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 Pawlowski9233 – Computational Sciences
Trilinos User Group Meeting
October 16, 2003
Continuation and Bifurcation Methods Using LOCA
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
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.
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
Tmax
Reaction Rate, r
r
Second parameter, h
1
1
3
Pseudo Arc-length Continuation Solves for Solution and Parameter Simultaneously
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
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
Bordering Algorithms Meet these Requirements
Full Newton Algorithm
Bordering Algorithm
Pseudo Arc-length Continuation
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
Abstraction of Continuation Process
Given initial guess , step size– Solve nonlinear equations to find 1st 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
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)
Group Vector
computeF() dot()
computeJacobian() scale()
computeNewton() norm()
applyJacobianInverse() update()
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 parametercomponent .
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
LOCA::Continuation::ArclengthGroup::applyJacobianInverse(const NOX::Abstract::Vector& input, NOX::Abstract::Vector& result) const {const LOCA::Continuation::ExtendedVector& con_input =
dynamic_cast<const LOCA::Continuation::ExtendedVector&>(input);LOCA::Continuation::ExtendedVector& con_result =
dynamic_cast<LOCA::Continuation::ExtendedVector&>(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()
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
NOX::Abstract::Group
Concrete group
Turning Point, Pitchfork Groups
LOCA::Continuation::AbstractGroup
LOCA::Bifurcation::TPBord::ExtendedGroup
LOCA::Bifurcation::TPBord::AbstractGroup• computeDJnDp()• computeDJnDxa()• applySingularJacobianInverse()
NOX::Abstract::Group
NOX::Abstract::Group
Concrete group
LOCA::Bifurcation::PitchforkBord::ExtendedGroup
LOCA::Continuation::AbstractGroup
Hopf Groups
LOCA::Continuation::AbstractGroup
LOCA::Bifurcation::HopfBord::ExtendedGroup
LOCA::Bifurcation::HopfBord::AbstractGroup• applyComplex()• applyComplexInverse()• computeDCeDp()• computeDCeDxa()
NOX::Abstract::Group NOX::Abstract::Group
Concrete group
LOCA::TimeDependent::AbstractGroup• computeMassMatrix()• applyMassMatrix()
LOCA::Continuation::AbstractGroup
LOCA::Bifurcation::TPBord::AbstractGroup
• 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
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)
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
3D Rayleigh-Benard Problem in 5x5x1 box
• MPSalsa• 208K unknowns• 16 Processors
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
