engleski

High order approximations of the operator Lyapunov equation have low rank (talk)

The feature underpinning the modern data driven approximation is the low rank structure of the solution (manifold) caused by the high order regularity of the dependence of the solution on the model parameters. We present a low-rank greedily adapted hp-finite element algorithm for computing an approximation to the solution of the Lyapunov operator equation AX +XA = bb^*. In the case in which the coefficient A is self-adjoint and positive definite, the Lyapunov equation has the unique positive and self-adjoint solution X. We interpret the problem of finding the low rank approximation of X as the problem of approximating the dominant eigenvalue cluster of a bounded self-adjoint operator. We show that there is a hidden regularity in eigenfunctions of the solution of the Lyapunov equation which can be utilized to justify the use of high order finite element spaces. We test our methods on several benchmark problems which test the influence of the critical restrictions of the theorems. For instance, to study the influence of the lower elliptic regularity we use an example of the Lyapunov equation whose coefficient is a Laplace operator defined on the dumbbell domain (two separate identical squares connected by a small bridge). Our numerical experiments indicate that we achieve eight figures of accuracy for computing the trace of the solution of the Lyapunov equation posed in this dumbbell-domain using a finite element space of dimension of only ten thousand degrees of freedom. Even more surprising is the observation that hp-refinement has an effect of reducing the rank of the approximation of the solution.

Lyapunov equation, hp adapted finite elements, low rank approximations, operator equations

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