最优控制问题谱方法及hp自适应谱元方法研究

The modelling and numerical simulation of various systems plays an important role in real-life applications and engineering design. Very often, mathematical models of complex systems result in partial differential equations. In most applications, our ultimate goal is not only the mathematical modelling and numerical simulation of the systems, but rather the optimization or optimal control of the considered process, such as the oil exploration, cooling of the products and pollutant diffusion. However, it is very difficult to compute the control problems analytically. Therefore, the efficient numerical methods are essential to successful applications of any optimal control problem...As we know, the finite element method is undoubtedly the most widely used numerical method in computing optimal control problems. Furthermore, the adaptive finite element method has been extensively investigated in last decade. Adaptive finite element approximation is among the most important means to boost the accuracy and efficiency of finite element discretizations. It ensures a higher density of nodes in a certain area of the given domain, where the solution is more difficult to approximate. However, the spectral method can enjoy great superiority and can be more efficient when the solutions of optimal control problems are smooth sufficiently, which can be met frequently in many applications. ..In this project, we will propose the spectral method and hp-adaptive spectral element discretization for various optimal control problems to analysis whether the hp-adaptive spectral element method can be more efficient than the adaptive finite element method for computing some optimal control problems. Spectral method, in the context of numerical schemes, was introduced and popularized by Orszag’s pioneer work. It enjoys a great superiority of fast convergence rate with a relatively small number of unknowns when the solutions are smooth, which is vital to successful approximation of optimal control problems. Spectral element method represents a special case of Galerkin methods in which the finite dimensional space of trial/test functions is made of continuous piecewise algebraic polynomials of high degree on each element of a partition of the computational domain. However, it is rational in applications of optimal control problems to demand the use of different scales in the model description, in particular higher grid resolution is necessary in the vicinity of the area where the solutions are not smooth. For these problems a uniform mesh could be unappropriate, especially when high order methods are used. These inspire us to develop hp-adaptive spectral element method for solve the optimal control problems...We shall do some research works to derive the optimality conditions briefly, design efficient and fast algorithms for discrete systems, and construct sharp a posteriori error estimators. The efficiency and reliability of the indicators will be proved, which confirm it can indicate the approximation error successfully. Thus it is hopeful to improve the numerical precision and decrease the computation amount by automatic distribution of the elements and polynomial degrees with the indication of a posteriori error estimators. To realize the wide applications of hp-adaptive spectral element method in approximating optimal control problems in production process, we will combine the rectangular partition and triangular partition for the irregular domain. A large amount of numerical experiments will be carried out, with particular attention to testing the efficiency of the proposed methods and the influence of various indicators developed in the hp-adaptive spectral element method.

偏微分方程最优控制问题已被广泛应用于石油合理开采、产品有序冷却及污染物扩散有效控制等应用背景。通常很难求得最优控制问题精确解，从而高效数值算法对最优控制成功应用至关重要。尽管有限元及自适应计算可成功应用于最优控制问题数值计算，但对于解充分光滑的最优控制问题来说谱方法有着独特的优势，谱元法结合了有限元灵活的网格剖分和谱方法的高精度，已成为求解偏微分方程重要数值方法之一。本项目旨在研究偏微分方程最优控制问题的谱方法分析与hp自适应谱元逼近，推导出相应的最优性条件，设计离散系统的快速求解算法，构建便于计算且可靠有效的后验误差估计子，确保这些估计子能很好指示误差，从而达到指示网格的剖分及网格上多项式次数的分布情况，使计算量尽可能少、逼近精度尽可能高；将对非规则区域上控制问题采用矩形剖分与三角形剖分相结合的策略以使得hp自适应谱方法能广泛应用于最优控制问题的求解，大量数值实验也将验证相应的理论结果。

偏微分方程最优控制问题在物理学、空气动力学、大气科学、化工产业和制造业等科学和工程问题上都有广泛应用，如低温超导激光能量爆破的控制、航空工业产品最优形状设计、油田二次注水优化开采和反应釜温度控制等，高效的数值方法对这些问题的理论研究和实际应用具有重要意义。本项目率先研究了偏微分方程最优控制问题谱方法和hp自适应谱元方法求解的理论分析和算法设计。.研究内容有：（1）首次研究了H^1范数状态受限椭圆和Stokes方程最优控制问题，以及控制状态双受限椭圆最优控制问题的谱方法，证明了离散解的先验误差估计和后验误差估计。（2）深入研究了控制受限非线性椭圆最优控制问题、逐点控制受限偏微分方程最优控制问题、积分状态受限椭圆和双调和方程最优控制问题、L^2范数状态受限椭圆最优控制问题、以及控制状态双受限椭圆最优控制问题的hp谱元方法，利用辅助方程、对偶技术、方程分解和插值证明了离散解的先验误差估计和后验误差估计。（3）研究了几类分数阶偏微分方程最优控制问题的谱方法和有限元方法，设计了快速求解算法；研究了具有随机场系数偏微分方程最优控制问题的谱方法和无网格方法，避免了对高维随机空间的网格剖分。此外，我们还研究了积分控制受限椭圆最优控制问题和逐点控制受限Neumann边界最优控制问题的有限元离散，证明了后验误差估计子的可靠性和有效性，设计了有限元自适应算法，分析了算法的收敛性及计算复杂度；研究了带多尺度系数偏微分方程最优控制问题多重调和样条的多尺度有限元方法，对比标准有限元方法，我们的计算量大大减少。.以上理论成果都通过了大量数值实验加以验证，证实了所构造方法的有效性，展示了所构建后验误差估计子能有效指示对解有奇异的地方进行网格加密，对解光滑的地方提高多项式次数，做到了以少量计算获得高阶逼近精度，推动了谱方法的自适应算法广泛应用于一般区域和解不光滑情形最优控制问题的求解。