Simplex interpolation

In many practical applications of nonlinear optimization, a linear approximation is iterated until some vector of system parameters has negligible norm. At each stage of such a process, previous steps provide a set of m n trial vectors in an n-dimensional parameter space, each associated with an output vector of gradients of the quantity to be optimized. The next iterative step is facilitated if a linear combination of such vectors can be found that produces an output vector of minimum norm. 

Linear interpolation can be described geometrically in terms of an m-simplex in the n-dimensional parameter space. An m-simplex in a hyperspace of dimension n m is a set of m + 1 points that do not lie in a subspace of dimension less than m. For example, a triangle is a 2-simplex and a tetrahedron is a 3-simplex. The interior of a simplex is the set of points X = x,/-,. such that 0 A, 1 and 

The simplest possible example is linear interpolation in a 1-simplex (x0, xi). The system of linear equations is 

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This method [75] uses simplex interpolation combined with an update step such as BFGS. The variational functional is... 

At each step of an iterative method for nonlinear optimization, the subsequent coordinate step Aq must be estimated. The vector Q, interpolated in a selected m-simplex of prior coordinate vectors, must be combined with an iterative estimate of the component of Aq orthogonal to the hyperplane of the simplex. 

Piecewise linear interpolation based on Delaunay triangulation is used to get the surface over the selected shape. A buffer zone is used for 1-dimensional simplexes. 

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