Construction of optimal background fields using semidefinite programming

Abstract

Quantitative analysis of systems exhibiting turbulence is challenging due to the lack of exact solutions and the cost of accurate simulations, but asymptotic or time-averaged properties can often be bounded rigorously using the background method. This rests on the construction of a background field for the system subject to a spectral constraint, which requires that a background-field-dependent linear operator has non-negative eigenvalues. This thesis develops techniques for the numerical optimisation of background fields and their corresponding bounds. First, bounds on the asymptotic energy of solutions of the Kuramoto–Sivashinsky equation are optimised by solving the Euler–Lagrange (EL) equations for the optimal background field using a time-marching algorithm. It is demonstrated that convergence to incorrect solutions occurs unless the derivation of the EL equations accounts for the multiplicity of eigenvalues in the spectral constraints. Second, semidefinite programmes (SDPs) are formulated to approximately solve optimisation problems subject to a class of integral inequalities on function spaces, to which spectral constraints can often be reduced. More precisely, inner and outer approximations of the feasible set of an integral inequality with one-dimensional compact integration domain, whose integrand is quadratic in the test functions and affine in the optimisation variables, are constructed using linear matrix inequalities. These SDP-based techniques, implemented in the MATLAB toolbox QUINOPT, are then utilised to bound the dissipation coefficient C_ε in stress-driven shear flows, and further improved to bound the Nusselt number Nu in Bénard-Marangoni convection at infinite Prandtl number. The results suggest that the existing analytical bounds on C_ε attain the optimal asymptotic scaling, while those on Nu may be lowered by a logarithmic factor upon constructing a non-monotonic background field. It is also concluded that semidefinite programming will offer an efficient, robust, and flexible framework to optimise background fields if the computational challenges presented by large-scale SDPs can be addressed.