Optimization of model reduction for linear ordinary differential equations.

摘要

A reduced model is a low dimensional approximation to a high dimensional system. We study model reduction via linear projection for large systems of linear ordinary differential equations arising from quadratic Hamiltonians. We show that several techniques, including Krylov subspace methods, a version of first order optimal prediction, and proper orthogonal decomposition, are expressible within a common projection framework: the projection is defined by the choice of subspace upon which we project. We study the problem of finding the subspace which minimizes the time averaged squared error of the resulting approximation. We prove that for short times the optimal subspace is a Krylov subspace generated from the initial conditions, and that for long times the optimal subspace converges to that eigenspace of the linear operator which is closest to the initial conditions. The rate of convergence depends only on the operator. The subspace defined by proper orthogonal decomposition converges to this same space, but the rate of convergence depends on the initial conditions as well as the operator.;We decompose the error of a reduction into two terms. One term measures the average of the distance between the exact solution and the subspace upon which we project. The other measures how close to invariance is the subspace with respect to the dynamics induced by the Hamiltonian. This decomposition helps to explain the long time behavior of the optimal subspace, and contrast the optimality properties of proper orthogonal decomposition and first order optimal prediction.