Guaranteed Estimates of Linear Continuous Functionals of Solutions and Right-hand Sides of the Helmholtz Equation in the Domains with Infinite Boundaries under Uncertainties
2013 (English)In: PIERS 2013 STOCKHOLM: PROGRESS IN ELECTROMAGNETICS RESEARCH SYMPOSIUM, 2013, 65-69 p.Conference paper (Refereed)
We consider the construction of guaranteed estimates of linear continuous function als of the unknown solutions and right-hand sides of the Helmholtz equation; the boundary value problems under study are associated with the wave diffraction by a bounded body D situated in a domain Omega is an element of R-n, n = 2, 3, whose boundary partial derivative Omega stretches to infinity (e.g., a wedge or a layer) and Green's function Phi(k) (x, y), (x, y is an element of Omega, x not equal y) corresponding to wave number k with k > 0 and boundary condition (I)k (x, y)vertical bar y is an element of Omega = 0 is known . Here, for a function u(y) defined in (Omega) over bar Bu(y)vertical bar(y is an element of partial derivative Omega) + beta partial derivative u(y)/partial derivative y vertical bar(y is an element of partial derivative Q), alpha, beta = 0, 1, alpha + beta = 1, v is outward normal to aft We assume that right-hand sides of the equations entering the problem statement are not known; the only available information is that they belong to a bounded set of the space of square-integrable functions. In order to solve these estimation problems we need additional data: observations in the form of certain linear transformations of the solution distorted by noise. The latter are realizations of the random fields with the unknown second moment functions belonging to a given bounded set in the appropriate functional space. The approach set forth in and developed in this study allows us to obtain optimal estimates of the unknown solution or righthand sides of the equations and linear functionals, i.e., estimates sought in the class of functionals linear with respect to observations for which the maximal mean-square estimation error taken over all elements belonging to the aforementioned sets takes minimal value. Such estimates are called minimax or guaranteed estimates. We obtain representations for these estimates and estimation errors in terms of solutions to certain integro-differential or integral equations in bounded subdomains of domain Omega \ D.
Place, publisher, year, edition, pages
2013. 65-69 p.
, Progress in Electromagnetics Research Symposium, ISSN 1559-9450
Research subject Mathematics; Mathematics
IdentifiersURN: urn:nbn:se:kau:diva-38684ISI: 000361384200010ISBN: 978-1-934142-26-4OAI: oai:DiVA.org:kau-38684DiVA: diva2:873348
Progress In Electromagnetics Research Symposium, AUG 12-15, 2013, Stockholm, SWEDEN