具有分数维Laplacian约束的最优控制问题的张量积快速解法
Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints
摘要：
本文介绍了在大的$n^{\otimes d}$张量积笛卡尔网格上离散化的约束条件下，用张量数值方法求解具有分数Laplacian型算子的$d$维最优控制问题.该方法是基于相应的分数差分拉普拉斯矩阵值函数的秩结构逼近。我们求解控制函数的方程，其中系统矩阵包括分数$d$维拉普拉斯算子及其逆算子的和。利用快速傅里叶变换（FFT）对张量网格上离散拉普拉斯算子的矩阵值函数进行对角化。然后，通过折叠相应的大对角矩阵的特征值得到的$d$维张量的低秩近似计算，这允许解决控制方程的张量结构的格式。通过对生成函数的拉普拉斯变换应用sinc求积逼近方法，证明了这类矩阵值函数的低秩正则逼近的存在性.控制函数的线性方程组由PCG迭代法求解，在每个迭代步骤中进行秩截断，其中预先计算低Kronecker秩预条件子。右手边、解向量和控制系统矩阵保持在秩结构张量格式中，这有益地降低了$O(n\log n)$的数值成本，优于3D情况下复杂度$O(n^3\log n)$的基于标准FFT的方法。二维和三维控制问题的数值试验证实了该方法在单变量网格尺寸$n$中的线性复杂度缩放。
英文摘要：
We introduce the tensor numerical method for solution of the $d$-dimensional optimal control problems with fractional Laplacian type operators in constraints discretized on large $n^{\otimes d}$ tensor-product Cartesian grids. The approach is based on the rank-structured approximation of the matrix valued functions of the corresponding fractional finite difference Laplacian. We solve the equation for the control function, where the system matrix includes the sum of the fractional $d$-dimensional Laplacian and its inverse. The matrix valued functions of discrete Laplace operator on a tensor grid are diagonalized by using the fast Fourier transform (FFT). Then the low rank approximation of the $d$-dimensional tensors obtained by folding of the corresponding large diagonal matrices of eigenvalues are computed, which allows to solve the governing equation for the control function in a tensor-structured format. The existence of low rank canonical approximation to the class of matrix valued functions involved is justified by using the sinc quadrature approximation method applied to the Laplace transform of the generating function. The linear system of equations for the control function is solved by the PCG iterative method with the rank truncation at each iteration step, where the low Kronecker rank preconditioner is precomputed. The right-hand side, the solution vector, and the governing system matrix are maintained in the rank-structured tensor format which beneficially reduces the numerical cost to $O(n\log n)$, outperforming the standard FFT based methods of complexity $O(n^3\log n)$ for 3D case. Numerical tests for the 2D and 3D control problems confirm the linear complexity scaling of the method in the univariate grid size $n$.
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