1. 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. Eric Phipps Andy Salinger, Roger Pawlowski 9233 – Computational Sciences Trilinos User Group Meeting October 16, 2003 Continuation and Bifurcation Methods Using LOCA

2. Why Do We Need Stability Analysis Algorithms for Large-Scale Applications? Nonlinear systems exhibit instabilities, e.g.: Multiple steady states Ignition Symmetry Breaking Onset of Oscillations Phase Transitions These phenomena must be understood in order to perform computational design and optimization. Established stability/bifurcation analysis libraries exist: AUTO (Doedel) CONTENT (Kuznetsov) MATCONT (Govaerts) We need algorithms, software, and experience to impact ASCI- and SciDAC-sized applications. Stability/bifurcation analysis provides qualitative information about time evolution of nonlinear systems by computing families of steady-state solutions. LOCA: Library of Continuation Algorithms

3. History LOCA library grew out of continuation code in MPSalsa –Andy Salinger, John Shadid, Roger Pawlowski, Louis Romero, Rich Lehoucq, Ed Wilkes, Beth Burroughs, Nawaf Bou-Rabee LOCA 1.0 released April 2002 –Written in C with wrapper functions for linking to application code –~200 downloads Complete rewrite in C++ around NOX framework began September 2002, part of Trilinos release September 2003.

4. LOCA: Library or Continuation Algorithms LOCA provides: Parameter Continuation: Tracks a family of steady state solutions with parameter Linear Stability Analysis: Calculates leading eigenvalues via Anasazi Bifurcation Tracking: Locates neutral stability point (x,p) and tracks as a function of a second parameter T max Reaction Rate, r r Second parameter, h 1 1 3

5. Pseudo Arc-length Continuation Solves for Solution and Parameter Simultaneously

6. Codimension 1 Bifurcations Turning Point Pitchfork Hopf Combustion Buckling of an Arch Buckling of a Beam Pattern formation Cell differentiation (morphogenesis) Vortex Shedding Predator-Prey models Puberty

7. LOCA Designed for Easy Linking to Existing Newton-based Applications Algorithmic choices for LOCA: Must work with iterative (approximate) linear solvers on distributed memory machines Non-Invasive Implementation (e.g. matrix blind) Should avoid or limit:  Requiring more derivatives  Changing sparsity pattern of matrix  Increasing memory requirements LOCA targets existing codes that are: Steady-State, Nonlinear Newton’s Method Large-Scale, Parallel

8. Bordering Algorithms Meet these Requirements Full Newton Algorithm Bordering Algorithm Pseudo Arc-length Continuation

9. Bordering Algorithms Meet these Requirements … but 4 solves of per Newton Iteration are used to drive singular! Turning Point Bifurcation Full Newton Algorithm Bordering Algorithm

10. Abstraction of Continuation Process Given initial guess, step size –Solve nonlinear equations to find 1 st point on curve –while !stop Compute predictor Compute predicted point Solve continuation equations for using as initial guess If successful –Postprocess (e.g., compute eigenvalues, output data) –Increase step size Else –Decrease step size –Restore previous solution End if If or or –stop = true –End while LOCA Stepper Predictor modules Step size modules NOX + continuation/ bifurcation groups

11. NOX implements various methods for solving Code to evaluate is encapsulated in a Group. NOX solver methods are generic, and implemented in terms of group/vector abstract interfaces: NOX solvers will work with any group/vector that implements these interfaces. NOX Nonlinear Solver (Kolda, Pawlowski, Hooper, Shadid) GroupVector computeF()dot() computeJacobian()scale() computeNewton()norm() applyJacobianInverse()update()

12. Idea: Given a vector to store and a group representing the equations, build an extended (“super”) group representing, e.g., pseudo arc-length continuation equations: and a super vector to store the solution component and parameter component. Super groups/vectors are generic: All abstract group/vector methods for super groups/vectors implemented in terms of methods of the underlying groups/vectors. Super groups are NOX groups: Extended nonlinear equations solved by most NOX solvers Super Vectors and Super Groups

13. LOCA::Continuation::ArclengthGroup::applyJacobianInverse(const NOX::Abstract::Vector& input, NOX::Abstract::Vector& result) const { const LOCA::Continuation::ExtendedVector& con_input = dynamic_cast (input); LOCA::Continuation::ExtendedVector& con_result = dynamic_cast (result); const NOX::Abstract::Vector& input_x = con_input.getXVec(); double input_p = con_input.getParam(); NOX::Abstract::Vector& result_x = con_result.getXVec(); double& result_p = con_result.getParam(); NOX::Abstract::Vector* b = input_x.clone(NOX::ShapeCopy); underlyingGroupPtr->applyJacobianInverse(input_x, result_x); underlyingGroupPtr->applyJacobianInverse(*dfdpVecPtr, *b); result_p = (predictorVecPtr->getXVec().dot(result_x) – input_p) / (predictorVecPtr->getXVec().dot(*b) – predictorVecPtr->getParam()); result_x.update(-result_p, *b, 1.0); delete b; } Arc-length Group applyJacobianInverse()

14. Continuation Groups LOCA::Continuation::ExtendedGroup LOCA::Continuation::NaturalGroup LOCA::Continuation::ArclengthGroup LOCA::Continuation::AbstractGroup setParam() getParam() operator = () computeDfDp() computeEigenvalues() printSolution() Mandatory Default implementation available Optional NOX::Abstract::Group Concrete group

15. Turning Point, Pitchfork Groups LOCA::Continuation::AbstractGroup LOCA::Bifurcation::TPBord::ExtendedGroup LOCA::Bifurcation::TPBord::AbstractGroup computeDJnDp() computeDJnDxa() applySingularJacobianInverse() NOX::Abstract::Group Concrete group LOCA::Bifurcation::PitchforkBord::ExtendedGroup LOCA::Continuation::AbstractGroup

16. Hopf Groups LOCA::Continuation::AbstractGroup LOCA::Bifurcation::HopfBord::ExtendedGroup LOCA::Bifurcation::HopfBord::AbstractGroup applyComplex() applyComplexInverse() computeDCeDp() computeDCeDxa() NOX::Abstract::Group Concrete group LOCA::TimeDependent::AbstractGroup computeMassMatrix() applyMassMatrix() LOCA::Continuation::AbstractGroup LOCA::Bifurcation::TPBord::AbstractGroup

17. Interfacing NOX to the application code is 90% of the work! For continuation turning point tracking pitchfork tracking at very minimum must be able to additionally set/retrieve parameter values, save complete state of system by copying group. For Hopf tracking, must implement a complex solve: Can overload many additional methods if better techniques are available block solves singular matrix solves estimating derivatives: Interfacing Application Codes to LOCA v2.0

18. LOCA’s Current Capabilities Single parameter continuation –Natural –Pseudo Arc-length Bifurcations –Turning point –Pitchfork –Hopf Predictors –Constant –Tangent –Secant –Random Step size control –Constant –Adaptive Artificial Homotopy Generic interface to Anasazi Native support for –LAPACK(all intefaces) –Epetra (all except Hopf)

19. Salinas Example: Bending a 1D Beam Example problem from Salinas test suite Original continuation run with 50 load steps NOX/LOCA interface written by Russell Hooper Variable step size algorithm reduced to 37 load steps

20. 3D Rayleigh-Benard Problem in 5x5x1 box MPSalsa 208K unknowns 16 Processors

21. Where We’re Going From Here Improve robustness –Better step size control –Improved bifurcation tracking algorithms –Debugging Incorporate –Multi-vector support –Multi-parameter continuation (Henderson, IBM) –Constraint enforcement –Automatic differentiation Impact more ASCI codes –Structural mechanics