针对具有二次矩阵束的线性系统族的模型降阶的单一LU分解方法
A single LU~factorization approach to model reduction for linear system families with a quadratic matrix pencil
AI总结：
针对系统矩阵依赖于族参数二次项的线性系统族，本文提出ODB正交导数基方法，通过一次LU分解结合解的导数递推，实现高效求解并大幅降低计算成本。
摘要：
为求解系统矩阵依赖于族参数二次项的大型线性系统族，本文提出一种新的求解方法。该方法利用解关于参数的导数，研究发现，在单个经特殊选取的参数值处计算得到的、阶数递增的累积解导数，可为整个线性系统族的解子空间提供逐次更优的近似。本文证明，这些导数可通过递推关系计算，每一步仅需求解同一矩阵的线性系统。为导数张成的子空间构造正交基，进而得到一种高效求解器，仅需一次（稀疏）LU分解（或迭代方案中的一次预条件子设置）即可求解整个线性系统族。本文将该新方法命名为ODB（正交导数基方法）。数值测试表明，与传统求解方法相比，ODB方法可大幅降低计算成本。
英文摘要：
To solve a family of large linear systems where the system matrix depends on the family parameter quadratically, we propose a new solution approach. Our method employs the derivatives of solutions with respect to the parameter. It turns out that the accumulated solution derivatives of increasing order, evaluated at a single specially chosen parameter value, provide successively better approximations to solution subspaces for the whole linear system family. We show that the derivatives can be computed by a recurrence, involving, at each step, solving a linear system with the same matrix. Constructing an orthonormal basis for the subspace spanned by the derivatives then leads to an efficient solver where just a single (sparse) LU factorization (or a single preconditioner setup in an iterative scheme) suffices to solve the whole linear system family. We call the new method ODB (orthonormal derivative basis method). As shown in numerical tests, compared to conventional solution methods, the ODB method allows to drastically reduce computational costs.
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