基于LSQR法的外部数值保角逆变换计算法
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摘要
本文提出了基于模拟电荷法的外部数值保角逆变换计算法.利用LSQR (Least Square QR-factorization)方法来求解基于模拟电荷法的外部数值保角逆变换中的约束方程,得到了电荷量和逆变换半径,进而构造了近似保角逆变换函数.数值实验证明本文提出的算法是有效的.
In this paper, a bidirectional method for the numerical conformal mapping of exterior domain is proposed. The LSQR(least square QR-factorization method) is used to solve the approximate inverse conformal mapping of the constraint equations. The maximum modulus theorem for regular function can be used to estimate error. Numerical examples are presented to illustrate the efficiency of the method.
In this paper, a bidirectional method for the numerical conformal mapping of exterior domain is proposed. The LSQR(least square QR-factorization method) is used to solve the approximate inverse conformal mapping of the constraint equations. The maximum modulus theorem for regular function can be used to estimate error. Numerical examples are presented to illustrate the efficiency of the method.
引文
[1]祝颖润,曹俊飞,王瑞庭.保形映射的一个充分必要条件[J].东北师大学报(自然科学版),2016, 1:159-160.
[2]朱满座.数值保角变换及其在电磁理论中的应用[D].西安:西安电子科技大学,2008.
[3]张家玲,吕毅斌.广义Willmore泛函及其极值子流形[J].昆明理工大学学报(自然科学版),2016, 5:138—142.
[4] Kythe P K. Computational conformal mapping[M]. Birkhauser:Boston, 1998.
[5] Dirscolland T A, Trefethen L N. Schwarz-Christoffel mapping[M]. Cambridge:Cambridge University Press, 2002.
[6] Sangawia A W K, Murid A H M, Nasser M M S. Annulus with circular slit map of bounded multiply connected regions via integral equation method[J]. Bull. Malays. Math. Sci. Soc., 2012,4(4):945-959.
[7] Kim J W, Lee D J. Optimized compact finite difference schemes with maximum resolution[J]. AIAA J., 1996, 34(5):887-893.
[8] Kokkinos C A, Papamichael N, Sideridis A B. An orthonormalization method for the approximate conformal mapping of multiply-connected domains[J]. IMAJ. Numer. Anal., 1990, 10(3):343-359.
[9] Papamichael N, Pritsker I E, Saff E B, et al. Approximation of conformal mappings of annular regions[J]. Numer. Math., 1997, 76(4):489-513.
[10] Lo W L, Wu N J, Chen C S, et al. Exact boundary derivative formulation for numerical conformal mapping method[J]. Math. Prob. Engin., 2016, 2:1-18.
[11] Singer H, Steinbigler H, Weiss P. A charge simulation method for calculation of high voltage field[J].IEEE Trans. Power Apparatus Syst., 1974, 9(3):1660-1668.
[12] Amano K. Numerical conformal mappings of exterior domains based on the charge simulationmethod(in Japanese)[J]. Trans. Inform. Proc. Soc. Japan, 1988, 29(1):62-72.
[13] Amano K. Numerical conformal mappings of interior domains based on the charge simulation(in Japanese)[J]. Trans. Inform. Proc. Soc. Japan, 1988, 29(7):697-699.
[14] Amano K. Dai O, Ogata H, et al. Numerical conformal mappings onto the linear slit domain[J].Japan J. Ind. Appl. Math., 2012, 7(29):165-186.
[15] Amano K. A charge simulation method for the numerical conformal mapping of interior,yexterior and doubly-connected domains[J]. J. Comput. Appl. Math., 1994, 53(3):354-361.
[16] Amano K. Numerical conformal mappings of interior both directions domains based on the charge simulation method(in Japanese)[J]. Trans. Inform. Proc. Soc. Japan, 1990, 5:623-632.
[17]吕毅斌,赖富明,王樱子,武德安.基于GMRES(m)法的双连通区域数值保角变换的计算法[J].数学杂志,2016, 36(6):1028-1034.
[18]王樱子,赖富明,吕毅斌等.基于Padé迭代法的数值保角变换计算法[J].数值计算与计算机应用,2016,37(4):299-306.
[19] Lu Y, Wu D, Wang Y, et al. The accuracy improvement of numerical conformal mapping using the modified Gram-Schmidt method[C]. The 19th International Conference on Industrial Engineering and Engineering Management, Berlin, Heidelberg:Springer, 2013:555-56.
[20] Lai F, Wang Y, Lu Y, et al. Improving the accuracy of the charge simulation method for numerical conformal mapping[J]. Math. Prob. Engin., 2017, 4:1-9.
[21] Paige C C, Saunders M A. LSQR:an algorithm for sparse linear equations and sparse least squares[J].Acm Trans. Math. Soft., 1982, 8(1):43-71.
[22]曹志浩著.变分迭代发[M].北京:科学出版社,2005.
[23]栾颖著.MATLAB R2013a求解数学问题[M].北京:清华大学出版社,2014.
[24]周茜,雷渊,乔文龙.一类线性约束矩阵不等式及其最小二乘问题[J].计算数学,2016, 2:171-186.
[2]朱满座.数值保角变换及其在电磁理论中的应用[D].西安:西安电子科技大学,2008.
[3]张家玲,吕毅斌.广义Willmore泛函及其极值子流形[J].昆明理工大学学报(自然科学版),2016, 5:138—142.
[4] Kythe P K. Computational conformal mapping[M]. Birkhauser:Boston, 1998.
[5] Dirscolland T A, Trefethen L N. Schwarz-Christoffel mapping[M]. Cambridge:Cambridge University Press, 2002.
[6] Sangawia A W K, Murid A H M, Nasser M M S. Annulus with circular slit map of bounded multiply connected regions via integral equation method[J]. Bull. Malays. Math. Sci. Soc., 2012,4(4):945-959.
[7] Kim J W, Lee D J. Optimized compact finite difference schemes with maximum resolution[J]. AIAA J., 1996, 34(5):887-893.
[8] Kokkinos C A, Papamichael N, Sideridis A B. An orthonormalization method for the approximate conformal mapping of multiply-connected domains[J]. IMAJ. Numer. Anal., 1990, 10(3):343-359.
[9] Papamichael N, Pritsker I E, Saff E B, et al. Approximation of conformal mappings of annular regions[J]. Numer. Math., 1997, 76(4):489-513.
[10] Lo W L, Wu N J, Chen C S, et al. Exact boundary derivative formulation for numerical conformal mapping method[J]. Math. Prob. Engin., 2016, 2:1-18.
[11] Singer H, Steinbigler H, Weiss P. A charge simulation method for calculation of high voltage field[J].IEEE Trans. Power Apparatus Syst., 1974, 9(3):1660-1668.
[12] Amano K. Numerical conformal mappings of exterior domains based on the charge simulationmethod(in Japanese)[J]. Trans. Inform. Proc. Soc. Japan, 1988, 29(1):62-72.
[13] Amano K. Numerical conformal mappings of interior domains based on the charge simulation(in Japanese)[J]. Trans. Inform. Proc. Soc. Japan, 1988, 29(7):697-699.
[14] Amano K. Dai O, Ogata H, et al. Numerical conformal mappings onto the linear slit domain[J].Japan J. Ind. Appl. Math., 2012, 7(29):165-186.
[15] Amano K. A charge simulation method for the numerical conformal mapping of interior,yexterior and doubly-connected domains[J]. J. Comput. Appl. Math., 1994, 53(3):354-361.
[16] Amano K. Numerical conformal mappings of interior both directions domains based on the charge simulation method(in Japanese)[J]. Trans. Inform. Proc. Soc. Japan, 1990, 5:623-632.
[17]吕毅斌,赖富明,王樱子,武德安.基于GMRES(m)法的双连通区域数值保角变换的计算法[J].数学杂志,2016, 36(6):1028-1034.
[18]王樱子,赖富明,吕毅斌等.基于Padé迭代法的数值保角变换计算法[J].数值计算与计算机应用,2016,37(4):299-306.
[19] Lu Y, Wu D, Wang Y, et al. The accuracy improvement of numerical conformal mapping using the modified Gram-Schmidt method[C]. The 19th International Conference on Industrial Engineering and Engineering Management, Berlin, Heidelberg:Springer, 2013:555-56.
[20] Lai F, Wang Y, Lu Y, et al. Improving the accuracy of the charge simulation method for numerical conformal mapping[J]. Math. Prob. Engin., 2017, 4:1-9.
[21] Paige C C, Saunders M A. LSQR:an algorithm for sparse linear equations and sparse least squares[J].Acm Trans. Math. Soft., 1982, 8(1):43-71.
[22]曹志浩著.变分迭代发[M].北京:科学出版社,2005.
[23]栾颖著.MATLAB R2013a求解数学问题[M].北京:清华大学出版社,2014.
[24]周茜,雷渊,乔文龙.一类线性约束矩阵不等式及其最小二乘问题[J].计算数学,2016, 2:171-186.