Projects per year
Abstract
We prove wellposedness results and a priori bounds on the solution of the Helmholtz equation ∇ (A∇u)+k ^{2}nu = f, posed either in ℝ ^{d} or in the exterior of a starshaped Lipschitz obstacle, for a class of random A and n; random data f, and for all k > 0. The particular class of A and n and the conditions on the obstacle ensure that the problem is nontrapping almost surely. These are the first wellposedness results and a priori bounds for the stochastic Helmholtz equation for arbitrarily large k and for A and n varying independently of k. These results are obtained by combining recent bounds on the Helmholtz equation for deterministic A and n and general arguments (i.e., not specific to the Helmholtz equation) presented in this paper for proving a priori bounds and wellposedness of variational formulations of linear elliptic stochastic PDEs. We emphasize that these general results do not rely on either the LaxMilgram theorem or Fredholm theory, since neither is applicable to the stochastic variational formulation of the Helmholtz equation.
Original language  English 

Pages (fromto)  5887 
Number of pages  30 
Journal  SIAM/ASA Journal on Uncertainty Quantification 
Volume  8 
Issue number  1 
Early online date  9 Jan 2020 
DOIs  
Publication status  Published  2020 
Keywords
 A priori bounds
 Helmholtz equation
 High frequency
 Nontrapping
 Random media
 Wellposedness
ASJC Scopus subject areas
 Statistics and Probability
 Modelling and Simulation
 Statistics, Probability and Uncertainty
 Discrete Mathematics and Combinatorics
 Applied Mathematics
Fingerprint
Dive into the research topics of 'The Helmholtz equation in random media: wellposedness and a priori bounds'. Together they form a unique fingerprint.Projects
 2 Finished

Fast solvers for frequencydomain wavescattering problems and applications
Graham, I., Gazzola, S. & Spence, E.
Engineering and Physical Sciences Research Council
1/01/19 → 31/12/22
Project: Research council

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council