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 highdimensional data assimilation problems, to operate in a matrixfree 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 limitedmemory 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 fourdimensional variational data assimilation (4DVar), 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 limitedmemory 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 GaussNewton procedure in incremental 4DVar with an approximation to the inverse Hessian generated using the three algorithms proposed.
Date of Award  25 Apr 2019 

Original language  English 

Awarding Institution   University Of Strathclyde


Sponsors  EPSRC (Engineering and Physical Sciences Research Council) 

Supervisor  Alison Ramage (Supervisor) & Philip Knight (Supervisor) 
