Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations

We study linear parabolic initial-value problems in a space-time variational formulation based on fractional calculus. This formulation uses "time derivatives of order one half" on the bi-infinite time axis. We show that for linear, parabolic initial-boundary value problems on $(0,\infty)$, the corresponding bilinear form admits an inf-sup condition with sparse tensor product trial and test function spaces. We deduce optimality of compressive, space-time Galerkin discretizations, where stability of Galerkin approximations is implied by the well-posedness of the parabolic operator equation. The variational setting adopted here admits more general Riesz bases than previous work; in particular, no stability in negative order Sobolev spaces on the spatial or temporal domains is required of the Riesz bases accommodated by the present formulation. The trial and test spaces are based on Sobolev spaces of equal order $1/2$ with respect to the temporal variable. Sparse tensor products of multi-level decompositions of the spatial and temporal spaces in Galerkin discretizations lead to large, non-symmetric linear systems of equations. We prove that their condition numbers are uniformly bounded with respect to the discretization level. In terms of the total number of degrees of freedom, the convergence orders equal, up to logarithmic terms, those of best $N$-term approximations of solutions of the corresponding elliptic problems.Comment: 26 page

Bounded real lemmas for positive descriptor systems

A well known result in the theory of linear positive systems is the existence of positive definite diagonal matrix (PDDM) solutions to some well known linear matrix inequalities (LMIs). In this paper, based on the positivity characterization, a novel bounded real lemma for continuous positive descriptor systems in terms of strict LMI is first established by the separating hyperplane theorem. The result developed here provides a necessary and sufficient condition for systems to possess H?H? norm less than ? and shows the existence of PDDM solution. Moreover, under certain condition, a simple model reduction method is introduced, which can preserve positivity, stability and H?H? norm of the original systems. An advantage of such method is that systems? matrices of the reduced order systems do not involve solving of LMIs conditions. Then, the obtained results are extended to discrete case. Finally, a numerical example is given to illustrate the effectiveness of the obtained results