Radu Serban

SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers

SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.

Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution

An adjoint sensitivity method is presented for parameter-dependent differential-algebraic equation systems (DAEs). The adjoint system is derived, along with conditions for its consistent initialization, for DAEs of index up to two (Hessenberg). For stable linear DAEs, stability of the adjoint system (for semi-explicit DAEs) or of an augmented adjoint system (for fully implicit DAEs) is shown. In addition, it is shown for these systems that numerical stability is maintained for the adjoint system or for the augmented adjoint system.

CVODES: The Sensitivity-Enabled ODE Solver in SUNDIALS

CVODES, which is part of the SUNDIALS software suite, is a stiff and nonstiff ordinary differential equation initial value problem solver with sensitivity analysis capabilities. CVODES is written in a data-independent manner, with a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods. It shares with the other SUNDIALS solvers several common modules, most notably the generic kernel of vector operations and a set of generic linear solvers and preconditioners. CVODES solves the IVP by one of two methods — backward differentiation formula or Adams-Moulton — both implemented in a variable-step, variable-order form. The forward sensitivity module in CVODES implements the simultaneous corrector method, as well as two flavors of staggered corrector methods. Its adjoint sensitivity module provides a combination of checkpointing and cubic Hermite interpolation for the efficient generation of the forward solution during the adjoint system integration. We describe the current capabilities of CVODES, its design principles, and its user interface, and provide an example problem to illustrate the performance of CVODES.Copyright

Sensitivity analysis of differential-algebraic equations and partial differential equations

Sensitivity analysis generates essential information for model development, design optimization, parameter estimation, optimal control, model reduction and experimental design. In this paper we describe the forward and adjoint methods for sensitivity analysis, and outline some of our recent work on theory, algorithms and software for sensitivity analysis of differential-algebraic equation (DAE) and time-dependent partial differential equation (PDE) systems.

Halo orbit mission correction maneuvers using optimal control

This paper addresses the computation of the required trajectory correction maneuvers for a halo orbit space mission to compensate for the launch velocity errors introduced by inaccuracies of the launch vehicle. By combining dynamical systems theory with optimal control techniques, we are able to provide a compelling portrait of the complex landscape of the trajectory design space. This approach enables automation of the analysis to perform parametric studies that simply were not available to mission designers a few years ago, such as how the magnitude of the errors and the timing of the first trajectory correction maneuver affects the correction @DV. The impetus for combining dynamical systems theory and optimal control in this problem arises from design issues for the Genesis Discovery Mission being developed for NASA by the Jet Propulsion Laboratory.

Error Estimation for Reduced-Order Models of Dynamical Systems

The use of reduced-order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most important, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error, while the derived bounds are within an order of magnitude.

Computational Algorithm for Dynamic Optimization of Chemical Vapor Deposition Processes in Stagnation Flow Reactors

There are potentially great benefits to developing materials processes that deliberately vary process conditions such as temper ature, or flow rates, during the course of the process. Transient processing holds the promise of reducing manufacturing cost and the possibility of producing material systems that would be infeasible to manufacture with steady processes. Once the notion of tra nsient processing is embraced, there is a need and opportunity to develop optimal trajectories through which the process will pro ceed. In this paper, a stagnation flow dynamic optimization algorithm for two chemical vapor deposition processes is demonstrated. The first example seeks to control film composition during the deposition of yttrium-barium-copper oxide films, in which a wafer temperature transient is imposed. Transient trajectories of precursor flow rates are determined by optimization, so that t he correct flux ratios of yttrium, barium, and copper atoms to the surface are maintained. The second example determines trajectorie s that minimize the cost associated with multiple competing objectives during the deposition of a copper film. Time varying traje ctories of copper precursor concentration and the inlet flow velocity are computed so as to minimize a composite cost function t hat considers precursor utilization and process throughput.

Kinematic and Kinetic Derivatives in Multibody System Analysis

Abstract Analytical formulas for kinematic and kinetic derivatives needed in multibody system analyses are derived. A broad spectrum of problems, including implicit numerical integration, dynamic sensitivity analysis, and kinematic workspace analysis, require evaluation of first derivatives of generalized inertia and force expressions and at least three derivatives of algebraic constraint functions. In the setting of a formulation based on Cartesian generalized coordinates with Euler parameters for orientation, basic identities are developed that enable practical and efficient computation of all derivatives required for a large number of multibody mechanical system analyses. The formulation is verified through application to a spatial slider crank mechanism and a 14 body vehicle model. Efficiency of computation using the expressions derived is compared with results obtained employing finite differences, showing significant computational advantage using the analytically derived expressions.

A Topology-Based Approach for Exploiting Sparsity in Multibody Dynamics in Cartesian Formulation*

In the present paper a new approach for solving for the accelerations and Lagrange multipliers when integrating multibody systems in descriptor form is given. When solving for these unknowns, the coef ficient matrix of the linear system to be solved has the particular form of an optimization matrix. Due to the conectivity among the bodies comprised within the system this matrix is likely to have a lar ge number of zero entries. Advantage is taken of both the special structure and the sparsity of the coef ficient matrix. Simple manipulations bring the

COOPT — a software package for optimal control of large-scale differential—algebraic equation systems

This paper describes the functionality and implementation of COOPT. This software package implements a direct method with modified multiple shooting type techniques for solving optimal control problems of large-scale differential–algebraic equation (DAE) systems. The basic approach in COOPT is to divide the original time interval into multiple shooting intervals, with the DAEs solved numerically on the subintervals at each optimization iteration. Continuity constraints are imposed across the subintervals. The resulting optimization problem is solved by sparse sequential quadratic programming (SQP) methods. Partial derivative matrices needed for the optimization are generated by DAE sensitivity software. The sensitivity equations to be solved are generated via automatic differentiation.