Efficient Gradient Computation Research Articles

Excited-State Forces within a First-Principles Green’s Function Formalism

We present a new first-principles formalism for calculating forces for optically excited electronic states using the interacting Green's function approach with the GW Bethe-Salpeter-equation method. This advance allows for efficient computation of gradients of the excited-state Born-Oppenheimer energy, allowing for the study of relaxation, molecular dynamics, and photoluminescence of excited states. The approach is tested on photoexcited carbon dioxide and ammonia molecules, and the calculations accurately describe the excitation energies and photoinduced structural deformations.

Design of Hermite Subdivision Schemes Aided by Spectral Radius Optimization

We present a method for constructing multivariate refinable Hermite interpolants and their associated subdivision algorithms based on a combination of analytical and numerical approaches. As the limit of a linear iterative procedure, the critical L2 Sobolev smoothness of a refinable Hermite interpolant is given by the spectral radius of a matrix dependent upon the refinement mask. The design question is, Given certain constraints (support size, symmetry type, refinement pattern, etc.), how can one choose the refinement mask so that the resulting refinable function has optimal smoothness? This question naturally gives rise to a spectral radius optimization problem. In general, the objective function is not convex and may not be differentiable, or even Lipschitz, at a local minimizer. Nonetheless, a recently developed robust solver for nonsmooth optimization problems may be applied to find local minimizers of the spectral radius objective function. In fact, we find that in specific cases that are of particular interest in the present context, the objective function is smooth at local minimizers and may be accurately minimized by standard techniques. We present two necessary mathematical tricks that make the method practical: (i) compression of matrix size based on symmetry and (ii) efficient computation of gradients of the objective function. We conclude by reporting some computational results.

Numerical parameter identification in multiphase flow through porous media

Multiphase flow is of high interest for the investigation of the behavior of waste in groundwater. The high nonlinearity of the model equations pose special problems. Here, a new parameter identification technique in this context is proposed which takes advantage of recently developed highly efficient numerical simulation techniques. It is based on a reduced Gauss–Newton technique in combination with an efficient gradient computation. Numerical experiments are performed for the McWhorter model problem.

Computing Gradients in Large-Scale Optimization Using Automatic Differentiation

The accurate and efficient computation of gradients for partially separable functions is central to the solution of large-scale optimization problems, because these functions are ubiquitous in large-scale problems. We describe two approaches for computing gradients of partially separable functions via automatic differentiation. In our experiments we employ the ADIFOR (automatic differentiation of Fortran) tool and the SparsLinC (sparse linear combination) library. We use applications from the MINPACK-2 test problem collection to compare the numerical reliability and computational efficiency of these approaches with hand-coded derivatives and approximations based on differences of function values. Our conclusion is that automatic differentiation is the method of choice, providing code for the efficient computation of the gradient without the need for tedious hand-coding.

Efficient computation of gradients and Jacobians by dynamic exploitation of sparsity in automatic differentiation

Automatic differentiation (AD) is a technique that augments computer codes with statements for the computation of derivatives. The computational workhorse of AD-generated codes for first-order derivatives is the linear combination of vectors. For many large-scale problems, the vectors involved in this operation are inherently sparse. If the underlying function is a partially separable one (e.g., if its Hessian is sparse), many of the intermediate gradient vectors computed by AD will also be sparse, even though the final gradient is likely to be dense. For large Jacobians computations, every intermediate derivative vector is usually at least as sparse as the least sparse row of the final Jacobian. In this paper, we show that dynamic exploitation of the sparsity inherent in derivative computation can result in dramatic gains in runtime and memory savings. For a set of gradient problems exhibiting implicit sparsity, we report on the runtime and memory requirements of computing the gradients with the ADIFOR (...

Robust inverse shape design in aerodynamics

The inverse problem being addressed is the construction of an aerodynamic airfoil shape. The objective of the present paper is: (i) to formulate the inverse problem such that the mathematical problem is well-posed both theoretically and numerically; (ii) to formulate the adjoint problem such that an efficient solution procedure can be constructed; (iii) to demonstrate how the solution procedure works. The inverse problem is posed as a minimization problem of an objective functional. The minimum of the functional corresponds to the attainment of a target velocity distribution by the aerodynamic airfoil shape. The design variables being considered are geometric parameters of the airfoil and an appropriately defined set of target velocity parameters which are introduced to assure the well-posedness of the problem. The minimization problem is solved by an optimization algorithm. Adjoint method is employed for an efficient computation of the objective functional gradient with respect to the geometric parameters. Numerical results demonstrate that the present inverse problem formulation is well-posed. The significance of the target velocity parameters in obtaining well-posedness is explained in terms of the Lighthill constraints.

Security constrained economic dispatch with participation factors based on worst case bus load variations

An efficient method is presented for finding the economic and steady-state secure base point generation levels to supply the present load. From the specified range of possible load variation anticipated for each bus during the interval before the next base point optimization, the generator participation factors are calculated to guarantee a secure supply to the worst case combination of bus load excursions. The base points are calculated using a linear but lossy system model coupled with an efficient gradient computation technique. The worst case bus load condition is found as an extension of the base point calculation.

Efficient computation of gradient and hessian of likelihood function in linear dynamic systems

This technical note describes a computationally efficient procedure to determine the first and second gradients of the likelihood function for parameter estimation in linear dynamic systems. The results presented here are extensions, of the sensitivity functions reduction procedure of [1]. An operation count shows the value of the new algorithm.