Parallel computation of invariant measures
- Univ. of Southern Mississippi, Hattiesburg, MS (United States)
A parallel numerical algorithm for computing invariant measures is presented. Let I{sup N} {triple_bond} [0,1]{sup N} be the unit N-cube in R N and let S : I{sup N}{r_arrow} I{sup N} be a nonsingular transformation, that is, S is Borel-measurable and m(A) = 0 implies m(S{sup -1}(A)) = 0, where m is the Lebesgue measure. The motivation of this study is the parallel computation of an absolutely continuous invariant measure {mu} under S, that is, {mu} {much_lt} m and {mu}(A) = {mu}(S{sup -1}(A)) for all Borel sets A {contained_in} I{sup N}. It is well-known that an absolutely continuous finite invariant measure {mu} can be obtained by computing a fixed density of the Frobenius-Perron operator Ps: L{sup 1} (I{sup N}) {r_arrow} L{sup 1}(I{sup N}) associated with S which is defined by (1) {integral}{sub A} P{sub S}fdm = {integral}{sub s-1(A)} fdm, {forall}f {element_of} L{sup 1} (I{sup N}). Using any suitable discretization scheme, the infinite dimensional eigenvector problem P{sub S}f = f in L{sup 1}(I{sup N}) can be approximated by an algebraic eigenvector problem P{sub l}f{sub l} = f{sub l} in {gradient}{sub l}, where P{sub l} is a finite approximation of P{sub s} associated with a finite element subspace {gradient}{sub l} of L{sup l} (I{sup N}) {intersection} L{sup {infinity}} (I{sup N}). It has been shown that for P{sub l} arising from Galerkin`s projection principle or the Markov finite approximation principle, there always exists a eigenvector f{sub l} to P{sub l}, and that a sequence of normalized eigenvectors (f{sub l}) converges to the density of an absolutely continuous probability invariant measure {mu} for a class of piecewise C{sup 2} expanding maps of I{sup N} under which the existence of {mu} is guaranteed by Gora-Boyarsky`s theorem which is reduced to Lasota-Yorke`s thoerem when N = 1.
- OSTI ID:
- 125517
- Report Number(s):
- CONF-950212-; TRN: 95:005768-0061
- Resource Relation:
- Conference: 7. Society for Industrial and Applied Mathematics (SIAM) conference on parallel processing for scientific computing, San Francisco, CA (United States), 15-17 Feb 1995; Other Information: PBD: 1995; Related Information: Is Part Of Proceedings of the seventh SIAM conference on parallel processing for scientific computing; Bailey, D.H.; Bjorstad, P.E.; Gilbert, J.R. [eds.] [and others]; PB: 894 p.
- Country of Publication:
- United States
- Language:
- English
Similar Records
Approximation of functions of variable smoothness by Fourier-Legendre sums
Confidential balls minimizing in mean convex functional