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基于奇异值分解的非线性系统流形解的数值计算
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Abstract:
本文利用边界矩阵技术，将非线性系统流形解的数值计算转化为增广非线性系统正则解的数值计算。利用雅可比矩阵的近似左零空间和右零空间构造增广矩阵，进而利用增广矩阵构造一族增广非线性系统，该族增广非线性系统中每个非线性系统均为给定非线性系统添加约束条件后的摄动系统，利用牛顿迭代更新增广非线性系统的解使得其逼近原非线性系统的流形解。
In this paper, the numerical computation of manifold solutions of nonlinear systems is transformed into the numerical computation of regular solutions of augmented nonlinear systems using the boundary matrix technique. The approximate left-zero space and right-zero space of Jacobi matrix are used to construct the generalized matrix, and then the generalized matrix is used to construct a family of generalized nonlinear systems, and each nonlinear system in the family of generalized nonlinear systems is a regenerative system after the constraints are added to the given nonlinear system, and Newton iteration is utilized to update the solution of the generalized nonlinear system to make it approximate to the manifold solution of the original nonlinear system.
| [1] | Allgower, E.L. and Georg, K. (1990) Numerical Continuation Methods: An Introduction. Springer. |
| [2] | Sommese, A.J. and Wampler, C.W. (2005) The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific Publishing. https://doi.org/10.1142/9789812567727 |
| [3] | Tenenbaum, J.B., Silva, V.D. and Langford, J.C. (2000) A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science, 290, 2319-2323. https://doi.org/10.1126/science.290.5500.2319 |
| [4] | Bates, D.J., Sommese, A.J., Hauenstein, J.D. and Wampler, C.W. (2013) Numerically Solving Polynomial Systems with Bertini. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611972702 |
| [5] | Hauenstein, J.D. and Sottile, F. (2012) Algorithm 921. ACM Transactions on Mathematical Software, 38, 1-20. https://doi.org/10.1145/2331130.2331136 |
| [6] | Kuznetsov, Y.A. (2004) Elements of Applied Bifurcation Theory. 3rd Edition, Springer. |
| [7] | O’Malley, R.E. (1991) Singular Perturbation Methods for Ordinary Differential Equations. Springer. |
| [8] | Raissi, M., Perdikaris, P. and Karniadakis, G.E. (2019) Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations. Journal of Computational Physics, 378, 686-707. https://doi.org/10.1016/j.jcp.2018.10.045 |
| [9] | Griewank, A. and Walther, A. (2008) Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. 2nd Edition, Society for Industrial and Applied Mathematics. |
| [10] | Deuflhard, P. (2004) Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer. |
| [11] | Kelley, C.T. (2003) Solving Nonlinear Equations with Newton's Method. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9780898718898 |
| [12] | Li, Z. and Sang, H. (2014) Verified Error Bounds for Singular Solutions of Nonlinear Systems. Numerical Algorithms, 70, 309-331. https://doi.org/10.1007/s11075-014-9948-3 |
| [13] | Li, Z., Wan, B. and Gao, R. (2017) Verified Error Bounds for Eigenvalues of Geometric Multiplicity Q and Corresponding Invariant Subspaces. Numerical Algorithms, 77, 631-652. https://doi.org/10.1007/s11075-017-0332-y |
