小波分析在常微分方程求解中的应用
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摘要
小波分析是一门新兴理论，它被广泛地应用于各个领域。作为80年代末期出现的时频分析工具，小波变换在信号与图像处理等领域里已经得到了成功的应用，并凭借其自身的诸多优点成为了JPEG2000的标准，基于多尺度分析的尺度函数和小波函数很好的分析特性和计算特性，充分利用这些特性以小波作为基函数离散微分方程，再来求解所得到的代数方程，这种小波有限元法最近得到人们的很大的关注。
本文在详细论述了M带多小波理论的基础上，研究了小波分析方法在常微分方程求解中的应用，讨论了常微分方程在小波基下离散成代数方程后，提出最佳M 项逼近的思想，并结合小波基的特性，对求解代数方程组的迭代法进行了改进，使得计算量大大减少，特别有利于求解大型方程组，从理论上分析了这一改进算法的可行性，并用数值实验证明了其合理性。
Wavelet analysis theory is a arisen science which was applied extensively to every domain. Being a time-frequency analysis tool in 1980s', wavelet transform has succeeded to be applied to the signal and image processing domain and been the criterion of JPEG 2000 by it's advantages. Based on Multiresolution Analysis, scale functions and wavelet functions have good analysis and computation characteristic which are been made the best of to discrete the differential equations. Then some algebra equations can be acquired. The method is named of wavelet finite element method has attracted many scholars' interests.
In this paper, after wavelet analysis theory being discussed clearly, the multiresolution analysis of M-band multiwavelet theory is introduced which contributed to multiresolution filter banks. The application of wavelet used to the solution of ordinary differential equations (ODE) and a modified algorithm which resolves ODE is proposed. After ODE being a discrete algebra equation systems by wavelet, the best M term approximation idea is combined with the wavelet characteristics. And the iterate method of equation systems is modified. The computation is reduced, the improved algorithm contributes to the resolution of large equation systems. The feasibility and reasonability are proved by theory analysis and numerical experiments.
本文在详细论述了M带多小波理论的基础上，研究了小波分析方法在常微分方程求解中的应用，讨论了常微分方程在小波基下离散成代数方程后，提出最佳M 项逼近的思想，并结合小波基的特性，对求解代数方程组的迭代法进行了改进，使得计算量大大减少，特别有利于求解大型方程组，从理论上分析了这一改进算法的可行性，并用数值实验证明了其合理性。
Wavelet analysis theory is a arisen science which was applied extensively to every domain. Being a time-frequency analysis tool in 1980s', wavelet transform has succeeded to be applied to the signal and image processing domain and been the criterion of JPEG 2000 by it's advantages. Based on Multiresolution Analysis, scale functions and wavelet functions have good analysis and computation characteristic which are been made the best of to discrete the differential equations. Then some algebra equations can be acquired. The method is named of wavelet finite element method has attracted many scholars' interests.
In this paper, after wavelet analysis theory being discussed clearly, the multiresolution analysis of M-band multiwavelet theory is introduced which contributed to multiresolution filter banks. The application of wavelet used to the solution of ordinary differential equations (ODE) and a modified algorithm which resolves ODE is proposed. After ODE being a discrete algebra equation systems by wavelet, the best M term approximation idea is combined with the wavelet characteristics. And the iterate method of equation systems is modified. The computation is reduced, the improved algorithm contributes to the resolution of large equation systems. The feasibility and reasonability are proved by theory analysis and numerical experiments.
引文
[1] 程正兴．小波分析算法与应用．西安交通大学出版社．
[2] 秦前清，杨宗凯．实用小波分析．西安电子科技大学出版社．
[3] Meyer Y. 小波与算子(Vol.1)．北京．世界图书出版社．1990．
[4] Daubechies I. Ten Lectures on Wavelets. CBMS-NSF Series On Applied Math. SIAM. 1992.
[5] V. Strela. Multiwavelet: Theory and applications. Ph D Thesis MIT. 1996.
[6] 程正兴，张玲玲．多小波分析与应用．工程数学学报，2001，18(1)：99-107．
[7] Chui C. K. and Lian J. A study on orthonormal multiwavelets [J]. J. Appl. Numer. Math. 1996: 20: 273-298.
[8] Plonka G. and Strela V. Construction of multiscaling functions with approximation and symmetry [J]. Siam J. Math. anal. 1998, 29:481-510.
[9] Hehong Zou, Tewfik Ahmed H. Discrete orthogoanl m-band wavelet decompositons [J]. IEEE Trans on Signal Processing. 1992, 4(4): 605-608.
[10] Peter Steffen. Theory of regular m-band wavelet bases [J]. IEEE Trans. on Signal Processing, 1993, 41(12): 3497-3511.
[11] Meyer Y. Ondelettes et functions splines. Semnaire EDP, Ecole Polytechnique. Paris. Dec. 1986.
[12] P. Steffen, P. Heller, and et. al. Theory of regular M-band wavelet bases. IEEE Trans. Signal Processing. 1993, 41: 3497-3511.
[13] T. N. T. Goodman, and S. L. Lee. Wavelet of Multiplicity r. Trans. Amer. Math. Soc. 1994, 342: 307-324.
[14] Asim Bhatti and Hǖseyin (?)zkaramanli. M-band multiwavelets from spline super functions with approximation order. IEEE ICASSP, Orlando, Florida. May 2002.
[15] H. Zou and A. H. Tewfik. Discrete orthogonal M-band wavelet decompositions. Proc. ICASSP. San Francisco, CA. 1992, 4: Ⅳ-605-Ⅳ-608.
[16] Geronimo J, Hardin D Massopust P R. Fractal functions and
wavelet expansions based on several functions [J]. J. Approx. Theory, 1994, 78: 373-401.
[17] Alpert B. A class of basis in L~2 for the sparse representation of intergral operators [J]. Siam J. Math. Naal, 1993, 24.
[18] DP Hardin, JA Marasovich. Biorthogonal multiwavelets on [-1, 1] [J]. Applied and computational harmonic analysis, 1999, 7(1).
[19] Asim Bhatti and Hǖseyin (?)zkaramanli. M-band multiwavelets from spline super functions with approximation order. IEEE ICASSP, Orlando, Florida. May 2002.
[20] W. Dahmen. Wavelet and multiscale methods for operator equations. Acta Numer. 2001, 6:55-228.
[21] W. Dahmen. Wavelet methods for PDEs-some recent development. Journal of computational and applied mathematics. 2001, 128: 133-185.
[22] A. Cohen, W. Dahmen and R. A. DeVore. Adaptive wavelet methods for elliptic operator equations comergence rates. Math. of Comp. 2001, 70: 27-75.
[23] Barinka, A. Barsch, T. Charton, etal. Adaptive wavelet schemes for elliptic problems implemention and numerical experiments. Siam J. Sci. Comput. 2001, 23(3): 910-939.
[24] D. Zhou. N. Chen and W. Cai. A fast wavelet collocation method for high-speed VLSI circuit simulation. Proc. ICCAD 95, San Jose, CA. 1995, 115-122.
[25] D. Zhou and W. Cai. A fast wavelet collocation method for higspeed circuit simulation. IEEE Trans. On Circuits And Systems-Ⅰ: Fundamental Theory And Applications. 1999, 46(8): 920-930.
[26] J. Wang. Cubic spline wavelet bases of Sobolev spaces and multilevel interpolation. Applied And Computational Harmonic Analysis. 1996, 23: 154-163.
[27] D. Zhou, W. Cai and W. Zhang. An adaptive wavelet method for nonlinear circuit simulation. IEEE Trans. On Circuits And Systems-Ⅰ: Fundamental Theory And Applications. 1999, 46(8): 931-938.
[28] W. Cai and J. Wang. Adaptive multiresolution collocation methods for initial boundary value problems of nonlinear PDEs. Siam Numer. Anal. 1996, 33(3): 937-970.
[29] R. Devore, B. Jauerth and V. Popov. Compression of wavelet decompositions. Amer. J. Math. 1992, 114: 737-785.
[30] A. Cohen, R. A. Devore and R. Hochmuth. Restricted nonlinear approximation. Siam J. Sci. Comput. 2000, 16: 85-113.
[31] S. Bertoluzza, M. Verani. Convergence of a nonlinear wavelet algorithm for the solution of PDEs. Applied Mathematics Letters. 2003, 16(1): 113-118.
[32] Mallat S. A theory of multiresolution signal decompositon: the wavelet representation. IEEE Trans. On Pattern And Machine Intell. 1989, 11: 674-693.
[33] Mallat S. Multiresolution representation and wavelets. PH. D. Thesis. Univ. of Pennsylvanis, Philadelphia, PA. 1988.
[34] Mallat S. Multiresolution approximation and wavelet orthonormal bases of L~2(R). Trans. Amer. Math. Soc. 1989, 315.
[35] Daubechies Ⅰ. Orthonormal bases of compactly supported wavelets. Comm. On Pure And Applied Math. 1988, 41.
[36] Daubechies Ⅰ. The wavelet transform: time-frequency localization and signal analysis. IEEE. Trans. Inform. Theory. 1990, 36.
[37] P. Levy. Le Mouvement Brownien. Mem. Sc. Math. Fasc. 1954, 126.
[38] Calderon. A. P. An atomic decompositon of distrubutions in parabolic H~P-spaces. Adv. In Math. 1977, 25.
[39] Calderon. A. P. Intermediate spaces and interpolation: the complex method. Studia Math. 1964, 24.
[40] GrossMann A. and Morlet J. Transforms associated to square integrable group representationsⅠ: general results. J. Mat. Phys. 26(10. 1985.
[41] GrossMann A. and Molet J, Decompositon of Hardy funcitons into square integrable wavelets of constant shape. SIAM. J. Math. Anal. 1984, 15.
[42] Meyer Y. Pseudodifferential operators. Proc. Symp. Pure. Math. 1985, 43.
[43] S. Jaffard. Wavelet methods for fast resolution of elliptic problems, SIAM J. Numer. Anal. 1992, 29(4).
[44] Chui K. 小波分析导论(程正兴译)．西安交通大学出版社．1992．
[45] 崔丽鸿程正兴．多小波与平衡多小波的理论和设计．工程数学学报，2001，18(5)：105-116．
[46] 李建平．小波分析与信号处理-理论、应用及软件实现．重庆出版社．
[2] 秦前清，杨宗凯．实用小波分析．西安电子科技大学出版社．
[3] Meyer Y. 小波与算子(Vol.1)．北京．世界图书出版社．1990．
[4] Daubechies I. Ten Lectures on Wavelets. CBMS-NSF Series On Applied Math. SIAM. 1992.
[5] V. Strela. Multiwavelet: Theory and applications. Ph D Thesis MIT. 1996.
[6] 程正兴，张玲玲．多小波分析与应用．工程数学学报，2001，18(1)：99-107．
[7] Chui C. K. and Lian J. A study on orthonormal multiwavelets [J]. J. Appl. Numer. Math. 1996: 20: 273-298.
[8] Plonka G. and Strela V. Construction of multiscaling functions with approximation and symmetry [J]. Siam J. Math. anal. 1998, 29:481-510.
[9] Hehong Zou, Tewfik Ahmed H. Discrete orthogoanl m-band wavelet decompositons [J]. IEEE Trans on Signal Processing. 1992, 4(4): 605-608.
[10] Peter Steffen. Theory of regular m-band wavelet bases [J]. IEEE Trans. on Signal Processing, 1993, 41(12): 3497-3511.
[11] Meyer Y. Ondelettes et functions splines. Semnaire EDP, Ecole Polytechnique. Paris. Dec. 1986.
[12] P. Steffen, P. Heller, and et. al. Theory of regular M-band wavelet bases. IEEE Trans. Signal Processing. 1993, 41: 3497-3511.
[13] T. N. T. Goodman, and S. L. Lee. Wavelet of Multiplicity r. Trans. Amer. Math. Soc. 1994, 342: 307-324.
[14] Asim Bhatti and Hǖseyin (?)zkaramanli. M-band multiwavelets from spline super functions with approximation order. IEEE ICASSP, Orlando, Florida. May 2002.
[15] H. Zou and A. H. Tewfik. Discrete orthogonal M-band wavelet decompositions. Proc. ICASSP. San Francisco, CA. 1992, 4: Ⅳ-605-Ⅳ-608.
[16] Geronimo J, Hardin D Massopust P R. Fractal functions and
wavelet expansions based on several functions [J]. J. Approx. Theory, 1994, 78: 373-401.
[17] Alpert B. A class of basis in L~2 for the sparse representation of intergral operators [J]. Siam J. Math. Naal, 1993, 24.
[18] DP Hardin, JA Marasovich. Biorthogonal multiwavelets on [-1, 1] [J]. Applied and computational harmonic analysis, 1999, 7(1).
[19] Asim Bhatti and Hǖseyin (?)zkaramanli. M-band multiwavelets from spline super functions with approximation order. IEEE ICASSP, Orlando, Florida. May 2002.
[20] W. Dahmen. Wavelet and multiscale methods for operator equations. Acta Numer. 2001, 6:55-228.
[21] W. Dahmen. Wavelet methods for PDEs-some recent development. Journal of computational and applied mathematics. 2001, 128: 133-185.
[22] A. Cohen, W. Dahmen and R. A. DeVore. Adaptive wavelet methods for elliptic operator equations comergence rates. Math. of Comp. 2001, 70: 27-75.
[23] Barinka, A. Barsch, T. Charton, etal. Adaptive wavelet schemes for elliptic problems implemention and numerical experiments. Siam J. Sci. Comput. 2001, 23(3): 910-939.
[24] D. Zhou. N. Chen and W. Cai. A fast wavelet collocation method for high-speed VLSI circuit simulation. Proc. ICCAD 95, San Jose, CA. 1995, 115-122.
[25] D. Zhou and W. Cai. A fast wavelet collocation method for higspeed circuit simulation. IEEE Trans. On Circuits And Systems-Ⅰ: Fundamental Theory And Applications. 1999, 46(8): 920-930.
[26] J. Wang. Cubic spline wavelet bases of Sobolev spaces and multilevel interpolation. Applied And Computational Harmonic Analysis. 1996, 23: 154-163.
[27] D. Zhou, W. Cai and W. Zhang. An adaptive wavelet method for nonlinear circuit simulation. IEEE Trans. On Circuits And Systems-Ⅰ: Fundamental Theory And Applications. 1999, 46(8): 931-938.
[28] W. Cai and J. Wang. Adaptive multiresolution collocation methods for initial boundary value problems of nonlinear PDEs. Siam Numer. Anal. 1996, 33(3): 937-970.
[29] R. Devore, B. Jauerth and V. Popov. Compression of wavelet decompositions. Amer. J. Math. 1992, 114: 737-785.
[30] A. Cohen, R. A. Devore and R. Hochmuth. Restricted nonlinear approximation. Siam J. Sci. Comput. 2000, 16: 85-113.
[31] S. Bertoluzza, M. Verani. Convergence of a nonlinear wavelet algorithm for the solution of PDEs. Applied Mathematics Letters. 2003, 16(1): 113-118.
[32] Mallat S. A theory of multiresolution signal decompositon: the wavelet representation. IEEE Trans. On Pattern And Machine Intell. 1989, 11: 674-693.
[33] Mallat S. Multiresolution representation and wavelets. PH. D. Thesis. Univ. of Pennsylvanis, Philadelphia, PA. 1988.
[34] Mallat S. Multiresolution approximation and wavelet orthonormal bases of L~2(R). Trans. Amer. Math. Soc. 1989, 315.
[35] Daubechies Ⅰ. Orthonormal bases of compactly supported wavelets. Comm. On Pure And Applied Math. 1988, 41.
[36] Daubechies Ⅰ. The wavelet transform: time-frequency localization and signal analysis. IEEE. Trans. Inform. Theory. 1990, 36.
[37] P. Levy. Le Mouvement Brownien. Mem. Sc. Math. Fasc. 1954, 126.
[38] Calderon. A. P. An atomic decompositon of distrubutions in parabolic H~P-spaces. Adv. In Math. 1977, 25.
[39] Calderon. A. P. Intermediate spaces and interpolation: the complex method. Studia Math. 1964, 24.
[40] GrossMann A. and Morlet J. Transforms associated to square integrable group representationsⅠ: general results. J. Mat. Phys. 26(10. 1985.
[41] GrossMann A. and Molet J, Decompositon of Hardy funcitons into square integrable wavelets of constant shape. SIAM. J. Math. Anal. 1984, 15.
[42] Meyer Y. Pseudodifferential operators. Proc. Symp. Pure. Math. 1985, 43.
[43] S. Jaffard. Wavelet methods for fast resolution of elliptic problems, SIAM J. Numer. Anal. 1992, 29(4).
[44] Chui K. 小波分析导论(程正兴译)．西安交通大学出版社．1992．
[45] 崔丽鸿程正兴．多小波与平衡多小波的理论和设计．工程数学学报，2001，18(5)：105-116．
[46] 李建平．小波分析与信号处理-理论、应用及软件实现．重庆出版社．