Research On Fast Algorithms And Their Application To Solar Force-free Magnetic Field

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(N³) to O(N²), 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.