A multilevel approach for approximating the inverse Hessian in four-dimensional variational data assimilation

Student thesis: Doctoral Thesis

Abstract

Data assimilation methods are routinely employed in many scientific and technological fields. The inverse Hessian, and inverse square root Hessian, are of importance in various aspects of these procedures. The Hessian vector product is usually defined in the context of geophysical and engineering applications by the sequential solution of a tangent linear and adjoint problem. However, there are no readily available routines for computing the inverse Hessian, and inverse square root Hessian, vector products.It is generally necessary, when solving high-dimensional data assimilation problems, to operate in a matrix-free environment. A suitable method for generating a compact representation of the inverse Hessian, and inverse square root Hessian, is required in such cases.A multilevel eigenvalue decomposition algorithm for constructing a limited-memory approximation to the inverse (and inverse square root) of any given symmetric positive definite matrix with eigenvalues clustered around unity is presented in this thesis. This algorithm is applied to the Hessian in the framework of incremental four-dimensional variational data assimilation (4D-Var), with the standard control variable transform implemented, in order to construct an approximation to the inverse Hessian. A novel decomposition of the Hessian as the sum of a set of local Hessians is introduced in this setting.Two practical variants of the multilevel eigenvalue decomposition algorithm for constructing a limited-memory approximation to the inverse (and inverse square root) of this Hessian are also presented. The accuracy of approximations to the inverse Hessian generated by applying the three algorithms proposed is investigated. The application considered in this thesis focuses on preconditioning the system of linear equations in the inner step of a Gauss-Newton procedure in incremental 4D-Var with an approximation to the inverse Hessian generated using the three algorithms proposed.
Date of Award1 Apr 2015
LanguageEnglish
Awarding Institution
  • University Of Strathclyde
SponsorsEPSRC (Engineering and Physical Sciences Research Council)
SupervisorAlison Ramage (Supervisor) & Philip Knight (Supervisor)

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