Let P (z) be a formal power series in z = (z1, · · · , zn) with o(P (z)) ≥ 2 and t a formal parameter which commutes with z. We say P (z) is HN (Hessian nilpotent) if its Hessian matrix HesP (z) = ( ∂ 2 P ∂zi∂zj ) is nilpotent. The deformed inversion pair Qt(z) of P (z) by definition is the unique Qt(z) ∈ C[[z, t]] with o(Qt(z)) ≥ 2 such that the formal maps Gt(z) = z + t∇Q(z) and Ft(z) = z − t...

We introduce a fast algorithm for entrywise evaluation of the Gauss--Newton Hessian (GNH) matrix fully connected feed-forward neural network. The has precomputation step and sampling step. While it generally requires $\mathcal{O}(Nn)$ work to compute an entry (and entire column) in GNH network with $N$ parameters $n$ data points, our reduces cost $\mathcal{O}(n+d/\epsilon^2)$ work, where $d$ is...

Abstract We extend the finite element method introduced by Lakkis and Pryer (SIAM J. Sci. Comput. 33(2): 786–801, 2011) to approximate solution of second-order elliptic problems in nonvariational form incorporate discontinuous Galerkin (DG) framework. This is done viewing “finite Hessian” as an auxiliary variable formulation. Representing Hessian a setting yields linear system same size having ...

Abstract We consider a scalar function depending on numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the value. The need to extract information Hessian or solve linear system having as coefficient arises in many research fields such optimization, Bayesian estimation, uncertainty quantification. From perspective memory efficiency, these tasks often e...

The damped Gauss-Newton (dGN) algorithm for CANDECOMP/PARAFAC (CP) decomposition can handle the challenges of collinearity of factors and different magnitudes of factors; nevertheless, for factorization of an N-D tensor of size I1 × · · · × IN with rank R, the algorithm is computationally demanding due to construction of large approximate Hessian of size (RT × RT ) and its inversion where T = n...

Let (X, g) be a d-dimensional compact Riemannian manifold and let f be a Morse function on X. That is, the set of critical points of f is a finite set {c1, . . . , ck} and the Hessian there are nondegenerate. Let dλ = e−λf/2deλf/2. Here d denotes the exterior differential operator on X. Taking an adjoint of dλ on L2(∧T ∗X, dx) (dx is the Riemannian volume), we see dλ = e λf/2d∗e−λf/2 explicitly...

We study fully nonlinear partial differential equations involving the determinant of the Hessian matrix of the unknown function with respect to a family of vector fields that generate a Carnot group. We prove a comparison theorem among viscosity suband supersolutions, for subsolutions uniformly convex with respect to the vector fields.

We study an influence of precise data on uncertainty of polarized parton distribution functions. This analysis includes the SLAC-E155 proton target data which are precise measurements. Polarized PDF uncertainties are estimated by using the Hessian matrix. We examine correlation effect between the antiquark and gluon uncertainties. It suggests that reducing the gluon uncertainty is needed to det...

Interpolation-based trust-region methods are an important class of algorithms for Derivative-Free Optimization which rely on locally approximating an objective function by quadratic polynomial interpolation models, frequently built from less points than there are basis components. Often, in practical applications, the contribution of the problem variables to the objective function is such that ...

Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertex coloring problems her...