Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration

under contract DE-AC04-94AL85000.

NOX: An Object-Oriented Nonlinear Solver Package

Roger Pawlowski, Tamara Kolda, Russell Hooper, and John Shadid

Sandia National Laboratories Albuquerque, NM

Trilinos User Group MeetingOctober 16th, 2003

Background• NOX is a Trilinos solver package built to:

– enable robust and efficient solutions to systems of nonlinear equations.

– provide a broad array of nonlinear algorithms through a single interface.

– rapid deployment of new solver technology into ASCI codes.– eliminate redundancies in ASCI code development.

• Started ~2.5 years ago. • Funding: ASCI Algorithms and LDRD (2.5 FTE).• Consists of two libraries:

– NOX – nonlinear algorithms– LOCA v2.0 – continuation and bifurcation algorithms (Eric Phipps)

• Website: software.sandia.gov/nox

In-NOX-ulated Applications

Code Application NOX Contact Application ContactXyce Electrical Circuit Simulation Kolda & Pawlowski Scott HutchinsonMPSalsa Chemically Reacting Flows Pawlowski Roger PawlowskiFEAP Solid Mechanics Hooper Mark AdamsCharon General PDE Solver Pawlowski Gary HennigPremo Compressible Flows Hooper & Pawlowski Tom SmithSIERRA Framework Hooper Alan Williams

Code Application NOX Contact Application ContactSalinas Solid Mechanics Hooper Kendall PiersonAdagio Solid Mechanics Hooper John MitchellALEGRA Radiation Transport Pawlowski Tom BrunnerGOMA Incompressible Flows Pawlowski Sam SubiaNEVADA Framework Pawlowski Tom Brunner

Fully Interfaced

Prototype Interface

Primary NOX Team: Tammy Kolda, Roger Pawlowski, Russ Hooper

Nonlinear Equations

Given

find for which

• F is a system of n nonlinear equations with n unknowns.

• x is the unknown or solution vector of size n.

MB f xc Bcd+=

Broyden’s Method

Newton’s MethodMN f xc

Jc d+=

Tensor Method MT f xc

Jcd 12---Tcdd+ +=

Nonlinear Solution Algorithms

Iterative Linear Solvers, Adaptive Forcing Terms

Line SearchInterval Halving

QuadraticCubic

More’-Thuente

HomotopyArtificial Parameter ContinuationNatural Parameter Continuation

Trust RegionDogleg

Inexact Dogleg

Globalizations

User Norms and Merit Functions

Stopping Criteria(Status Test)

Example: Newton’s Method for F (x) = 0

• Choose an initial guess x0

• For k = 0,1,2,...– Compute Fk = F (xk)– Compute Jk where

(Jk )ij = F i(xk)/x j

– Let dk = -Jk-1 Fk

– (Optional) Let k be a calculated step length

– Set xk+1 = xk + kdk

– Test for Convergence or Failure

Calculatingthe Direction

Damping orLine Search

Iterate Control(Solver)

Building Blocks of NOX

NOX Solvers(Iterate Control)

1. LineSearchBased • Compute Direction• Compute Step Length• Scale direction by step length• Update solution• Check Convergence

2. TrustRegionBased (dogleg)• Computes Newton Direction• Computes Cauchy Direction• Adjust trust region radius• Update solution• Check Convergence

3. InexactTrustRegionBased (Prerelease)

4. TensorBased (Prerelease)5. UserDefined

Solver is determined by passing a parameter list to a handle for the class:

NOX::Solver::Manager solver(group, StatusTests, ParameterList);int status = solver.solve();

Derived from NOX::Solver::Genericbool  reset(NOX::Abstract::Group& grp, NOX::StatusTest::Generic& tests, NOX::Parameter::List& params) bool  reset(NOX::Abstract::Group& grp, NOX::StatusTest::Generic& tests) NOX::StatusTest::StatusType getStatus() NOX::StatusTest::StatusType iterate() NOX::StatusTest::StatusType solve() const NOX::Abstract::Group& getSolutionGroup() const const NOX::Abstract::Group& getPreviousSolutionGroup() const int getNumIterations() const const NOX::Parameter::List& getParameterList() const

NOX Directions• Steepest Descent

• Newton• Requires a linear solve• Adaptive forcing terms

• Broyden• Modified Newton with

rank-1 updates to the most recently computed Jacobian

• Nonlinear-CG (Prerelease)

• Tensor (Prerelease)

User Defined DirectionsDerived from NOX::Direction::GenericPassed into the solver via parameter list using a direction template ctor.

User Defined Merit FunctionDerived from NOX::Parameter::MeritFunctionPassed into the solver via parameter list using the NOX::Parameter::Arbitrary entry

Direction is determined and constructed in a handle called a “Manager”:

NOX::Direction::Manager dir(PrintParams, DirectionParams);

NOX Line Searches

• Full Step• Constant step size.• Defaults to 1.0.

• Backtrack• interval halving

• Polynomial Interpolation• Quadratic and Cubic interpolation• Requires a “Sufficient Decrease”

Condition

• More’-Thuente• Polynomial Interpolation• Sufficient Decrease and Curvature

Conditions

• Nonlinear-CG – (Prerelease) • Curvilinear (Prerelease)

User Defined Line SearchesDerived from NOX::LineSearch::GenericPassed into the solver via parameter list using a template constructor.

User Defined Merit FunctionDerived from NOX::Parameter::MeritFunctionPassed into the solver via parameter list using the NOX::Parameter::Arbitrary entry

User Defined NormsDerived from NOX::Parameter::UserNormPassed into the solver via parameter list using the NOX::Parameter::Arbitrary entry

Line Searches are determined and constructed in a handle called a “Manager”:

NOX::LineSearch::Manager ls(PrintParams, LineSearchParams);

Mix-n-Match Solver Algorithms

• Higly versatile code environment• Solver

– Line Search Based– Trust Region Based– Tensor Based

• Direction– Newton– Broyden– Steepest Descent– Tensor– User-Defined

• Line Search / Damping– Full Step– Backtrack– Polynomial/Quadratic– More’-Thuente– Curvilinear– User-Defined

Newton

Trust-Region Method

Tensor Method

Newly-DeployedBroyden Method

Quadratic Line Search

Stopping Criteria (StatusTests)

Highly Flexible Design: Users build a convergence test hierarchy and registers it with the solver (via solver constructor or reset method).

– Norm F: {Inf, One, Two} {absolute, relative}– Norm Update X: {Inf, One, Two}– Norm Weighted Root Mean Square (WRMS)

– Max Iterations – Failure test if solvers reaches max iters– FiniteValue – Failure test that checks for NaN and Inf on – Stagnation – Failure test that triggers if the convergence rate

fails a tolerance check for n consecutive iterations.

– Combination: {AND, OR}– Users Designed: Derive from NOX::StatusTest::Generic

Return TypesUnconvergedConverged*Failed*Unevaluated

Building a Status Test

• Fail if value of becomes Nan or Inf

NOX::StatusTest::FiniteValue finiteValueTest;

FiniteValue: finiteValueTest

• Fail if we reach maximum iterations

• Converge if both:

MaxIters: maxItersTest

NOX::StatusTest::MaxIters maxItersTest(200);

normFTest

NOX::StatusTest::NormF normFTest();normWRMSTest

NOX::StatusTest::NormWRMS normWRMSTest();

Combo(AND): convergedTest

NOX::StatusTest::Combo convergedTest(NOX::StatusTest::Combo::AND);

Combo(OR)allTests

NOX::StatusTest::Combo allTests(NOX::StatusTest::Combo::OR);allTests.addStatusTest(finiteValueTest);allTests.addStatusTest(maxItersTest);allTests.addStatusTest(convergedTest);

convergedTest.addStatusTest(normFTest);convergedTest.addStatusTest(normWRMSTest);

Status Tests Continued

User Defined are Derived from NOX::StatusTest::GenericNOX::StatusTest::StatusType checkStatus(const NOX::Solver::Generic &problem) NOX::StatusTest::StatusType checkStatusEfficiently(const NOX::Solver::Generic &problem, NOX::StatusTest::CheckType checkType) NOX::StatusTest::StatusType getStatus() const ostream& print(ostream &stream, int indent=0) const

-- Status Test Results --**...........OR Combination -> **...........AND Combination -> **...........F-Norm = 5.907e-01 < 1.000e-08 (Length-Scaled Two-Norm, Absolute Tolerance) **...........WRMS-Norm = 4.794e+01 < 1 (Min Step Size: 1.000e+00 >= 1) (Max Lin Solv Tol: 1.314e-15 < 0.5) **...........Finite Number Check (Two-Norm F) = Finite **...........Number of Iterations = 2 < 200

-- Final Status Test Results --Converged....OR Combination -> Converged....AND Combination -> Converged....F-Norm = 3.567e-13 < 1.000e-08 (Length-Scaled Two-Norm, Absolute Tolerance) Converged....WRMS-Norm = 1.724e-03 < 1 (Min Step Size: 1.000e+00 >= 1) (Max Lin Solv Tol: 4.951e-14 < 0.5) ??...........Finite Number Check (Two-Norm F) = Unknown ??...........Number of Iterations = -1 < 200

NOX Framework

SolverLayer

Abstract Vector & Abstract Group

AbstractLayer

Solvers- Line Search

- Trust Region

Directions- e.g., Newton

Line Searches- e.g., Polynomial

Status Tests- e.g., Norm F

• Don’t need to directly access the vector or matrix entries, only manipulate the objects.

• NOX uses an abstract interface to manipulate linear algebra objects.• Isolate the Solver layer from the linear algebra implementations used by

the application.• This approach means that NOX does NOT rely on any specific linear

algebra format.• Allows the apps to tailor the linear algebra to their own needs!

– Serial or Parallel– Any Storage format: LAPACK, PETSc, Epetra, User Defined

NOX::Abstract::Vector• Initialization

– x = y – x = |y|– xi = 1/yi for i = 1 to n– xi = for i = 1 to n

• Length of Vector

• Scaling– x = x– xi = xiyi for i = 1 to n

• Update– x = a + x, – x = a + b + x

• Norm– k x k1, k x k2 , k x k1– k x kw (weighted norm)

• Dot– x ¢ y

• Clone (create a copy)– y = x

The solver is not allowed nor does it need explicit

access to the vector – just the ability to manipulate

it.

NOX::Abstract::Group

• x = Iterate – Initialize– Update– Access

• F = F (x)– Compute– Access

• J = Jacobian of F at x– Compute– Apply / Apply Transpose– Apply Inverse– Apply Preconditioning

• n = Newton Vector – Compute n = -J-1F to specified

tolerance– Access

• g = Gradient of kF (xk)k2

– g = JTF– Access

• Clone

The linear solver and application interface are combined into

the Group. Vectors x, F, n, g are accessed

as NOX::Abstract::Vecto

r’s. Matrix J is never

directly accessed!

Code Demonstration

NOX Framework

SolverLayer

Abstract Vector & Abstract Group

AbstractLayer

Linear AlgebraInterface

Implementations- EPetra- PETSc- LAPACK- USER DEFINED

EPetra Dependent Features- Matrix-Free Newton-Krylov- Preconditioning- Graph Coloring / Finite Diff.

Solvers- Line Search

- Trust Region

Directions- e.g., Newton

Line Searches- e.g., Polynomial

Status Tests- e.g., Norm F

ApplicationInterface

Layer

User Interface- Compute F- Compute Jacobian- Compute Preconditioner

Linear Algebra Support Features

• NOX’s support libraries define a concrete implementation of the Abstract Vector and Group.

• We only require the user to implement a minimal interface derived from NOX::<Package>::Interface.

bool computeF(const Epetra_Vector &x, Epetra_Vector &f, FillType flag=F)

bool computeJacobian(const Epetra_Vector &x, Epetra_Operator &Jac) bool computePrecMatrix(const Epetra_Vector &x, Epetra_RowMatrix &M) bool computePreconditioner(const Epetra_Vector &x, Epetra_Operator &M)

Group(NOX::Parameter::List& printingParams, NOX::Parameter::List& linearSolverParams, NOX::Epetra::Interface& i, NOX::Epetra::Vector& x, Epetra_Operator& J)

Group(NOX::Parameter::List& printingParams, NOX::Parameter::List& linearSolverParams, NOX::Epetra::Interface& i, NOX::Epetra::Vector& x, Epetra_Operator& J, Epetra_Operator& M)

The Epetra “Goodies”• Matrix-Free Newton-Krylov Operator

• Derived from Epetra_Operator• Can be used to estimate Jacobian action on a

vector • NOX::Epetra::MatrixFree

• Finite Difference Jacobian• Derived from an Epetra_RowMatrix • Can be used as a preconditioner matrix• NOX::Epetra::FiniteDifference

• Graph Colored Finite Difference Jacobian• Derived from NOX::Epetra::FiniteDifference• Fast Jacobian fills – need connectivity/coloring

graph• (NOX::Epetra::FiniteDifferenceColoring)

• Full interface to AztecOO using NOX parameter list• Preconditioners: internal AztecOO, Ifpack, User defined • Scaling object

Jy F x y+ F x –-----------------------------------------=

JjF x ej+ F x –

-------------------------------------------=

Homotopy Algorithms

LOCA: Library of Continuation Algorithms

• Bifurcation and stability analysis package.

• Tightly coupled to NOX to reuse application interface.

• Provided as a single package.• Stepper object provides a robust

and intelligent step control for parameter continuation.

• Eric Phipps

Parameter ContinuationSolve a series of problems thatprogressively become thesystem of interest based on a parameter, :Natural: FH (x) = F(x,)Artificial: FH (x) = F(x) + (1 – )G(x)

0 1

F(x)

G(x) easy

NOX Specific Configure Options

• Compiling NOX library in Trilinos: --enable-nox

• Compiling prerelease code: --enable-prerelease

• Using nox built-in linear algebra support:LAPACK: --enable-nox-lapack

--enable-nox-lapack-examplesEpetra: --enable-nox-epetra

--enable-nox-epetra-examplesPETSc: --enable-nox-petsc

--enable-nox-petsc-examples

• Compiling the test suite:--enable-tests--enable-nox-tests

• Compiling LOCA library in Trilinos/NOX: --enable-loca

Future Work

• Test Suite Development – Darin Diachin• Re-work of the NOX::Epetra support library for efficiency -

Pawlowski• TSF support library – Pawlowski, Hooper• Tensor algorithm development – Brett Bader, Tammy Kolda• Broyden algorithm development – Kolda, Pawlowski• Multi-Physics Solvers – Strong code coupling

Multi-Physics Solvers• MultiPhysics – Bill Spotz and Alfred Lorber

• rapid prototyping environment to explore solver coupling algorithms.• PyNOX - Python interface to call NOX solver.

• SIERRA – Russell Hooper, John Shadid and Roger Pawlowski• Implementation of a Jacobian-Free coupling algorithm in SIERRA.

Loose Coupling Full Coupling

kk1k

k1kT

kkTT

kk1k

kkT

kkTT

CCC

C,TRCJ

TTT

C,TRTJ

kC

kT

k

k

kCC

kCT

kTC

kTT

RR

CT

JJJJ

kkk RUJ

Jacobian-Free Coupling

URpURJp

URpMURpJM

11

with preconditioner:

k

CC

kTT

J00J

MAnalytic

ORFD Coloring (Epetra)

TCTxCD

tC 2

2

2

2

)x()x,0(T6.0)1,t(T)0,t(T

)x()x,0(C

6.00.2)1,t(C)0,t(C

T)1(CTxTD

tT 2

2

2

1

Brusselator Examplein NOX

(Russell Hooper)

Summary• Key Features

– Object-oriented C++ library– Large number of cutting-edge algorithms– Can use any linear algebra implementation– Easy interface – User Flexibility: Add solvers, directions, line searches,

convergence tests

• NOX Support libraries– Fast integration – reduced interface– parallel iterative solvers (AztecOO, PETSc)– preconditioners (Ifpack, ML, Aztec, PETSc)– Matrix-Free Newton-Krylov– Matrix Estimation: Finite Difference/Coloring

NOX Contributors (Ideas, Code, and Testing)

Sandia– Tammy Kolda *– Roger Pawlowski *– Russ Hooper *– John Shadid– Todd Coffey *– Andy Salinger (LOCA) *– Eric Phipps (LOCA) *– Bill Spotz *– Mike Heroux (Trilinos) – Scott Hutchinson (Xyce)– Rob Hoekstra (Xyce)– Eric Keiter (Xyce)– Alan Williams (SIERRA)– Tom Smith (Premo)– Alfred Lorber (Premo)– Tom Brunner (Alegra)

Academics– Homer Walker, WPI– Joe Simonis, WPI– Tim Kelley, NCSU– Bobby Schnabel, UCB– Richard Byrd, UCB– Ryan McKenzie, Kentucky– Craig Douglas, Kentucky

* NOX Developer

NOX InformationTammy Kolda: tgkolda@sandia.gov

Roger Pawlowski: rppawlo@sandia.govDownload: http://software.sandia.gov/nox