This thesis demonstrated how the Newton and BFGS algorithm could be used to solve the standard VGM relations and least-squares formulation respectively, for arbitrary forms of the _ parameter, including nonlinear _ s, appearing in the VMM. The proposed procedure were also shown t
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This thesis demonstrated how the Newton and BFGS algorithm could be used to solve the standard VGM relations and least-squares formulation respectively, for arbitrary forms of the _ parameter, including nonlinear _ s, appearing in the VMM. The proposed procedure were also shown to be able to handle arbitrary projectors that are compatible with the VGM. When applied to the advection-di_usion equation, Burgers' equation and Stokes equations the algorithms always reached the speci_ed stopping criteria and did not exceed the maximum alloted iterations. Additionally the increase in computational e_ort required was shown to be limited. However, it was shown that the Newton procedure for the VGM could not be used in cases where the local VGM residuals had varying signs. In that case the BFGS and least-squares VGM were successful. Application of the proposed procedures to di_erent SGS models showed that nonlinear models tended to outperform linear models in terms of the L2 and projected error. It was also shown that the parametrization of an SGS model can greatly inuence how well the VGM will be able to optimize its coe_cients.
