A semigroup/Laplace transform approach to approximating flows

Abstract

It is well known that all flows in a state space O induce a semigroup of linear operators on an appropriately chosen vector space of functions (observables) from O into a vector space Z (observations). After choosing appropriate continuity assumptions on the flow, the associated semigroup will be strongly continuous and will have a linear, infinitesimal generator A. The purpose of this dissertation is to explore approximation methods for linear semigroups and/or Laplace transform inversion methods in order to reconstruct the flow starting with the linear generator A . In preparing for these investigations, we collect some of the essential approximation theorems of semigroup theory and improve a recent generalization of the Trotter-Kato Theorem due to McAllister, Neubrander, Riser, and Zhuang. Moreover, we show that rational Laplace transform inversions of order m are exact for all polynomials of degree less than m. We will demonstrate that the flow can be efficiently reconstructed whenever the generator A of the induced semigroup has a resolvent that can be efficiently computed or approximated. We demonstrate this for flows solving nonlinear first order ordinary differential equations x'(t) = a(x(t)), x(s)= w and the induced generator (Af)(s)=a(s)f'(s) and for flows solving non-autonomous linear first order ordinary differential equations u'(t) = a(t)u(t), u(s)= w and the induced generator(Af)(s) = f'(s)+a(s)f(s). As a by-product of our investigation, we find a numerically efficient way to compute the inverse of increasing real-valued functions. Finally, we explore whether linear semigroup approximation methods can be used efficiently to approximate solutions of non-autonomous Cauchy problems u'(t) = A(t)u(t), u(s) = x in terms of the generator (Af)(s) = f'(s) + A(s)f(s) of the induced linear operator semigroup. As we will see, the Lie-Trotter approach suggested by G. Nickel seems to be the only efficient way to find the solutions of the non-autonomous problems in terms of the semigroup generated by A.