有关太阳无力磁场的快速算法应用研究
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摘要
半个多世纪以来,经过许多天体物理学家和应用数学家的共同努力,使得有关太阳无力磁场的研究一直长盛不衰、结出硕果无数,其数学和物理内容之丰富多彩委实令人惊叹。这其中最重要的一个问题就是通过数值方法求解太阳无力磁场方程的边值问题。我国学者颜毅华研究员独树一帜,推导出太阳无力磁场方程边值问题的边界积分公式,并用边界元方法进行数值计算,在国际天体物理学界赢得了广泛的认可,特别对解释日冕物理作出了杰出的贡献。本文紧密围绕颜毅华研究员开拓的太阳无力磁场边界积分方法开展工作,致力于全面系统地发展求解Yan方程的快速算法,以便进一步开拓Yan方程的应用空间,同时拉动相关快速算法的发展。本文完成如下工作:
1.将求解非对称线性方程组的现代迭代算法广义最小残量法引入太阳无力磁场边界元外推计算中,以便高效地求解其中的边界元方程组。以著名的Low无力场为基础,实施了高强度的数值实验,以便揭示广义最小残量法相对于太阳无力磁场边界元方程组的收敛性。特别地,详细考察了相关参量对收敛速度的影响,为在实际计算中确定相应的参数提供了依据。依赖于所取的求解器参数,当问题的阶数达到12321阶以上时,新程序的效率比原始程序至少提高1000倍;当精度要求比较松弛时,效率提高可达9000倍,能有力地支持边界元外推的日常使用。
2.对太阳常α无力磁场外推的Chiu-Hilton方法和Yan方法作了定量性能评估。这里的两个边界积分公式通过不同的机制使用横场信息:Chiu-Hilton方法以隐性方式(至少局部地)使用横场,Yan方法以显性方式使用全部横场。就所考虑的例子而言,在边界条件充足的情形下,两种方法均能以合理的精度在有效域内重构解析磁场;而以显性方式使用全部横场的Yan方法更具有优势,能在较小的视场观测条件下,使重构的磁场符合关键指标。由于最新研究表明存在能够表示为常α无力磁场的有力磁场,这里的结果对发展更高级的有力磁场外推时选择相应常α无力磁场求解器提供了依据。
3.提出计算太阳常α无力磁场的修正Nystrom外推方法,旨在提高生成矩阵以及磁场外推这两个阶段的计算效率。该算法避免了边界元方法中的插值基函数和局部坐标,并通过引入对偶网格来离散边界积分公式中的积分变量和参数变量,从而克服核函数的奇异性。对边界元方法和修正Nystrom方法中涉及的计算量作了算术估计。数值算例表明当边界数据的分辨率较细时,两种外推方法的精度相当,而修正Nystrom外推程序的总体效率比新的边界元外推程序提高了大致7倍。
4.将快速小波变换应用于求解太阳无力磁场边界元方程组。首先,为了使得快速小波变换适应非二进阶的系数矩阵,提出二进分块快速小波变换,有效地扩大了快速小波变换在物理问题中的应用范围。其次,为了求解大规模情形下的太阳无力磁场边界元方程组,提出紧压缩算法,其关键想法是将稠密矩阵逐块地生成、变换、压缩。解决了标准算法占用大量额外内存的缺陷,同时具有标准算法的简单性和通用性。这样,即使当系数矩阵的规模超出内存以及虚拟内存若干倍时,紧压缩算法仍然可以高效地运转,只要稠密系数矩阵具有足够的小波可压缩性。再次,给出了稀疏解在一定条件下的小波域误差估计公式,为选择适当的阈值提供了线索。第四,针对问题涉及到很多参量,而阈值稀疏化是非线性算子,提出OTDT分析作为对整个问题进行系统全面数值实验的通用框架,为确定优化参量提供参照依据。最后,将OTDT分析和紧压缩算法成功地用于求解太阳无力磁场边界元方程组。
After developing for more than a half century, the researches on the solar force-free magnetic fields have kept being active and productive due to the constant efforts of astrophysicists and applied mathematicians, resulting in an astonishing and abundant outcome of mathematics and physics. Amongst these, one of the most important problem is to solve the boundary value problem for the solar force-free magnetic field equation. Since the 1990's, Yan and colleagues have formulated a kind of boundary integral representations for the linear or non-linear solar force-free magnetic field with finite energy content in the semi-infinite space above the Sun, and applied a well-established numerical method - boundary element method (BEM) - to compute the magnetic field above the active regions on the Sun, which has become well-known in the international astrophysics community, due to its outstanding contribution on interpreting corona physics. The present work devotes to developing fast algorithms for solving Yan's equation, in order to exploit new application domain for Yan's methods, which may in turn promote the development of relevant fast algorithms. The main contributions are listed as follows:
1. The generalized minimal residual method (GMRES) is introduced into the BEM extrapolation of the solar force-free magnetic fields, in order to efficiently solve the associated BEM system of linear equations, which was previously solved by the Gauss elimination method with full pivoting. Being a modern iterative method for non-symmetric linear systems, the GMRES method reduces the computational cost for the BEM system from O(N~3) to O(N~2), where N is the number of unknowns in the linear system. Intensive numerical experiments are conducted on a well-known analytical model of the force-free magnetic field to reveal the convergence behaviour of the GMRES method subjected to the BEM systems. The impacts of the relevant parameters on the convergence speed are investigated in detail. Taking the Krylov dimension to be 50 and the relative residual bound to be 10~(-6) (or 10~(-2)), the GMRES method is at least 1000 (or 9000) times faster than the full pivoting Gauss elimination method used in the original BEM extrapolation code, when N is greater than 12321, according to the CPU timing information measured on a common desktop computer for the model problem. This achievement may promote the routine use of the BEM extrapolation.
2. The performances of two newly implemented codes for extrapolating the solar linear force-free magnetic fields, originally proposed by Chiu and Hilton, and Yan, respectively, are evaluated by measuring their quantified responses to the lower boundary vector field on a finite region due to analytical models. The codes are based on two boundary integral formulas with different mechanisms in exploiting the transverse boundary field: one exploits, implicitly, at least a local portion of the transverse boundary field, while the other exploits explicitly the whole transverse boundary field, in addition to the vertical field component. Relative to the present test cases: both of the codes could re-produce the analytical model fields with reasonable accuracy within the valid domain, provided sufficient lower boundary data being available; the code exploiting explicitly all three components of the boundary field due to Yan has the merit of requiring relatively smaller field of view in order to achieve reasonable accuracy in crucial metric quantities.
3. The modified Nystr(o|¨)m method is proposed as an efficient alternative algorithm to the boundary element method for extrapolating the constant a force-free magnetic field, based on a boundary integral formulation established by Yan and colleagues. The boundary integral formulation can be directly discretized by a particular scheme of the modified Nystr(o|¨)m method designed for the 3D field extrapolation, without involvement of any interpolating base functions or local coordinate transforms, while the diagonal singularity of the kernel function is overcome by introducing two distinct grids for discritizing the integrand variables and the parameter variables in the boundary integral formulation. The amount of computations involved in the boundary element method and the modified Nystr(o|¨)m method is arithmetically estimated and compared. Numerical experiments show that the accuracy of the two extrapolation methods is comparable to each other when the resolution of the boundary data is fine, while the efficiency of the modified Nystr(o|¨)m extrapolation code is about seven times higher than the rebuilt boundary element extrapolation code.
4. The fast wavelet transform is applied to solve the BEM system due to Yan's integral formula. First of all, the fast wavelet transform based on the binary partition techniques are proposed, in order to apply an 'almost' fast wavelet transform to matrices with non-dyadic sizes, which have effectively enlarged its problem domain in physical applications. Second, we propose the so-called 'compact compression algorithm' to deal with the BEM system due to Yan's integral formula. The key idea is to generate, transform, and compress the dense coefficient matrix in a block-wise manner. This strategy reserves the merit of simpleness and generality of the 'standard algorithm', while remedies the limitation of occupying a great deal of additional memory inherited by the 'standard algorithm'. Therefore, provided the dense coefficient matrix possesses a high degree of wavelet compressibility, the compact compression algorithm will remain efficient even if the coefficient matrix becomes too large to be accommodated in the memory. Third, we propose an error estimation formula for the FWT-based solvers of a class of dense linear equations, especially those arising from the numerical resolution of some kind of integral equations, which provides a clue to determining the value of the threshold such that the corresponding approximate solution has a desired accuracy. Forth, since many parameters are involved in the problem, with the threshold operation being non-linear, we propose the 'OTDT analysis' as a general framework for conducting systematic numerical experiments for the problem under consideration, which may provide empirical bases for determining the various parameters in a nearly optimal manner. Finally, we have applied the OTDT analysis and the compact compression algorithm to solve the BEM system due to Yan's integral formula.
1.将求解非对称线性方程组的现代迭代算法广义最小残量法引入太阳无力磁场边界元外推计算中,以便高效地求解其中的边界元方程组。以著名的Low无力场为基础,实施了高强度的数值实验,以便揭示广义最小残量法相对于太阳无力磁场边界元方程组的收敛性。特别地,详细考察了相关参量对收敛速度的影响,为在实际计算中确定相应的参数提供了依据。依赖于所取的求解器参数,当问题的阶数达到12321阶以上时,新程序的效率比原始程序至少提高1000倍;当精度要求比较松弛时,效率提高可达9000倍,能有力地支持边界元外推的日常使用。
2.对太阳常α无力磁场外推的Chiu-Hilton方法和Yan方法作了定量性能评估。这里的两个边界积分公式通过不同的机制使用横场信息:Chiu-Hilton方法以隐性方式(至少局部地)使用横场,Yan方法以显性方式使用全部横场。就所考虑的例子而言,在边界条件充足的情形下,两种方法均能以合理的精度在有效域内重构解析磁场;而以显性方式使用全部横场的Yan方法更具有优势,能在较小的视场观测条件下,使重构的磁场符合关键指标。由于最新研究表明存在能够表示为常α无力磁场的有力磁场,这里的结果对发展更高级的有力磁场外推时选择相应常α无力磁场求解器提供了依据。
3.提出计算太阳常α无力磁场的修正Nystrom外推方法,旨在提高生成矩阵以及磁场外推这两个阶段的计算效率。该算法避免了边界元方法中的插值基函数和局部坐标,并通过引入对偶网格来离散边界积分公式中的积分变量和参数变量,从而克服核函数的奇异性。对边界元方法和修正Nystrom方法中涉及的计算量作了算术估计。数值算例表明当边界数据的分辨率较细时,两种外推方法的精度相当,而修正Nystrom外推程序的总体效率比新的边界元外推程序提高了大致7倍。
4.将快速小波变换应用于求解太阳无力磁场边界元方程组。首先,为了使得快速小波变换适应非二进阶的系数矩阵,提出二进分块快速小波变换,有效地扩大了快速小波变换在物理问题中的应用范围。其次,为了求解大规模情形下的太阳无力磁场边界元方程组,提出紧压缩算法,其关键想法是将稠密矩阵逐块地生成、变换、压缩。解决了标准算法占用大量额外内存的缺陷,同时具有标准算法的简单性和通用性。这样,即使当系数矩阵的规模超出内存以及虚拟内存若干倍时,紧压缩算法仍然可以高效地运转,只要稠密系数矩阵具有足够的小波可压缩性。再次,给出了稀疏解在一定条件下的小波域误差估计公式,为选择适当的阈值提供了线索。第四,针对问题涉及到很多参量,而阈值稀疏化是非线性算子,提出OTDT分析作为对整个问题进行系统全面数值实验的通用框架,为确定优化参量提供参照依据。最后,将OTDT分析和紧压缩算法成功地用于求解太阳无力磁场边界元方程组。
After developing for more than a half century, the researches on the solar force-free magnetic fields have kept being active and productive due to the constant efforts of astrophysicists and applied mathematicians, resulting in an astonishing and abundant outcome of mathematics and physics. Amongst these, one of the most important problem is to solve the boundary value problem for the solar force-free magnetic field equation. Since the 1990's, Yan and colleagues have formulated a kind of boundary integral representations for the linear or non-linear solar force-free magnetic field with finite energy content in the semi-infinite space above the Sun, and applied a well-established numerical method - boundary element method (BEM) - to compute the magnetic field above the active regions on the Sun, which has become well-known in the international astrophysics community, due to its outstanding contribution on interpreting corona physics. The present work devotes to developing fast algorithms for solving Yan's equation, in order to exploit new application domain for Yan's methods, which may in turn promote the development of relevant fast algorithms. The main contributions are listed as follows:
1. The generalized minimal residual method (GMRES) is introduced into the BEM extrapolation of the solar force-free magnetic fields, in order to efficiently solve the associated BEM system of linear equations, which was previously solved by the Gauss elimination method with full pivoting. Being a modern iterative method for non-symmetric linear systems, the GMRES method reduces the computational cost for the BEM system from O(N~3) to O(N~2), where N is the number of unknowns in the linear system. Intensive numerical experiments are conducted on a well-known analytical model of the force-free magnetic field to reveal the convergence behaviour of the GMRES method subjected to the BEM systems. The impacts of the relevant parameters on the convergence speed are investigated in detail. Taking the Krylov dimension to be 50 and the relative residual bound to be 10~(-6) (or 10~(-2)), the GMRES method is at least 1000 (or 9000) times faster than the full pivoting Gauss elimination method used in the original BEM extrapolation code, when N is greater than 12321, according to the CPU timing information measured on a common desktop computer for the model problem. This achievement may promote the routine use of the BEM extrapolation.
2. The performances of two newly implemented codes for extrapolating the solar linear force-free magnetic fields, originally proposed by Chiu and Hilton, and Yan, respectively, are evaluated by measuring their quantified responses to the lower boundary vector field on a finite region due to analytical models. The codes are based on two boundary integral formulas with different mechanisms in exploiting the transverse boundary field: one exploits, implicitly, at least a local portion of the transverse boundary field, while the other exploits explicitly the whole transverse boundary field, in addition to the vertical field component. Relative to the present test cases: both of the codes could re-produce the analytical model fields with reasonable accuracy within the valid domain, provided sufficient lower boundary data being available; the code exploiting explicitly all three components of the boundary field due to Yan has the merit of requiring relatively smaller field of view in order to achieve reasonable accuracy in crucial metric quantities.
3. The modified Nystr(o|¨)m method is proposed as an efficient alternative algorithm to the boundary element method for extrapolating the constant a force-free magnetic field, based on a boundary integral formulation established by Yan and colleagues. The boundary integral formulation can be directly discretized by a particular scheme of the modified Nystr(o|¨)m method designed for the 3D field extrapolation, without involvement of any interpolating base functions or local coordinate transforms, while the diagonal singularity of the kernel function is overcome by introducing two distinct grids for discritizing the integrand variables and the parameter variables in the boundary integral formulation. The amount of computations involved in the boundary element method and the modified Nystr(o|¨)m method is arithmetically estimated and compared. Numerical experiments show that the accuracy of the two extrapolation methods is comparable to each other when the resolution of the boundary data is fine, while the efficiency of the modified Nystr(o|¨)m extrapolation code is about seven times higher than the rebuilt boundary element extrapolation code.
4. The fast wavelet transform is applied to solve the BEM system due to Yan's integral formula. First of all, the fast wavelet transform based on the binary partition techniques are proposed, in order to apply an 'almost' fast wavelet transform to matrices with non-dyadic sizes, which have effectively enlarged its problem domain in physical applications. Second, we propose the so-called 'compact compression algorithm' to deal with the BEM system due to Yan's integral formula. The key idea is to generate, transform, and compress the dense coefficient matrix in a block-wise manner. This strategy reserves the merit of simpleness and generality of the 'standard algorithm', while remedies the limitation of occupying a great deal of additional memory inherited by the 'standard algorithm'. Therefore, provided the dense coefficient matrix possesses a high degree of wavelet compressibility, the compact compression algorithm will remain efficient even if the coefficient matrix becomes too large to be accommodated in the memory. Third, we propose an error estimation formula for the FWT-based solvers of a class of dense linear equations, especially those arising from the numerical resolution of some kind of integral equations, which provides a clue to determining the value of the threshold such that the corresponding approximate solution has a desired accuracy. Forth, since many parameters are involved in the problem, with the threshold operation being non-linear, we propose the 'OTDT analysis' as a general framework for conducting systematic numerical experiments for the problem under consideration, which may provide empirical bases for determining the various parameters in a nearly optimal manner. Finally, we have applied the OTDT analysis and the compact compression algorithm to solve the BEM system due to Yan's integral formula.
引文
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