Battelle for the US Department of Energy under Contract DE-AC05-76RL01830.

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Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Wenrui Hao

  2. Computational Mathematics and Simulation Science, École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Jan Hesthaven

  3. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Guang Lin

  4. Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing, 100124, China

    Bin Zheng

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  1. Wenrui Hao
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Hao, W., Hesthaven, J., Lin, G. et al. A Homotopy Method with Adaptive Basis Selection for Computing Multiple Solutions of Differential Equations. J Sci Comput 82, 19 (2020). https://doi.org/10.1007/s10915-020-01123-1

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