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Continuation and Bifurcation Methods Using LOCA. Eric Phipps Andy Salinger, Roger Pawlowski Sandia National Laboratories Trilinos Workshop at Copper Mountain March 30, 2004.

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continuation and bifurcation methods using loca

Continuation and Bifurcation Methods Using LOCA

Eric Phipps

Andy Salinger, Roger Pawlowski

Sandia National Laboratories

Trilinos Workshop at Copper Mountain

March 30, 2004

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.

why do we need stability analysis algorithms for large scale applications
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

LOCA: Library of Continuation Algorithms

We need algorithms, software, and experience to impact ASCI- and SciDAC-sized applications.

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)

Stability/bifurcation analysis provides qualitative information about time evolution of nonlinear systems by computing families of steady-state solutions.

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


Complete rewrite in C++ around NOX framework began September 2002, part of Trilinos release September 2003.

loca library or continuation algorithms






Reaction Rate, r

Second parameter, h

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 (Thornquist, Lehoucq)
  • Bifurcation Tracking: Locates neutral stability point (x,p) and tracks as a function of a second parameter
codimension 1 bifurcations
Codimension 1 Bifurcations

Turning Point

  • 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
LOCA Designed for Easy Linking to Existing Newton-based Applications

LOCA targets existing codes that are:

  • Steady-State, Nonlinear
  • Newton’s Method
  • Large-Scale, Parallel

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
bordering algorithms meet these requirements9
Bordering Algorithms Meet these Requirements

Full Newton Algorithm

Turning Point Bifurcation

… but 4 solves of per Newton Iteration are used to drive singular!

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


Decrease step size

Restore previous solution

End if

If or or

stop = true

End while

Abstraction of Continuation Process

LOCA Stepper

Predictor modules





Step size modules

nox nonlinear solver kolda pawlowski hooper shadid
NOX Nonlinear Solver (Kolda, Pawlowski, Hooper, Shadid)

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


super vectors and super groups
Super Vectors and Super Groups

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

continuation groups
Continuation Groups







  • setParam()
  • getParam()
  • operator = ()
  • computeDfDp()
  • computeEigenvalues()
  • printSolution()


Default implementation



Concrete group

interfacing application codes to loca v2 0
Interfacing Application Codes to LOCA v2.0
  • 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:
loca s current capabilities
Single parameter continuation


Pseudo Arc-length


Turning point








Step size control



Artificial Homotopy

Generic interface to Anasazi

Native support for

LAPACK (all intefaces)

Epetra (all except Hopf)

LOCA’s Current Capabilities
continuation example chan problem
Continuation Example: Chan Problem





turning point continuation example
Turning Point Continuation Example



structural mechanics example salinas bending a 1d beam
Structural Mechanics Example: Salinas 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 b nard problem in 5 x 5 x 1 box 208k unknowns 16 processors
3D Rayleigh-Bénard Problem in 5x5x1 box(208K unknowns, 16 processors)

MPSalsa (Shadid et al., SNL):

  • Incompressible Navier-Stokes
  • Heat and Mass Transfer, Reactions
  • Unstrucured Finite Element (Galerkin/Least-Squares)
future development
Improve robustness

Better step size control

Improved bifurcation tracking algorithms


More features for homotopy


Multi-vector support

Multi-parameter continuation (Henderson, IBM)

Constraint enforcement

Automatic differentiation

Future Development