模型降阶中投影方法的误差公式的推广与应用

模型降阶有很多重要的应用背景，例如天气预报、分子动力学和国际空间站状态控制等。有理Krylov子空间投影法是模型降阶的主流方法之一。它的数学基础矩匹配定理保证:降阶系统的传递函数是原系统传递函数的有理插值多项式。这是个带有特殊结构的有理插值问题，之前没有插值余项公式。.申请人刚刚得到了该降阶方法的显式误差公式，它是这个有理插值问题的余项公式。原有的理论和算法都是基于矩匹配定理得到，现在理论分析工具有了明显进步，显见一些以往比较难处理的问题会被解决。新的误差公式在误差估计、算法的收敛性分析和收敛速度分析上有明显优势。项目拟定的研究目标主要集中在这些方面，具体为：.1.把新的误差公式推广到更复杂的多输入多输出系统的模型降阶上。2.改进H2范数最优模型降阶的IRKA算法，设计出二次收敛的算法。3.应用误差公式对前人计算系统H无穷范数的算法进行收敛速度分析，以及设计更好的算法。

Model order reduction has many applications, such as weather prediction, molecular dynamics and the state controller of the international space station. Rational Krylov subspace projection methods are one of the mainstream methods. The methods are based on the moments matching theorem. The theorem implies the transfer function of the reduced system is a rational interpolating polynomial of the one of the origin system. This is a rational polynomial interpolation problem, which has special structures. But the interpolation remainder was unknown. .Recently, we obtain the explicit expression of the approximation error. The new error formula is actually the remainder of Hermitian interpolation formula. The previous theories and algorithms are based on the moments matching theorem. Now, we acquire a new theoretical analysis tool. Thus, some of thorny problems will be solved. The new error formula is quite suitable for error bounds estimation, convergence analysis and convergence speed analysis..Based on the new result, we intend to do the following researches: 1. Generalize the new error formula onto the approximation of the multi-input-multi-output system. 2. Find faster algorithms than algorithm IRKA, which computes H2-optimal model order reduction. 3. Estimate the convergence rate of the rational Krylov subspace method, which computes the H infinity norm of the dynamical system. Design more efficient algorithms about it.