有效條件數於無網格法輻射基底函數及基本解之應用

Abstract

本論文幾乎完全關注有效條件數在 RBF 和 MFS 中的應用，以幫助科學家和工程師成功應用這些無網格方法。 自由和問題相關的參數，包括 RBF 的類型、RBF 無網格方法中的形狀參數以及 MFS 中源（中心）點的位置仍然是懸而未決的研究課題。在本論文中，我們建議使用不確定性原理 (UP) 來克服所有挑戰。 [14]中的不確定性（權衡）原則指出“不可能構造同時保證良好穩定性和小誤差的徑向基函數”。這意味著在準確性和穩定性之間存在權衡。穩定性可以通過線性系統的調節來測量。因此，產生病態線性系統的形狀參數值預計會產生小誤差，反之亦然。然而，標準（常規）條件數並不是對線性系統實際條件的良好估計。另一方面，有效條件數被認為是對實際條件數的更好估計。基於我們在 RBF 無網格方法和基本解方法中的數值觀察，觀察到有效條件數與誤差之間存在很強的數值相關性。因此，在不確定性原則的推動下，我們建議根據有效條件數搜索最不穩定的設置，以實現 UP 承諾的高度準確的結果。這些設置可以通過更好的 RBF 類型、良好的形狀參數（在 RBF 中）和源點的適當位置（在 MFS 中）來設置。此外，引入了三種方法來提高所提出方法的效率。 最後，將結果與著名的留一法交叉驗證 (LOOCV) 方法進行比較。 在大多數情況下觀察到更好的準確性，同時在所有情況下都獲得了更高的效率。

In recent decades radial basis functions (RBFs) have been proven to be valuable tools in scientific computing areas like (i) Computational mechanics problems, including stress analysis and elasticity, (ii) Fluid dynamics Problems, including reaction-diffusion, shallow water equations, and convection advection problems, (iii) Image analysis, computer graphics, animated deformations and including shape modeling, and (iv) Economics problems, including pricing. These applications are connected through common mathematical problems and concepts like (i) Recovery of functions data, (ii) Meshfree or meshless methods for solving partial differential equations (PDEs), (iii) Inverse and ill–posed problems, (iv) Learning algorithms, and (v) Neural networks, which can be handled easily and successfully by radial basis functions. We focus on meshless methods for (i) Reconstruction of multivariate functions, and (ii) Solving partial differential equations. To do so, a few core techniques, including the Kansa method and method of fundamental solutions are needed. This dissertation focuses almost entirely on applications of the Effective Condition Number in RBF and MFS} to help scientists and engineers apply these meshless methods successfully. Free and problem-dependent parameters including the type of RBF, a shape parameter in RBF meshless methods, and the location of source (center) points in MFS still remain outstanding research topics. In this dissertation, we propose to overcome all challenges using the Uncertainty Principle (UP). The Uncertainty (Trade-off) Principle in radial basis functions was first proposed by Schaback in 1995 [14] and states that ``It is impossible to construct radial basis functions which guarantee good stability and small errors at the same time". This means that there is a trade-off between accuracy and stability. Stability can be measured by the conditioning of the linear system. As such, a shape parameter value yielding an ill-conditioned linear system is expected to produce a small error and vice versa. However, the standard (conventional) condition number is not a good estimate of the real conditioning of a linear system. On the other hand, the Effective Condition Number is known as a better estimate of the real condition number. Based on our numerical observations in RBF meshless methods and method of fundamental solutions a strong numerical correlation between the Effective Condition Number and error is observed. Therefore, motivated by the Uncertainty Principle, we propose to search for the most unstable settings in terms of the Effective Condition Number to achieve the highly accurate results promised by the UP. These settings can be set up by a better type of RBF, a good shape parameter (in RBF), and proper locations of source points (in MFS). In addition, three methods are introduced to increase the efficiency of the proposed (i) The approximation of the minimal singular value, and (ii) An efficient search algorithm. Note that Different strategies are applied to different methods. In the end, the results are compared with the well-known leave-one-out cross-validation (LOOCV) method. Better accuracy was observed for most cases while higher efficiency was obtained for all cases.