I

THE WORKSHOP ON ITERATIVE METHODS FOR

LARGE SCALE NONLINEAR PROBLEMS TE-FC-03- q5EK 25 255

Homer F. Walker Department of Mathematics and Statistics

Utah State University Logan, Utah 84322-3900

Michael Pernice Utah Supercomputing Institute

University of Utah, Salt Lake City, UT 84112

12/95/84

Research Report 12/95/84, Department of Mathematics and Statistics, Utah State University, December, 1995.

DISCLAIMER

Portions of this document may be iliegible in electronic image products. Images are produced from the best available original document.

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof. nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any spe- cific commercial product, process, or service by trade name, trademark, manufac- turer, or otherwise does not necessarily constitute or imply its endorsement, Ttcom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not neccssarily state or reflect those of the United States Government or any agency thereof.

T h e Workshop o n Iterative Methods for Large Scale Nonlinear Problems

The Workshop on Iterative Methods for Large Scale Nonlinear Problems was held September 14-16, 1995 on the campus of Utah State University, Logan, Utah. The aim of the workshop was to bring together researchers working on large scale applications with numerical specialists of various kinds. Applications that were addressed included reactive flows (combustion and other chemicaJly reacting flows, tokamak modeling), porous media flows, cardiac modeling, chemical vapor deposition, image restoration, macromolecular modeling, and population dynamics. Numerical areas included Newton iterative (truncated Newton) methods, Krylov subspace methods, domain decomposition and other preconditioning methods, large scale optimization and optimal control, and parallel implementations and software.

There were 34 participants. Of these, 25 were from universities and 9 were from national laboratories. There were 10 graduate students and 3 postdoctoral associates as well as 21 more senior investigators. The informal workshop atmosphere, with ample free time and no concurrent talks, was intended to promote the easy exchange of ideas among these diverse participants.

This report offers a brief summary of workshop activities and information about the partici- pants. Interested re rs are encouraged to look into an online proceedings available at

http://wwrr.usi.utah.edu/loganqroceedings

In this, the material offered here is augmented with hypertext abstracts that include links to locations such as speakers’ home pages, PostScript copies of talks and papers, cross-references to related talks, and other information about topics addressed at the workshop.

Special thanks are extended to the following for their generous financial support, without which the workshop could not have taken place: the U.S. Department of Energy, the National Science Foundation, the Utah State University Office of Research, and the Utah Supercomputing Institute.

Homer Walker and Michael Pernice Conference Organizers

Contents

page Workshop schedule ............................................................ 2 Titles and abstracts ........................................................... 4 List of participants and e-mail addresses ...................................... 17 Mailing addresses of participants ............................................. 18

1

WORKSHOP ON ITERATIVE M E T H O D S for

LARGE SCALE NONLINEAR PROBLEMS

SCHEDULE

Wednesday, September 13 6:OO - 1O:OO Social, Room 302, University Inn

Thursday, September 14

8 :OO-9 : 30 8: 00-8 : 20

8:20 8:30 9:oo 9:30

1O:OO-10:30

10:30 11:oo 11:30

12100-2:45

2:45

3:15

3~45-4:OO

4:OO 4:30

5100 - 6:OO

6:OO

Late registration Coffee, juice, pastry

Session 1, Room 307-309 ECC*, Homer Walker, chair Welcoming remarks, Dr. Peter Gerity, USU Vice President for Research James Keener, Cardiac Arrhythmias Juan C. Meza, Optimal Control of Chemical Vapor Deposition Reactors John N. Shadid, Solution of Complex Chemically Reacting Flows on MP machines

Break Session 2, Room 307-309 ECC*, Juan C. Meza, chair

Philip J. Smith, ation for Non-Linear Methods from Cambustion Applications Michael Pernice, Nonlinear Iterative Methods for Chemically Reacting Flows Lois Curfman McInnes, Software for the Scalable Solution of Nonlinear Equations

Break

C. T. Kelley, Accurate and Economical Solution of the Pressure Head Form of Richards' Equation by the Method of Lines Hector Klie, Krylov-Secant Preconditioners

Break Session 4, Room 307-309 ECC*, Peter N. Brown, chair

Tony F. Chan, A Nonlinear Primal-Dual Method for TV-based Image Restomtion Jorge J. Mor&, Smoothing Techniques for Macromolecular Global Optimization

Social, Room 302, University Inn

Banquet, USU Alumni House After-dinner address, James A. MacMahon, Dean, USU College of Science

Session 3, Room 307-309 ECC', Michael Pernice, chair

~

*Eccles Conference Center

2

Friday, September 15 8:OO-8:30 Coffee, juice, pastry

Session 5, Room 307-309 EGG*, Jorge J. Mor&, chair 8:30

9:00

9:30

Peter White, Dynamics of Mass Attack, Spatial Invasion of Pine Beetles into Lodgepole Forests: A Numerical Study Matthias Heinkenschloss, The Design of SQP Algorithms for the Solution of Optimal Control Problems Luis Vicente, Analysis of Inexact SQP Algorithms

1O:OO-10:30 Break Session 6, Room 307-309 EGG*, C. T. Kelley, chair

10:30 David E. Keyes, Newton-Krylov-Schwarz: An Implicit Solver for CFD Applications 11:OO Dana A. Knoll, Nonlinear Iterative Methods Applied to the Tokamak Edge Plasma Fluid

Equations 11:30 Peter N. Brown, Preconditioned Krylov Methods in Tokamak Edge Plasma Modeling

12:OO-4:15 Break 4A5-5:15 Poster session, Guinavah-Malibu picnic ground, Logan Canyon

Kristina Bogar, Numerical Methods for the Propagation of Action Potentials in Cardiac T - zssue Paul R. McHugh, Application of Newton-Krylov Algorithms in Computational Fluid Dynamics Carol A. San Soucie, A Two-Grid Finite Difference Scheme for Nonlinear Parabolic Equations Barry Smith, More Details on PETSc 2.0 Ray S. Tuminaro, AZTEC: A Parallel Itemtive Package for Solving Linear Systems Arising in Newton-Krylov Methods Silvia Veronese, Computational Methods for Non-linear Waves in Anisotropic Reacting Media Homer Walker, A Globalized Newton-Krylov Method for Reacting Flows in SALSA

5:30 Cookout avah-Malibu picnic ground, Logan Canyon

Saturday, September 16 8: 154245 Coffee, juice, pastry

Session 7, Room 307-309 ECC*, Raytcho Lazarov, chair 8:45 Xiao-Chuan Cai, Overlapping Schwarz Methods for the Implicit Solution of Compressible

Flow Problems on Unstructured Meshes 9:15 Tarek Mathew, Solution of Advection Diflusion and Parabolic Equations by the Schwarz

Alternating Method

9:4*5-10:15 Break Session 8, Room 307-309 EGG*, David E. Keyes, chair

10:15 10:45

11:30 - 12:30

Xuejun Zhang, Two Level Preconditioners for Elliptic Finite Element Problems Raytcho Lazarov, Domain Splitting Algorithm for Mixed Finite Element Approxima- tions Buffet lunch, West Colony Room, Taggart Student Center

*Eccles Conference Center

3

WORKSHOP ON ITERATIVE METHODS for

LARGE SCALE NONLINEAR PROBLEMS

TITLES A N D ABSTRACTS

The following are in alphabetical order of presenting authors (underlined).

Kristina Bogar, Department of Mathematics, University of Utah, Salt Lake City, Utah, 84112, bogar@math.utah. edu

A Numerical Study of Implicit Explicit Methods for Nonlinear Parabolic Wave Propagation Problems

In this study, a numerical scheme €or nonlinear parabolic wave propagation is sought which is computationally efficient without compromising accuracy. The models considered in this study consist of a system of coupled nonlinear parabolic equations and ODEs. Explicit methods have frequently been used to solve these systems since they are easy to implement. However, due to stability problems with explicit methods, severe time step restrictions are imposed. To d o w for larger time steps, implicit methods have been u . In the context of nodinear parabolic wave propagation, these methods require that a nonlinear system of equations be solved for each time step. The decrease in CPU time, resulting from a larger time step, may not compensate for the extra work required to solve this nonlinear system. Due to these concerns, hybrid methods, referred to as implicit-explicit (IMEX) schemes, have been proposed. With an IMEX method, an implicit scheme is used for the diffusion operator and an explicit scheme is used for the time discretization. These methods, in contrast to the fully implicit methods, require a linear system instead of a nonlinear system be solved at each time step which leads to an increase in computational speed over a fully implicit method. IMEX methods also allow for larger time steps than the explicit methods. Such methods have been used in conjunction with spectral methods for fluid flow problems and recently in reaction diffusion problems for pattern formation. For the problems in this study, an IMEX scheme is used for the parabolic equations and the variables governed by the ODEs are updated with an implicit scheme. A comparison between the fully implicit scheme and the implicit-explicit scheme in one spatial dimension will be presented. The two dimensional case is under investigation.

4

Peter N. Brown, Center for Computational Science and Engineering, L-316, Lawrence Livermore National Laboratory, P. 0. Box 808, Livermore, California 94551, pnbrownQlln1 .gov -

Preconditioned Krylov Met hods in Tokamak Edge Plasma Modeling

Incomplete factorization techniques are used as preconditioners in an iterative solution of the linear systems arising in the solution of Tokamak edge plasma models. Newton-Krylov methods are used as the main nonlinear iteration, enhanced by global convergence strategies. Various reordering strategies of the linear equations and unknowns are used to reduce fill-in and improve the overall effectiveness of the incomplete factorization techniques. In addition, implicit scaling of the nonlinear equations and unknowns is used to enhance the robustness of the linear and nonlinear iterations.

- Xiao-Chuan Cai and Marcus Sarkis, Department of Computer Science, University of Coforado a t Boulder, Boulder, Colorado 80309, caiOcs . Colorado. edu, msarkisQcs . Colorado. edu

Overlapping Schwarz Methods for the Implicit Solution of Compressible Flow Problems o n Unstructured Meshes

In this presentation, we discuss some recent results on the use of overlapping Schwarx methods for the numerical solution of unsteady Navier-Stokes equations. The partial differential equations are discretized by combined finite element/finite volume methods on unstructured meshes. The discrete algebraic equations are then solved, at every time step, by overlapping Schwarz precondi- tioned Krylov space methods. We shall report several simulations €or problems defined in complex geometries, and €or both subsonic and transonic flows.

- Tony F. Chan',Gene H. Golub2, and Pep Muletlb, 'Department of Mathematics, UCLA, 405 Hifgard Avenue, Los Angefes, California 90024, Department of Computer Science, Stanford University, Stanford, California 94305, and University of V'encia, Spain, chanOmath .ucla. edu

A Nonlinear Primal-Dual Method for TV-based Image Restoration

During some phases of the manipulation of an image some random noise is usually introduced. The presence of this noise makes difficult and inaccurate the latter phases of the image processing.

The algorithms for noise removal have been mainly based on least squares. The output of these Lz-based algorithms will be a continuous function, which cannot obviously be a good approximation to our original image if it contains edges. To overcome this difficulty a technique based on the minimization of the Total Variation norm subject to some noise constraints is proposed in Rudin, Osher and Fatemi 92, where it is also proposed a time marching scheme to solve the associated Euler-Lagrange equations.

One of the difficulties of solving the Euler-Lagrange equations is the presence of a highly nclnlinear term, which causes convergence difficulties for Newton's method even when combined

5

with a globalization technique such as a line search. We propose here an algorithm based on Newton’s method that borrows some ideas from primal-dual optimization methods of Conn and Overton 94 and Anderson 94 and that has proved to be much more robust and efficient than the usual implementation of Newton’s method.

William Gropp, Lois Curfman McInnes, and Barry Smith, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4844, groppamcs . an1 .gov , curfmanQmcs.anl.gov, bsmith@mcs.anl.gov

Software for the Scalable Solution of Nonlinear Equations

The portability, versatility, and scalability of traditional software for the solution of large- scale systems of nonlinear equations and unconstrained minimization problems are restricted when considering parallel architectures. For example, most software for the solution of nonlinear (and linear) systems of equations has been written in the form of subroutine libraries using particular data structures for the solution of a single problem. However, modern PDE solvers for large-scale applications typically require the solution of one or more nonlinear systems at each time step, and efficient performance often requires the reuse of data structure information from previous time steps or linear solves.

PETSc, the Portable, Extensible Toolkit for Scientific computation, is a library for the numer- ical solution of partial differential equations on high-performance parallel (and sequential) comput- ers. The package has been completely redesigned, incorporating many extensions, revisions, and additions to previous versions. The basic “model problems” for PETSc are now time-dependent , multicomponent, nonlinear PDEs. The nonlinear solvers within PETSc are built in an object- oriented framework that reuses data structure information whenever possible, thereby enabling more efficient solution than is possible by conventional means. In addition, our design enables easy experimentation with Newton-Krylov variants, data structures, convergence strategies, etc.

Our presentation will focus on the PETSc design philosophy and its benefits in providing a uniform and versatile framework for developing optimization software and solving large-scale nonlinear problems.

6

- Matt hias Heinkenschloss, Interdisciplinary Center for Applied Mathematics and Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 - U531, heinkenQmath . v t . edu

T h e Design of SQP Algorithms for t h e Solution of Optimal Control Problems

The structure of optimal control problems arising from the partioning of variables into states and controls makes SQP methods an attractive tool for the solution of these large scale problems. However, the design of SQP algorithms that are usable and effective has to allow for user specified, problem dependent linear system solvers and has to incorporate the scaling of the problem, which is iilsually determined by the infinite dimensional formulation of the optimal control problem. A proper handling of the scaling avoids artificial ill-conditioning and is the foundation of mesh in- dependent convergence behavior. This talk will introduce a class of trust-region interior-point sequential quadratic programming algorithms for the solution of minimization problems with non- linear equality constraints and simple bounds on some of the variables. The algorithms keep strict feasibility with respect to the bound constraints, allow the user to provide the linear or nonlinear algebra routines to handle the equality constraints, and use trust-region techniques to ensure global convergence.

James - Keener, Department of Mathematics, University of Utah, Salt Lake City, Utah, 84112, keanerQmath .ut&. edu

Cardiac Arrhythmias

I will give a brief introduction to some of the physiological, mathmematical and numerical issues involved in understanding the evolution, subsequent dynamics and termination of fatal cardiac arrhythmias.

- C. T. Kelley, Department of Mathematics, Box 8205, North Carolina State University, Raleigh, North Carolina 27695-8205, TimKelley@ncsu . edu

Accurate and Economical Solution of t h e Pressure Head Form of Richards’ Equation by the Method of Lines

The pressure head based form of Richards’ equation is difficult to solve accurately using stan- dard time integration methods, one symptom being mass balance errors that grow as the integration progresses. A differential algebraic equations implementation of the method of lines, using DASSL, can give solutions that are accurate, have good mass balance properties, and are competitive in coat with standard approaches. Our approach requires modification of the internal iterations in the integrator to take ill-conditioning and sharp moving fronts into account. Our presentation will focus on these modifications and proposed methods to turn them on and off as the steep fronts pass through the region of interest.

7

David E. Keyes, Computer Science Department, Old Dominion University, Norfolk, VA 23519- 0162, keyesOcs . odu. edu, and Senior Research Associate, ICASE, MS 132C, NASA Langley Re- search Center Hampton, Virginia 23681-0001, keyesQicase. edu

Newton-Krylov-Schwarz: A n Implicit Solver for CFD Applications

Newton-Krylov methods and Krylov-Schwarz (domain decomposition) methods have begun to become established in computational fluid dynamics (CFD) over the past decade. The former employ a Krylov method, such as the generalized minimal residual method, inside of Newton’s method in a Jacobian-free manner, through directional differencing. The latter employ overlapping Schwarz-type decomposition to derive a preconditioner for the Krylov accelerator that relies primar- ily on local information, for parallelism. They may be composed as Newton-Krylov-Schwarz (NKS) methods, which seem particularly well suited for solving nonlinear elliptic systems in high-latency distributed-memory environments.

We describe recent numerical simulations with Newton-Krylov-Schwarz methods in CFD car- ried out at ICASE/NASA Langley, emphasizing the effect of a coarse grid, the preconditioning of a higher-order discrete operator with a lower-order discrete operator, and comparisons with multigrid and standard defect-correction approaches.

HQctor KlieI, Marcelo Ramkl , and Mary F. Wheeler2, Computational and Applied Math- ematics Department, MS 134, Rice University, P. 0. Box 1892, Houston, Texas 77251-1892, and

Texas Institute for Computational and Applied Mathematics, TAY 2.400, University of Texas a t Austin, Austin, Texas 78712, klieQmasc6.rice.edu,

Krylov-Secant Preconditioners

Secant methods have been traditionally conceived as an alternative way to solve nonlinear equations at a lower computational cost. On the other hand, inexact Newton solvers have been increasing in popularity to tackle large scale problems. Due to its robustness, GMRES has been the method of choice in most implementations and the basis of theoretical analysis of local and global convergence of inexact Newton methods. In this work we combine secant methods and inexact Newton methods to generate a new family of preconditioners for large scale problems. Our approach exploits the current Krylov basis generated by GMRES as useful information in forthcoming Newton iterations. The strategy relies on rank-one updates to the Hessenberg matrix that results from the application of the Arnoldi process. In this way we reproduce, but not explicitly compute, the corresponding secant update of the current Jacobian matrix. In order to improve the effectiveness of the method we incorporate recent advances in the solution of nonsymmetric systems with multiple right-hand sides to accomodate the value of the function at new points leading to the solution of the nonlinear system of equations. Preliminary computational experiments show encouraging results in the application of our preconditioners to model nonlinear problems and in the context of large multiphase problems in reservoir simulation.

8

Dana A. Knoll and Paul R. McHugh, Idaho National Engineering Laboratory,P.O. Box 1625, Idaho Falls, Idaho 83415-3808, nolainel . gov

Nonlinear Iterative Methods Applied to the Tokamak Edge Plasma Fluid Equations

The tokamak edge plasma fluid equations are a highly nonlinear system of two-dimensional coiivection-diffusion-reaction partial differential equations that describe the boundary layer of a tokamak fusion reactor. These equations describe charged particle mass, momentum, and energy tramport from the reactor core to vessel structure. They are characterized by multiple time and spatial scales, and have highly anisotropic transport coefficients. We use Newton% method to lin- earize the nonlinear system of equations resulting from the implicit, finite volume discretization of the governing partial differential equations. The resulting linear systems are neither symmet- ric nor positive definite, and are poorly conditioned. Preconditioned Krylov iterative techniques are employed to solve these linear systems. We have investigated both standard and matrix-free Newton-Krylov implementations. Additionally, domain decomposition preconditioners, namely ad- ditive and multiplicative Schwarz methods, have been investigated and compared with Incomplete Lower-Upper factorization (ILU) preconditioning. We will present results based on algorithmic research over the past four years. Our Newton-Krylov algorithm, including global convergence techniques, will be outlined. Performance comparisons of different Krylov algorithms and different prcaconditioners will briefly be discussed. The majority of the presentation will be devoted to global convergence issues. Currently our global convergence techniques are a combination of damped it- eration, mesh sequencing, pseudo-transient continuation, and physics parameter continuation. We will discuss our implementation of these ideas and provide performance results.

Hongsen Chen and Raytcho Lazarov, Institute for Scientific Computation, Texas A&M Uni- versity, College Station, Texas 77845, Raytcho . LazarovcPmath. t a u . edu

Domain Splitting Algorithm for Mixed Finite Element Approximations to Parabolic Problems

We consider domain splitting techniques for mixed finite element approximations to linear paxabolic initial-boundary value problems. In contrast with the usual overlapping domain deco- mosition method this technique leads to noniterative algorithms, i.e., the subdomain problems are solved independently and the solution in the whole domain is obtained by restriction of the subdo- main solutions and averaging along the common boundaries. This method uses essentially the fact that after discretizing the time variable, an elliptic problem with large coefficient in front of the zero order term is obtained. The solutions of such problems exhibit boundary layer with thickness proportional to the square root of the time discretization parameter. Intuitively, any error in the boundary conditions will decay exponentially and, therefore, a reasonable overlap will produce a sufficiently accurate method. We prove that such splitting method is stable in L2-norm and has the same accuracy as the global method. Extensions to nonlinear problems are suggested and discussed. Keywords Parabolic problem, mixed finite element method, domain decomposition, error esti- mate, stability S ubject classification (AMS/MOS) 65M55,65N307 65Y05

9

Tarek Mathew, Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071- 3036,

Solution of Advection Diffusion and Parabolic Equations by t he Schwarz Alternating Method

We first study the solution of two types of singular perturbation advection diffusion equations by the Schwarz alternating method. The first singular perturbation equation corresponds to an advection dominated equation, while the second one corresponds to an identity dominated equation. For both of these problems, for a suitable choice of subdomains, the Schwarz method can be shown to converge at a rate independent of the diffusion, mesh size and other parameters.

As an application, we study the solution of implicit discretizations of parabolic equations by the Schwarz alternating method. We show that provided the subdomains have overlap of size T P + ~ / ~ , where T is the time step, then one iteration of the Schwarz method is sufficient per time step to maintain the accuracy and stabilty of the original scheme. The convergence of the modified scheme is examined in the maximum norm for non-self-adjoint problems, and in an energy norm for self-adjoint problems.

P a u l R. McHugh and Dana A. Knoll, Idaho National Engineering Laboratory, P.O. Box 1625, Idaho Falls, Idaho 8341 5-3808, ughQinel . gov

Application of Newton-Krylov Algorithms in Computational Fluid Dynamics

Newton-Krylov algorithms are investigated for solving strongly coupled, nonlinear systems of partial differential equations arising in the field of computational fluid dynamics. Primitive variable forms of the steady incompressible and compressible Navier-Stokes and energy equations that describe the flow of a laminar Newtonian fluid in two-dimensions are specifically considered. Both reacting and non-reacting low Mach number compressible flow applications are studied. Nu- merical solutions are obtained by first integrating over discrete finite volumes that compose the computational mesh. The resulting system of nonlinear algebraic equations are linearized using Newton’s method. Preconditioned Krylov subspace based iterative algorithms then solve these lin- ear systems on each Newton iteration using inexact Newton convergence criteria. Selected Krylov algorithms include the Arnoldi-based Generalized Minimal RESidual (GMRES) algorithm, and the Lanczos-based Conjugate Gradients Squared (CGS), Bi-CGSTAB, and Transpose-Free Quasi- Minimal Residual (TFQMR) algorithms. Both Incomplete Lower-Upper (ILU) factorization and domain-based additive and multiplicative Schwarz preconditioning strategies are employed. Nu- merical techniques such as mesh sequencing, adaptive damping, pseudo-transient relaxation, and parameter continuation are used to improve the solution efficiency, while algorithm implementation is simplified using a numerical Jacobian evaluation. The capabilities of standard Newton-Krylov algorithms are demonstrated via solutions to both incompressible and compressible flow problems. Incompressible flow problems include natural convection in an enclosed cavity, and mixed/forced convection past a. backward facing step. Compressible flow problems include low Mach number flow past a backward facing step and combusting flow through a channel. Additionally, matrix-free

10

Newton- K r ylov implement ations are constructed by approximating the Jacobian- vec tor products appearing in the Krylov algorithms with finite difference approximations (i.e. nonlinear Krylov iteration). Performance of the matrix-free implementation is found to depend upon problem size, problem nonlinearity, and Krylov algorithm selection. Practical matrix-free Newton-Krylov imple- mentations are demonstrated in solving the low Mach number combustion model problem, during which expensive preconditioner evaluations are amortized over many Newton steps.

_. J. C. Meza a n d T. D. Plantenga, Sandia National Laboratories, P.O. Box 969, MS 9214, Livermore, California 94551-0969, meza(9ca. sandia . gov

Optimal Control of Chemical Vapor Deposition Reactors

The optimal design and control of a chemical vapor deposition (CVD) reactor is an important issue in the semiconductor industry. In the optimal control problem, the god is to determine a set of parameters that will yield a target temperature distribution in a CVD reactor under a given set of constraints. Through the use of thermal analysis simulation codes it is easy to predict the temperature distributions inside a CVD reactor for a given set of model parameters. In this talk, we examine the issues of using optimization methods for the optimd control of CVD reactors described by such analysis codes in which the objective function is expensive, noisy, and derivatives are not readily available. We will present numerical results from several models that indicate that this approach can yield excellent predictions.

-

- Jorge J. Mor6 and Zhijun Wu, Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, more(9antares . m c s . a n 1 . gov

Smoothing Techniques for Macromolecular Global Optimization

We discuss recent developments on the solution of global optimization problems that arise in ma,cromolecular modelling. Our approach is based on transforming the function into a smoother function with fewer minimizers, and using an optimization algorithm on the transformed func- tions, tracing the minimizers back to the original function. We motivate and define the Gaussian transform, and outline some of the basic properties of this transformation.

We consider two different dasses of problems: distance geometry problems that arise in the interpretation of NMR data and in the determination of protein structure, and molecular cluster problems that arise in the determination of stable configurations of ionic systems. We present numerical results obtained with the IBM SP parallel system at Argonne’s High-Performance Com- puting Research Facility.

-

11

S. Kumar’, M. Pernice2, R. Rawat’, P. Smith’, H. F. Walker3, and I;. Zhou2, ‘Department of Chemical and Fuels Engineering and Utah Supercomputing Institute, Univer- sity of Utah, Salt Lake City, Utah 84112, and 3Mathematics and Statistics Department, Utah State University, Logan, Utah 84322-3900, usimapOsneffels .us i . U t a h . edu

Nonlinear Iterative Methods for Chemically Reacting Flows

Simulation of chemically reacting flows presents significant challenges to computational sci- entists and engineers. Chemically reacting flows are governed by systems of nonlinear partial differentia3 equations that couple fluid dynamics with turbulent mixing and reaction models, com- plex chemical kinetic mechanisms, particle dispersion models, and radiative heat transfer. These simulations can be used by the chemical process industry to increase yield and reduce emissions in reactors such as process heaters, mixing tanks, and catalytic reactors. However the size and complexity of the problem limit the applicability of our current methods, motivating exploration of techniques that are more robust and better suited to large-scale parallelism.

In this talk we describe our initial experiences with a globalized inexact Newton-Krylov- Schwarz method on simple model problems, and efforts to extend the approach to efficiently handle three dimensional problems, Initial results of applying a globalized nonlinear Krylov accelerator to our existing method are also discussed.

Clint N. Dawson, Carol A. San Soucie, and Mary F. Wheeler, Texas Institute for Com- putational and Applied Mathematics, TAY 2.400, University of Texas at Austin, Austin, Texas 78712, carol@t icam. ut exas. edu

A Two-Grid Finite Difference Scheme for Nonlinear Parabolic Equations

We present a two level finite difference scheme for the appro-ation of the nonlinear heat equation. Discrete inner products and the lowest order Raviart-Thomas approximating space are used in the expanded mixed method in order to develop the scheme. Analysis of the scheme is given assuming an implicit time discretization. In this two level method, the full nonlinear problem is solved on a “coarse” grid of size H. The nonlinearities are expanded about the coarse grid solution and an appropriate interpolation operator is used to provide values of the coarse grid solution on a “fine” grid of size h. The resulting linear but nonsymmetric system is solved on the fine grid. Some a priori error estimates are derived which show that the discrete L”(L2) and L 2 ( H 1 ) errors are O(h2 + H4-d/2 + At), where d 2 1 is the spatial dimension. Extensions to this research are given for Richards’ equation, a nonlinear parabolic equation arising in flow through porous media.

12

- J. N. Shadid, H. K. Moffat, A. G . Salinger, K. D. Devine, S. A. Hutchinson, a n d G. L. Hennigan, Sandia National Laboratories, M S 1110, PO Box 5800, Albuquerque, New Mexico 87185, jnshadi@cs . sandia.gov

Solution of Complex Chemically Reacting Flows on MP machines

An understanding of the nonlinear interactions of momentum, heat, and mass transfer along with nonequilibrium chemical reactions is central to the design and optimization of chemical reactors and materials processing systems. Large changes in a system's response to a small change in operating conditions, problems with process scale-up, and unsteady dynamic behavior can all result form nonlinearity. An understanding of these interactions can be used to obtain stable operating parameters, improvements in design, and more reliable control systems.

Computational simulations can be used to obtain information about such systems. However these simulations require significant resources that are now only available on massively parallel (MP) computers. In this talk we will describe a new MP unstructured finite element reacting flow code, SALSA, which has been de oped to simulate laminar variable-density reacting flow problems. This code is based on fully licit time integration, an inexact Newton method with backtracking and advanced preconditioned Krylov solvers. This talk will provide a brief overview of 'important solution strategies along with preliminary results on the simulation of chemical vapor deposition (CVD) reactors for production of Silicon Carbide (Sic) and Gallium Arsenide (GaAs).

This work was supported by the Office of Scientific Computing, U.S. Department of Energy and was performed at Sandia National Laboratories operated for the U.S. Department of Energy un,der contract No. DE-AC04-94AL85000.

William Gropp, Lois Curfman McInnes, and Barry Smith, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4844, groppQmcs . a n 1 . gov , curfmanQmcs . an1 . gov , bsmithQmcs . an1 . gov

More Details on PETSc 2.0

PETSc 2.0 provides parallel tools for the numerical solution of partial differential equations, including linear and nonlinear solvers. In this poster session we will extend the information present- ed in the talk "Software for the Scalable Solution of Nonlinear Equations." In particular, we will discuss the source code for a complete example that demonstrates our easy-to-use parallel sparse matrix assembly routines and distributed arrays.

13

Philip J. Smith, Department of Chemical and Fuels Engineering, University of Utah, Salt Lake City, Utah 84112, smithQopus .ut&. edu

Motivation for Non-Linear Methods from Combustion Applications

We have spent 20 years developing numerical combustion simulation tools for engineering applications to industrial problems. The development, initiated as a scientific and educational ex- perience, has progressed into a useful engineering tool which we have been using to directly assist in solving industrial problems. The applications have been in two areas: 1) engineering problem solving for existing equipment, 2) design or retrofit of new systems. This presentation will show practical examples of each type and identify how these applications have provided the motivation for further research in non-linear methods. These combustion applications couple multiple physical and chemical processes. Trade-offs are often made between numerical and model accuracy, solution robustness, computer storage requirements, and computational run times in order to obtain engi- neering solutions to the problem at hand. These trade-offs will be illustrated with specific examples. We are currently exploring ways of using these simulations for optimization and non-linear on-line computer control of combustion systems.

Scott A. Hutchinson, John N. Shadid, and Ray S . Turninaro, Sandia National Laborato- ries, MS 1110, PO Box 5800, Albuquerque, New Mexico 87185, tuminaroQcs .sandia.gov

AZTEC: A Parallel Iterative Package for Solving Linear Systems Arising in Newton-Krylov Methods

We describe a parallel linear system package, AZTEC, that is being used within a nonlinear finite element Newton-Krylov solver. The package incorporates a number of parallel iterative methods (eg. GMRES, biCGSTAB, CGS, TFQMR) and preconditioners (e.g. Jacobi, Gauss- Seidel, polynomial, domain decomposition with LU or ILU within subdomains). Additionally, AZTEC allows for the reuse of previous preconditioning factorizations within the Newton scheme. Currently, this package is being used to solve highly nonlinear chemically reactive flow problems.

In this poster we emphasize the parallel programming ease and the overall efficiency of AZTEC. Ease-of-use is attained using the notion of a global distributed matrix. The global distributed matrix allows a user to specify pieces (different rows for different processors) of his application matrix exactly as he would in the serial setting. Efficiency is achieved by using a transformation function which rewrites the user supplied matrix into one more convenient for efficient distributed memory computing (locally numbered entries, ghost variables, grouped messages). Additional performance is attained using efficient dense matrix algorithms for block sparse matrices.

AZTEC can be used on the Intel Paragon, nCUBE 2, IBM SP2, individual workstations (e.g. SUN and SGI) and uses the MPI message passing system.

14

- Silvia Veronese'*2 and Hans G. Othmer', 'Department of Mathematics and Utah Supercom- puting Institute, University of Utah, Salt Lake City, Utah, 84112, silviaQosiris.usi.utah. edu, othmerQmath . U t a h . edu

Computational Methods for Non-linear Waves in Anisotropic Reacting Media

A wide variety of problems in computational biology and engineering give rise to systems of non-linear parabolic equations or coupled parabolic and elliptic systems. Frequently the type of solutions sought are traveling waves. In the context of computational cardiodynamics the resolution needed to capture the important features of these propagating waves is of the order of a tenth of a millimeter on 2- and 3- dimensional domains spanning several centimeters. Although the theory of wave propagation in anisotropic media is reasonably well developed, little has been done on the development of efficient and scalable algorithms for computing such traveling waves.

We present several different algorithms for these problems. We describe a hybrid Alternating Direction Implicit (ADI)/ Multigrid algorithm for two-dimensional problems that involve coupled parabolic and elliptic systems and show that it scales optimally on shared-memory architectures. We also discuss a class of methods based on an AD1 splitting algorithm in 3 space dimensions tha.t has been successfully applied to model problems of wave propagation using FitzHugh-Nagumo dynamics. We present preliminary performance results for these algorithms and compare them with other methods.

_. Luis Vicente, Computational & Applied Mathematics Department, MS 234, Rice University, P.O. Box 1892, Houston, Texas 77251-1892,lvicenteQcaam.rice. edu

Analysis of Inexact SQP Algorithms

Sequential quadratic programming (SQP) algorithms for the solution of nonlinear program- ming problems require derivative evaluations, the solution of linearized constraint equations, and null space computations. For many engineering applications the constraints come from the dis- cretization of partial differential equations. In such cases the calculation of derivative information and the solution of linearized equations is the dominate cost of the procedure. Furthermore, we are interested in cases when it is impractical to assume, as conventional optimization codes do, that the optimization algorithm can specify how these computations axe done in order to incorporate the linear algebra procedures into the optimization algorithm. Often, the solution of linear systems and directional derivatives are computed inexactly, yielding nonzero residuals.

We address the effect of these sources of inexactness on the practical and theoretical global convergence of SQP algorithms using trust regions. We give practical rules to control the size of the residuals of these inexact calculations. We show that if the size of the residuals is of the order of both the size of the constraints and the trust-region radius, then the SQP algorithms are globally first-order convergent.

We show the impact of inexactness in different variants of SQP methods by reporting on numerical experiments with optimal control problems governed by nonlinear partial differential equations.

15

J o h n N. Shadid', Ray S . Tuminarol, and Homer F. Walker2, 'Sandia National Laborato- ries, MS 1110, PO Box 5800, Albuquerque, New Mexico 87185, and 2Mathematics and Statistics Department, Utah State University, Logan, Utah 84322-3900, walkeramath. usu. edu

A Globalized Newton-Krylov Method for Reacting Flows in SALSA

SALSA is a code for simulating chemically reacting flows on massively parallel computers. It can be used to solve either steady-state problems or time-dependent problems using fully implicit methods. In either case, very large nonlinear systems must be solved, and Newton-Krylov methods are natural solution methods. We decribe the implementation in SALSA of a particular Newton- Krylov method globalized by backtracking and give preliminary test results on two model problems.

Peter White, Mathematics and Statistics Department, Utah State University, Logan, Utah 84322- 3900, pwhite@math.usu.edu

Dynamics of Mass Attack, Spatial Invasion of Pine Beetles into Lodgepole Forests: A Numerical Study

The mountain pine beetle (MPB) is a major cause of destruction of valuable wood resources in the western states. Modeling the spatial redistribution of the MPB could be an important part of shaping forest management decisions in the future. A system of partial differential equations has been developed to model the spatial redistribution of MPD. In this talk a numerical scheme for solving this system will be presented. An Adams predictor-corrector algorithm used on both one and two space dimensions and a method for simulating "transparent" boundary conditions will be discussed.

Xuejun Zhang, Department of Mathematics, Texas A&M University, College Station, TX 77843, Xue jun . ZhangQmath . tarnu. edu

Two Level Preconditioners for Elliptic Finite Element Problems

In this talk, I will discuss how to construct preconditioners for an elliptic finite element problem which utilize preconditioners or solvers for simpler discretizations. An abstract theory for analyzing these methods will be presented. Applications of this technique to fourth order problems will be discussed. One consequence of the theory shows that the biharmonic equation discretized by one finite element can be preconditioned by any other element and both the approximation element and the preconditioning element can be nonconforming. I will concentrate on the application of the theory to the two-level additive Schwarz methods for the biharmonic conforming finite element problem.

16

List of participants and e-mail addresses:

1. Banoczi, Jim 2. Bogar, Kristina 3. Brown, Peter 4. Cai, Xaio-Chuan 5. Chan, Tony 6. Choi, Tony 7. Curfman McInnes, Lois 8. Heinkenschloss, Matthias 9. Keener, Jim

10. Kelley, Tim 11. Keyes, David 12. Klie, Hector 13. Knoll, Dana 14. Kumar, Seshadri 15. Lazarov, Raytcho 16. Mathew, Tarek 17. McHugh, Paul 18. Meza, Juan 19. Mod, Jorge 20. Pernice, Mike 21. Rawat, Rajesh 22. San Soucie, Carol 23. Sakis, Marcus 24. Shadid, John 25. Smith, Barry 26. Smith, Phil 27. Strong, David 28. Tocci, Mike 29. Tuminaro, Ray 30. Veronese, Silvia 31. Vicente, Luis 32. Walker, Homer 33. White, Peter 34. Zhang, Xuejun

jmbanocz@unity.ncsu.edu . bogar@math.utah.edu pnbrown@llnl.gov cai @ s chw ar z . cs .color ado.edu chan@mat h .ucla.edu t dchoi@unity.ncsu.edu curfman@mcs.anl.gov heinkenQmath.vt .edu keener@mat h.utah.edu TimXelley @ncsu.edu keyes@icase.edu klie@rice.edu nol@inel.gov skumar @opus .ut ah.edu raytcho.lazarov@math.tamu.edu mathewQledaig.uwyo.edu ugh@inel.gov meza@ca.sandia.gov more@antares.mcs.ad.gov usimap@sneffels.usi.utah.edu rawat @opus.utah.edu carol@ ticam.utexas.edu msarkis@cs.colorado.edu jnshadi@cs.sandia.gov bsmith@mcs.anl.gov smith@opus.utah.edu dstrong@math.ucla.edu mdtocci@unity.ncsu.edu rs tumin@cs .sandia.gov silvia@osiris.usi.utah.edu lvicent e@rice.edu walker@math.usu.edu pwhite@math.usu.edu xzhang@math.tamu.edu

17

Jim Banoczi Department of Mathematics

North Carolina State University Raleigh, NC 27695-8205

B~~ 8205

Kristina Bogar Department of Mathematics University of Utah Salt Lake City, U T 84112

Dr. Peter N. Brown Center for Computational Sci. & Eng.

Lawrence Livermore National Lab. P. 0. Box 808 Livermore, CA 94551

L-316

Professor Xiao-Chuan Cai Department of Computer Science University of Colorado Boulder, CO 80309-0430

Professor Tony F. Chan Department of Mathematics UCLA 405 Hilgard Ave. Los Angeles, CA 90024

Tony Choi Department of Mathematics Box 8205 North Carolina State University Raleigh. NC 27695-8205

Dr. Lois Curfman Mclnnes Math. & Comp. Sci. Division Argonne National Laboratory 9700 South Cass Avenue Argonne, IL 60439-mcinnes

Professor Mat t hias H ein kensch . ~ s s Department of Mathematics VPI & su Blacksburg, VA 24061-0123

Professor James Keener Department of Mathematics 102 Widstoe Building University of Utah Salt Lake City, UT 84112

Professor C. T. Kelley Department of Mathematics Box 8205 North Carolina State University Raleigh, NC 27695-8205

Professor David Keyes Department of Computer Science Old Dominion University Norfolk, VA 23529

Hector Klie Computational & Applied Math., MS 134 Rice University P. 0. Box 1892 Houston, TX 77251-1892

Dr. Dana Knoll Idaho National Engineering Laboratory P.O. 80x 1625. M.S. 3895 ’

Idaho Falls, ID 83415

Seshadri Kumar Chemical & Fuels Engineering Department 3290 Merrill Engineering Building University of Utah Salt lake City, Utah 84112

Professor R. D. Lazarov Department of Mathematics Texas A&M University College Station, TX 77843

Professor Tarek Mathew Department of Mathematics University of Wyoming Laramie. WY 82071-3036

18

Dr. Paul McHugh Idaho National Engineering Laboratory P.O. Box 1625, M.S. 3895 Idaho Falls, ID 83415

Dr. Juan C. Meza Sandia National Laboratories Center for Computational Engineering Division 8117 P.O. Box 969 Livermore, CA 94551-0969

Dr. John N. Shadid Sandia National Laboratory Dept. 1422. MS 1110 P.O. Box 5800 Albuquerque NM 87185-1110

Dr. Barry F. Smith MCS Division Argonne National Laboratory 9700 South Cass Ave. Argonne IL 60439-4844

Dr. Jorge J. Mor6 Applied Mathematics Argonne National Laboratory 9700 South Cass Avenue Argonne, IL 60439

Dr. Michael Pernice Utah Supercomputing Institute 85 Student Services Building University of Utah Salt Lake City, UT 84112

Professor Philip J. Smith Chemical & Fuels Engineering Department 3290 Merrill Engineering Building University of Utah Salt Lake City, Utah 84112

David Strong Department of Mathematics UCLA 405 Hilgard Ave. Los Angeles. CA 90024

Rajesh Rawat Mike Tocci Chemical & Fuels Engineering Department 3290 Merrill Engineering Building University of Utah Salt Lake City, Utah 84112

Department of Mathematics Box 8205 North Carolina State University Raleigh, NC 27695-8205

Carol San Soucie Texas Inst. for Comp. & Appl. Math. TAY 2.400 University of Texas at Austin Austin, TX 78712

Dr. Marcus Sarkis Department of Computer Science University of Colorado Boulder, CO 80309-0430

Dr. Raymond S. Tuminaro S andia National Laboratory Dept. 1422. MS 1130 P.O. Box 5800 Albuquerque NM 87185-1110

Dr. Silvia Veronese Utah Supercomputing Institute University of Utah Salt Lake City, UT 84112

19

Luis Vicente Computational & Applied Math., MS 134 Rice University P. 0. Box 1892 Houston. TX 77251-1892

Professor Homer F. Walker Mathematics and Statistics Department Utah State University Logan, UT 84322-3900

Professor Peter White Mathematics and Statistics Department Utah State University Logan, UT 84322-3900

Professor Xuejun Zhang Department of Mat hema tics Texas A&M University College Station, TX 77843