To improve the condition number of the Hessian, we advocate the use of a change of polynomial basis, defined using a Cholesky factorization of the Hessian, that permits solution of problems nearer to the boundary of the realizable set.
The closure is only well-defined on the set of moments that are realizable from a positive underlying distribution, and as the boundary of the realizable set is approached, the dual problem becomes increasingly difficult to solve due to ill-conditioning of the Hessian matrix.
We also advocate a fixed quadrature scheme, rather than adaptive quadrature, since the latter introduces unnecessary expense and changes the computationally realizable set as the quadrature changes.
In fact, the set of realizable provability principles is the intersection of \(\mathsf{GL}\) and \(\mathsf{S4}\) (Goris 2007).
The difficulty for such a limiter lies in the computation of the intersection of a ray with the set of realizable moments.
In some instances, one can show that a particular untyped realizability interpretation matches a particular typed one, in the sense that they give the same set of realizable formulae.
These interrelations are used to delineate a compact and convex space that bounds the set of all physically realizable 2-point correlations called the 2-point correlations hull.
During the conference we also announced a set of ambitious but realizable goals for the way we source our coffee, how we help to sustain our environment, and what we can do to work with, and continue to contribute to, the communities in which we do our business and the world as a whole.
To model the turbulent stress tensor, the last nonlinear term in Eq. (2) and hence provide closure of the above open set of governing equations, the realizable k-epsilon turbulence model [26, 27] is used in our particular problem.
The following result shows that any given set of real numbers is SCRS realizable.
