论文标题
哈密顿扩展的Krylov子空间法
The Hamiltonian Extended Krylov Subspace Method
论文作者
论文摘要
用于构建扩展Krylov子空间$ \ Mathcal {k} _ {r，s} = \ operatorName {range} \ {u，hu，hu，hu，hu，h^2u，$ h^2u，$ h^ldots，$ h^ldots，$ h^^u，$ h^^u，hu， \ ldots，h^{ - 2s} u \}，$其中$ h \ in \ mathbb {r}^{2n \ times 2n} $是衍生的大（且稀疏的）汉密尔顿矩阵（用于$ r = s+1 $或$ r = s $）。令人惊讶的是，这允许最多涉及以前生成的基础向量的短期复发。将$ h $投影到子空间$ \ Mathcal {k} _ {r，s} $产生一个小的汉密尔顿矩阵。可以使用所得的HEK算法来大约$ f（h）u $，其中$ f $是将hamiltonian矩阵$ h $映射到的函数，例如，a（skew-）hamiltonian或simplectic矩阵。数值实验表明，与使用其他（结构提供）Krylov子空间方法相比，与HEK算法相比，将$ f（h）u $与HEK算法相比具有竞争力。
An algorithm for constructing a $J$-orthogonal basis of the extended Krylov subspace $\mathcal{K}_{r,s}=\operatorname{range}\{u,Hu, H^2u,$ $ \ldots, $ $H^{2r-1}u, H^{-1}u, H^{-2}u, \ldots, H^{-2s}u\},$ where $H \in \mathbb{R}^{2n \times 2n}$ is a large (and sparse) Hamiltonian matrix is derived (for $r = s+1$ or $r=s$). Surprisingly, this allows for short recurrences involving at most five previously generated basis vectors. Projecting $H$ onto the subspace $\mathcal{K}_{r,s}$ yields a small Hamiltonian matrix. The resulting HEKS algorithm may be used in order to approximate $f(H)u$ where $f$ is a function which maps the Hamiltonian matrix $H$ to, e.g., a (skew-)Hamiltonian or symplectic matrix. Numerical experiments illustrate that approximating $f(H)u$ with the HEKS algorithm is competitive for some functions compared to the use of other (structure-preserving) Krylov subspace methods.
