一类用于代数Riccati方程的Petrov-Galerkin Krylov方法
A class of Petrov-Galerkin Krylov methods for algebraic Riccati equations
AI总结：
针对大规模连续时间代数Riccati方程，提出基于块有理Krylov子空间的Petrov-Galerkin投影方法，可高效构建低秩近似解、计算残差范数并通过特征值截断进一步降秩，给出了数值算例。
摘要：
本文提出了一类基于（块）有理Krylov子空间的投影方法，用于求解大规模连续时间代数Riccati方程（CARE）$0 = \mathcal{R}(X) := A^HX + XA + C^HC - XBB^HX$，其中$A$为大型稀疏矩阵，$B$和$C$为满低秩矩阵。将CARE投影到块有理Krylov子空间$\mathcal{K}_j$上，该子空间由形如$(A^H - s_kI)^{-1}C^H$的块张成，其中$s_k, k = 1, \ldots, j$为位移参数。所考虑的投影无需正交，由与$\mathcal{K}_j$相关的块有理Arnoldi分解中出现的矩阵构建。对得到的投影Riccati方程求解小型方阵厄米特解$Y_j$，随后构造$X$的厄米特低秩近似$X_j = Z_jY_jZ_j^H$，其中$Z_j$的列张成$\mathcal{K}_j$。残差范数$\\\\|R(X_j )\\\\|_F$可通过一个易获取的$2p \times 2p$矩阵的范数高效计算。我们建议通过截去$X_j$的小特征值进一步降低近似解$X_j$的秩，该截断近似解可解释为投影到$\mathcal{K}_j$的一个子空间上的Riccati残差的解，这为我们提供了一种高效计算所得残差范数的方法。文中给出了数值算例。
英文摘要：
A class of (block) rational Krylov subspace based projection method for solving large-scale continuous-time algebraic Riccati equation (CARE) $0 = \mathcal{R}(X) := A^HX + XA + C^HC - XBB^HX$ with a large, sparse $A$ and $B$ and $C$ of full low rank is proposed. The CARE is projected onto a block rational Krylov subspace $\mathcal{K}_j$ spanned by blocks of the form $(A^H - s_kI)^{-1}C^H$ for some shifts $s_k, k = 1, \ldots, j.$ The considered projections do not need to be orthogonal and are built from the matrices appearing in the block rational Arnoldi decomposition associated to $\mathcal{K}_j.$ The resulting projected Riccati equation is solved for the small square Hermitian $Y_j.$ Then the Hermitian low-rank approximation $X_j = Z_jY_jZ_j^H$ to $X$ is set up where the columns of $Z_j$ span $\mathcal{K}_j.$ The residual norm $\|R(X_j )\|_F$ can be computed efficiently via the norm of a readily available $2p \times 2p$ matrix. We suggest to reduce the rank of the approximate solution $X_j$ even further by truncating small eigenvalues from $X_j.$ This truncated approximate solution can be interpreted as the solution of the Riccati residual projected to a subspace of $\mathcal{K}_j.$ This gives us a way to efficiently evaluate the norm of the resulting residual. Numerical examples are presented.
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