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## Publications

Publications (225)

Molecular dynamics simulations often classically evolve the nuclear geometry on adiabatic potential energy surfaces (PESs), punctuated by random hops between energy levels in regions of strong coupling, in an algorithm known as surface hopping. However, the computational expense of integrating the geometry on a full-dimensional PES and computing th...

In the construction of reduced-order models for dynamical systems, linear projection methods, such as proper orthogonal decompositions, are commonly employed. However, for many dynamical systems, the lower dimensional representation of the state space can most accurately be described by a \textit{nonlinear} manifold. Previous research has shown tha...

We give conditions under which the generalized Davidson algorithm for eigenvalue computations is mesh-independent. In this case mesh-independence means that the iteration statistics (residual norms, convergence rates, for example) of a sequence of discretizations of a problem in a Banach space converge the statistics for the infinite-dimensional pr...

The potential energy surface (PES) describes the energy of a chemical system as a function of its geometry and is a fundamental concept in computational chemistry. A PES provides much useful information about the system, including the structures and energies of various stationary points, such as local minima, maxima, and transition states. Construc...

Load imbalance plagues domain decomposed Monte Carlo calculations when sources are not uniform. Parallel efficiency for domain decomposed Monte Carlo transport calculations improves through a nonuniform allocation of processors over subdomains. We optimize the allocation with runtime diagnostics collected during a calibration step, then complete th...

Molecular dynamics (MD) simulations with full-dimensional potential energy surfaces (PESs) obtained from high-level ab initio calculations are frequently used to model reaction dynamics of small molecules (i.e., molecules with up to 10 atoms). Construction of full-dimensional PESs for larger molecules is, however, not feasible since the number of a...

Nondilute transport in porous media results in fronts that are much sharper in space and time than the corresponding transport of a conservative, nonreactive dilute species. A thermodynamically constrained averaging theory model for such situations is developed. A novel closure scheme is formulated, which is cross-coupled between flow and transport...

This article is about numerical methods for the solution of nonlinear equations. We consider both the fixed-point form $\mathbf{x}=\mathbf{G}(\mathbf{x})$ and the equations form $\mathbf{F}(\mathbf{x})=0$ and explain why both versions are necessary to understand the solvers. We include the classical methods to make the presentation complete and dis...

In this paper we explore the convergence properties of deterministic direct search methods when the objective function contains a stochastic or Monte Carlo simulation. We present new results for the case where the objective is only defined on a set with certain minimal regularity properties. We present two numerical examples to illustrate the ideas...

This paper evaluates the performance of multiphysics coupling algorithms applied to a light water nuclear reactor core simulation. The simulation couples the k-eigenvalue form of the neutron transport equation with heat conduction and subchannel flow equations. We compare Picard iteration (block Gauss-Seidel) to Anderson acceleration and multiple v...

Iron(II) polypyridine complexes have the potential for numerous applications on a global scale, such as sensitizers, sensors, and molecular memory. The excited-state properties of these systems, particularly the intersystem crossing (ISC) rates, are sensitive to the choice of ligands and can be significantly altered depending on the coordination en...

A generally applicable calibration technique for digitally reconfigurable self-healing radio frequency integrated circuits based on a hybrid of the Nelder-Mead and Hooke-Jeeves direct search algorithms is presented. The proposed algorithm is applied to the multiobjective problem of gain error and phase error minimization for a self-healing phase ro...

Anderson(m) is a method for acceleration of fixed point iteration which stores m+1 prior evaluations of the fixed point map and computes the new iteration as a linear combination of those evaluations. Anderson(0) is fixed point iteration. In this paper we show that Anderson(m) is locally r-linearly convergent if the fixed point map is a contraction...

A parallel implementation of an eigensolver designed for electronic structure
calculations is presented. The method is applicable to computational tasks that
solve a sequence of eigenvalue problems where the solution for a particular
iteration is similar but not identical to the solution from the previous
iteration. Such problems occur frequently w...

We analyze the behavior of inexact Newton methods for problems where the nonlinear residual, Jacobian, and Jacobian-vector products are the outputs of Monte Carlo simulations. We propose algorithms which account for the randomness in the iteration, develop theory for the behavior of these algorithms, and illustrate the results with an example from...

In this paper we present a new implementation of Smolyak's sparse grid interpolation algorithm designed for dynamical simulations. The implementation is motivated by an application to quantum chemistry where the goal is to simulate photo-induced molecular transformations. A molecule conforms to a geometry that minimizes its potential energy, and ma...

In this article we present a hybrid deterministic/Monte Carlo algorithm for computing the dominant eigenvalue/eigenvector pair for the neutron transport k-eigenvalue problem in multiple space dimensions. We begin by deriving the Nonlinear Diffusion Acceleration method (Knoll, Park, and Newman, 2011; Park, Knoll, and Newman, 201210.
Park, H., Knoll,...

Computing the potential energy of an N-atom molecule is an expensive optimization process of 3N - 6 molecular coordinates, so following reaction pathways as a function of all 3N - 6 coordinates is unfeasible for large molecules. In this paper, we present a method that isolates d < 3N - 6 molecular coordinates and continuously follows reaction paths...

The Boltzmann transport equation is used to model the neutron flux in a nuclear reactor. The solution of the transport equation is the neutron flux, which depends on a large number of material cross sections that can be on the order of thousands. These cross sections describe various types of possible interactions between neutrons, such as fission,...

In this paper we discuss a parallel hybrid deterministic/Monte Carlo (MC) method for the solution of the neutron transport equation in two space dimensions. The algorithm uses an NDA formulation of the transport equation, with a MC solver for the high-order equation. The scalability arises from the concentration of work in the MC phase of the algor...

Given a legacy dynamic simulator of a chemical process plant, we construct a computational procedure that can be “wrapped around” the simulator to compute its steady states (both stable and unstable) and their dependence on input parameters. We apply this approach to the Tennessee Eastman (TE) challenge problem presented by Downs and Vogel, who als...

We consider the simulation of steady-state variably saturated groundwater flow using Richards’ equation (RE). The difficulties associated with solving RE numerically are well known. Most discretization approaches for RE lead to nonlinear systems that are large and difficult to solve. The solution of nonlinear systems for steady-state problems can b...

This paper describes a calibration technique for noisy and nonconvex circuit responses based on the Nelder-Mead direct search algorithm. As Nelder-Mead is intended for unconstrained optimization problems, we present an implementation of the algorithm which is suitable for bounded and discretized RFIC calibration problems. We apply the proposed algo...

A new model for studying the behavior of nanoscale tunneling devices has been developed in C++ using the Wigner-Poisson formulation. This model incorporates the parallel solvers of Sandia National Lab's Trilinos software with the efficient use of parallel data structures to create a code that scales well to a high number of processors. It also inco...

We analyze and extend an explicit pseudo-transient continuation algorithm proposed by Han and Han. We present a convergence proof, extend the method to bound-constrained optimization, and proposed an improved step-size control strategy.

In this paper we describe a hybrid deterministic/Monte Carlo algorithm for neutron transport simulation. The algorithm is based on nonlinear accelerators for source iteration, using Monte Carlo methods for the purely absorbing high-order problem and a Jacobian-free Newton- Krylov iteration for the low-order problem. We couple the Monte Carlo soluti...

Simulations of light-induced molecular conformational transformations have traditionally been limited to a single degree of freedom because of the complexity of potential energy calculations. We propose a method of simulation that incrementally builds a surrogate for the potential energy function by computing gridpoints in parallel. We incorporate...

We describe a method for using interpolatory models to accurately and efficiently simulate molecular excitation and relaxation. We use sparse interpolation for efficiency and local error estimation and control for robustness and accuracy.

In this paper we discuss a GPU implementation of a hybrid deterministic/Monte Carlo method for the solution of the neutron transport equation. The key feature is using GPUs to perform a Monte Carlo transport sweep as part of the evaluation of the nonlinear residual and Jacobian-vector product. We describe the algorithm and present some preliminary...

This study develops a lumped cardiovascular-respiratory system model that incorporates patient-specific data to predict cardiorespiratory response to hypercapnia (increased CO(2) partial pressure) for a patient with congestive heart failure (CHF). In particular, the study focuses on predicting cerebrovascular CO(2) reactivity, which can be defined...

On the occasion of his 65th birthday, we briefly recount Dan Sorensen’s profound contributions to optimization, numerical linear algebra, and model order reduction for dynamical systems.

We develop a new Proper Orthogonal Decomposition (POD) reduced order model for saturated groundwater flow, and apply that model to an inverse problem for the hydraulic conductivity field. We use sensitivities as the POD basis. We compare the output when the optimizer uses the reduced order model against results obtained with a full PDE based model....

The next generation of electronic devices will be developed at the nanoscale and molecular level, where quantum mechanical effects are observed. These effects must be accounted for in the design process for such small devices. One prototypical nanoscale semiconductor device under investigation is a resonant tunneling diode (RTD). Scientists are hop...

In this paper, we present a new approach for finding a stable solution of a system of nonlinear equations arising from dynamical systems. We introduce the concept of stability functions and use this idea to construct stability solution models of several typical small signal stability problems in dynamical systems. Each model consists of a system of...

The conformational dynamics of molecules that arise due to light-induced transitions are critically important in many biochemical reactions, and therefore dictate the functionality of many types of biological sensors. Therefore, researchers of biological science and biological-inspired technology often need to prescribe the molecular geometry of th...

We examine the local convergence of the Levenberg—Marquardt method for the solution of nonlinear least squares problems that are rank-deficient and have nonzero residual. We show that replacing the Jacobian by a truncated singular value decomposition can be numerically unstable. We recommend instead the use of subset selection. We corroborate our r...

We examine the local convergence of the Levenberg-Marquardt method for the solution of nonlinear least squares problems that are rank-deficient and have nonzero residual. We show that replacing the Jacobian by a truncated singular value decomposition can be numerically unstable. We recommend instead the use of subset selection. We corroborate our r...

A more efficient and accurate discretization of the Wigner-Poisson model for double barrier resonant tunneling diodes is presented. This new implementation uses nonuniform grids and higher order numerical methods to improve the accuracy of the solutions at a significantly lower computational cost. Using the new implementation, devices with short an...

In this paper we describe a parallel algorithm for generating interpolatory approximations to molecular potential energy surfaces. We show how that algorithm can be applied to efficiently model a transition from a stable ground state, to an excited state, and finally to a different stable ground state.

This study presents an analysis of a cerebral autoregulation (CA) model developed by Ursino and Lodi Ursino and Lodi (1997). We have used this model to analyze non-invasive measurements of cerebral blood flow velocity (CBFV) and arterial blood pressure obtained during postural change from sitting to standing for a healthy young subject. This paper...

An engineering and mathematics research project is performed to develop the science base and simulation capability for the study, analysis and design of highly-integrated and molecular-based information processing systems. Specifically, this research investigate MQCA architectures as a potential new paradigm for information processing through the d...

This paper demonstrates how pseudo-transient continuation improves the efficiency and robustness of a Newton iteration within a non-linear transient elasticity simulation. Pseudo-transient continuation improves efficiency by enabling larger time steps than possible with a Newton iteration. Robustness improves because pseudo-transient continuation r...

In this short note we observe that results of Dennis and Audet extend naturally to a wide variety of deterministic sampling methods. For bound-constrained problems, we show that any method based on coordinate search which includes a sufficiently rich set of directions, for example random directions at each state of the sampling, will, when applied...

The use of temporary transfers, such as options and leases, has grown as utilities attempt to meet increases in demand while reducing dependence on the expansion of costly infrastructure capacity (e.g., reservoirs). Earlier work has been done to construct optimal portfolios comprising firm capacity and transfers, using decision rules that determine...

Rosenbrock methods are popular for solving a stiff initial-value problem of ordinary differential equations. One advantage
is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as
backward difference formulas or implicit Runge–Kutta methods. In this article, we introduce a trust-regi...

This study shows how sensitivity analysis and subset selection can be employed in a cardiovascular model to estimate total systemic resistance, cerebrovascular resistance, arterial compliance, and time for peak systolic ventricular pressure for healthy young and elderly subjects. These quantities are parameters in a simple lumped parameter model th...

In this paper, we propose a continuous Newton-type method in the form of an ordinary differential equation by combining the negative gradient and Newton's direction. It is shown that for a general function f (x), our method converges globally to a connected subset of the stationary points of f (x) under some mild conditions; and converges globally...

Management decisions involving groundwater supply and remediation often rely on optimization techniques to determine an effective strategy. We introduce several derivative-free sampling methods for solving constrained optimization problems that have not yet been considered in this field, and we include a genetic algorithm for completeness. Two well...

Recently, we developed a thermodynamically optimized integral equation method which has been successfully tested on both simple and homonuclear diatomic Lennard-Jones fluids [J. Chem. Phys. 2007, 126, 124107]. The systematic evaluation of correlation functions required by the optimization of the chemical potential has shown a clear need for more ef...

We propose and analyze a pseudo-transient continuation algorithm for dynamics on subsets of RN . Examples include certain flows on manifolds and the dynamic formulation of bound-constrained optimization problems. The method gets its global convergence properties from the dynamics and inherits its local convergence properties from any fast locally c...

We bound the condition number of the Jacobian in pseudo arclength continuation problems, and we quantify the effect of this condition number on the linear system solution in a Newton GMRES solve. In pseudo arclength continuation one repeatedly solves systems of nonlinear equations $F(u(s),\lambda(s))=0$ for a real-valued function $u$ and a real par...

Many optimization algorithms exploit parallelism by calling multiple independent instances of the function to be minimized, and these function in turn may call off-the-shelf simulators. The I/O load from the simulators can cause problems for an NFS file system. In this paper we explore efficient parallelization in a parallel program for which each...

We analyze a preconditioner for the time-independent Wigner-Poisson equations for a resonant tunneling diode and present a numerical example of a continuation study which supports the theory. The application of the preconditioner transforms the equations into a compact fixed point problem for the Wigner distribution. After discretization, the Jacob...

In this paper we show that the convergence behavior of the DIviding RECTangles (DIRECT) algorithm is sensitive to additive scaling of the objective function. We illustrate this problem with a computation and show how the algorithm can be modified to eliminate this sensitivity.

Spatially distributed problems are often approximately modeled in terms of partial differential equations (PDEs) for appropriate coarse-grained quantities (e.g., concentrations). The derivation of accurate such PDEs starting from finer scale, atomistic models, and using suitable averaging is often a challenging task; approximate PDEs are typically...

This paper presents new methods for finding dynamically stable solutions of systems of nonlinear equations. The concepts of stability functions and the so-called stable solutions are defined. Based on those new concepts, two models of stable solutions and three stability functions are proposed. These stability functions are semismooth. Smoothing te...

The next generation of electronic devices will be developed at the nanoscale and molecular level, where quantum mechanical effects are observed. These effects must be accounted for in the design process for such small devices. One prototypical nanoscale semiconductor device under investigation is a resonant tunneling diode (RTD). Scientists are hop...

We will discuss a parametric study of the solution of the Wigner-Poisson equations for resonant tunneling diodes. These structures exhibit self-sustaining oscillations in certain operating regimes. We will describe the engineering consequences of our study and how it is a significant advance from some previous work, which used much coarser grids. W...

Most cities rely on firm water supply capacity to meet demand, but increasing scarcity and supply costs are encouraging greater use of temporary transfers (e.g., spot leases, options). This raises questions regarding how best to coordinate the use of these transfers in meeting cost and reliability objectives. This paper combines a hydrologic-water...

The Ornstein-Zernike (OZ) equations (1) can be used to nd probability distributions of atoms in uid states where the primary unknowns are the radial pair correlation function and the direct correlation function. In order to specify conditions of the state, certain parameters such as density and temperature are chosen. While one can nd a single solu...

Rosenbrock methods are popular for solving stiff initial value problems for ordinary differ- ential equations. One advantage is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as backward difference formulas and implicit Runge-Kutta methods. In this paper, we introduce some trust...

The traditional implementation of resonant tunneling diodes (RTD) as a high-frequency power source always requires the utilization of negative-differential resistance (NDR). However, there are inherent problems associated with effectively utilizing the two-terminal NDR gain to achieve significant levels of output power. This paper will present a ne...

Pseudo-transient continuation is a Newton-like iterative method for computing steady-state solutions of differential equations in cases where the initial data are far from a steady state. The iteration mimics a temporal integration scheme, with the time step being increased as steady state is approached. The iteration is an inexact Newton iteration...

The goal of a hydraulic capture model for remediation purposes is to design a well field so that the direction of groundwater flow is altered, thereby halting or reversing the migration of a contaminant plume. Management strategies typically require a well design that will contain or shrink a plume at minimum cost. Objective functions and constrain...

Non-Lipschitz continuous nonlinearities arise frequently in models for groundwater flow and species transport. The van Genuchten and Mualem PSK relations for unsaturated flow and the Freundlich equilibrium expressions in reactive transport are examples. Numerical methods such as nonlinear solvers based on Newton's method, error estimators for diffe...

Resonant tunneling diodes (RTDs) are ultra-small semiconductor devices that have potential as very high fre-quency oscillators. To describe the electron transport within these devices, the Wigner-Poisson Equations are used. These equations incorporate quantum mechanics to describe how the electron distribution changes in time due to kinetic energy,...

This paper is concerned with the algorithmic behavior of the DIRECT (DIviding RECTangles) algorithm. We show that DIRECT is sensitive to additive scaling, and this sensitivity can affect convergence. We present a modified version of the algorithm, and illustrate the effectiveness of our modification with numerical results.

We report on a multilevel method for the solution of Ornstein-Zernike equations and related systems of integro-algebraic equations. Our approach is based on an extension of Atkinson-Brakhage method, with Newton-GMRES method used as the coarse mesh solver. We report on several numerical experiments to illustrate the effectiveness of the method. The...

Problems involving the management of groundwater resources occur routinely, and management decisions based upon optimization approaches offer the potential to save substantial amounts of money. However, this class of application is notoriously difficult to solve due to non-convex objective functions with multiple local minima and both nonlinear mod...

We study how the Newton-GMRES iteration can enable dynamic simulators (time-steppers) to perform fixed-point and path-following computations.For a class of dissipative problems, whose dynamics are characterized by a slow manifold, the Jacobian matrices in such computations are compact perturbations of the identity. We examine the number of GMRES it...

The DIRECT algorithm is a deterministic sampling method for bound constrained Lipschitz continuous optimization. We prove a subsequential convergence result for the DIRECT algorithm that quantifles some of the convergence observations in the literature. Our results apply to several variations on the original method, including one that will handle g...

This paper presents theoretical results on instability processes that occur in the positive-differential-resistance region of nanoscale tunneling structures and reports on efforts to development advanced numerical techniques for use in future optimization studies. These results were obtained from numerical implementations of the Wigner-Poisson elec...