Second partial derivative test

Abstract: A numerical model can be fit to data by minimizing a positive quadratic function of the differences between the data and their model counterparts. The rate at which algorithms for computing the best fit to data converge depends on the size of the condition number and the distribution of eigenvalues of the Hessian matrix, which contains the second derivatives of this quadratic function. The inverse of the Hessian can be identified as the covariance matrix that establishes the accuracy to which the model state is determined by the data; the reciprocals of the Hessian's eigenvalues represent the variances of linear combinations of variables determined by its eigenvectors. The aspect of the model state that are most difficult to compute are those about which the data provide the least information. A unified formalism is presented in which the model may be treated as providing either strong or weak constraints, and methods for computing and inverting the Hessian matrix are discussed. Examples are given of the uncertainties resulting from fitting an oceanographic model to several different sets of hypothetical data.

Abstract: Alternatively we may write (1.3) Fk[u] = [Du]k, where [A]k denotes the sum of the k× k principal minors of an n× n matrix A. Our purpose in this paper is to extend the definition of the Fk to corresponding classes of continuous functions so that Fk[u] is a Borel measure and to consider the Dirichlet problem in this setting. A function u ∈ C(Ω) is called k-convex (uniformly k-convex) in Ω if Fj [u] ≥ 0 (> 0) for j = 1, . . . , k. The operator Fk

Abstract: The tangent stiffness matrix (i.e., the Hessian) for nonlinear structural models in computational solid mechanics is typically computed by linearization of the weak form of the equilibrium equations via the directional derivative formula. Depending on the specific mechanical model, away from equilibrium this procedure will in general yield a nonsymmetric tangent stiffness matrix. By contrast, if the directional (Gateaux) derivative is replaced by the covariant derivative (relative to a certain Riemannian metric) an intrinsic definition of the Hessian is obtained which is always symmetric away from equilibrium. It is shown that by endowing the rotation group with the standard bi-invariant metric, the resulting (symmetric) Hessian is simply the symmetrization of the usual expression obtained via the Gateaux derivative. In a finite element context, this property provides a rigorous justification for the common practice of symmetrizing the ‘seemingly nonsymmetric tangent’ away from equilibrium. This apparently ad hoc symmetrization procedure yields, in fact, the actual Hessian.

Abstract: In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C1,1 functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gateaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems.