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Solving Bordered Systems of Linear Equations for Large-Scale Continuation and Bifurcation Analysis

Solving Bordered Systems of Linear Equations for Large-Scale Continuation and Bifurcation Analysis. Eric Phipps and Andy Salinger Applied Computational Methods Department Sandia National Laboratories 9 th Copper Mountain Conference on Iterative Methods April 7, 2006.

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Solving Bordered Systems of Linear Equations for Large-Scale Continuation and Bifurcation Analysis

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  1. Solving Bordered Systems of Linear Equations for Large-ScaleContinuation and Bifurcation Analysis Eric Phipps and Andy Salinger Applied Computational Methods Department Sandia National Laboratories 9th Copper Mountain Conference on Iterative Methods April 7, 2006 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.

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

  3. LOCA: Library of Continuation Algorithms • Application code provides: • Nonlinear steady-state residual and Jacobian fill: • Newton-like linear solves: • 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 External force 1 External force 3 1 Second parameter

  4. 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

  5. Turning Point Bifurcation Full Newton Algorithm Bordering Algorithm Bordering Algorithms Meet These Requirements … but 4 solves of J per Newton iteration are used to drive J singular!

  6. These Techniques Have Been Applied To Many Interesting Problems Capillary Condensation Flow in CVD Reactor Yeast Cell-Cycle Control Buckling of Garden Hose Block Copolymer Self-Assembly Propane&Propylene Combustion

  7. LOCA has been rewritten as part of the Trilinos framework • Collection of parallel solver packages, with a common build process, made interoperable where appropriate (Heroux et al) • Better algorithms can be implemented with tighter coupling to linear algebra Trilinos 6.0 Released: September 1, 2005 • Sandia’s parallel iterative solvers are made available to LOCA users 2004 • LOCA can build upon the existing interface between application codes and the NOX nonlinear solver HPC Software Challenge 2004

  8. Pseudo-Arclength Continuation Bordering Algorithm Constraint Following Turning Point Identification Bordered Systems of Equations Solving bordered systems of equations is a ubiquitous computation: • Only requires solves of J but • Requires m+l linear solves • Has difficulty when J is nearly singular

  9. Solving Bordered Systems via QR Extension of Householder pseudo-arclength technique by Homer Walker1 Rearranged Bordered System Compact WY Representation2 QR Factorization where then Write P is nxn, nonsingular, rank m update to J 1H.F Walker, SIAM J. Sci. Comput., 1999 2R. Schreiber, SIAM J. Stat. Comput., 1989

  10. Snap-through Buckling of a Shallow Cap(66K tri-shell elements, 200K unknowns, 16 procs) • Salinas (Reese et al., SNL): • Unstructured finite element, linear elasticity • Corotational formulation for beams and shells (C. Felippa, CU-Boulder) • Updated Lagrangian for solid elements • Analytic, Sparse Jacobian • Fully Coupled Newton Method (NOX) • GMRES (Aztec) with RILU(k) Preconditioner (Ifpack) • Distributed Memory Parallelism

  11. Turning Point Bifurcation Full Newton Algorithm Bordering Algorithm Turning Point IdentificationMoore-Spence Formulation … but 4 solves of J per Newton iteration are used to drive J singular!

  12. RILU fill factor: 6 • RILU overlap: 6 • Krylov space: 2000 Snap-through Buckling of a Shallow CapTurning Point Bordering Method

  13. Modified Turning Point Bordering Algorithm Solve 5 bordered systems of equations using QR approach Then

  14. Snap-through Buckling of a Shallow CapModified Turning Point Bordering Method • RILU fill factor: 6 • RILU overlap: 6 • Krylov space: 2000

  15. Minimally Augmented Turning Point Formulation Given and , let then There are constants such that Standard formulation: Note for Newton’s method: 3 linear solves per Newton iteration (5 for modified bordering)! For symmetric problems reduces to 2 solves.

  16. Minimally Augmented Turning Point Formulation for Large-Scale Problems With iterative solvers, so define where Also

  17. Snap-through Buckling of a Shallow CapMinimally Augmented Formulation • RILU fill factor: 6 • RILU overlap: 6 • Krylov space: 2000 • Modified bordering • Minimally augmented

  18. Moore-Spence w/Bordering: 7 total linear solves Pseudo-Arclength Equations Minimally Augmented: 3 total linear solves Pseudo-Arclength Continuation of Turning Points Newton Solve

  19. Snap-through Buckling of a Symmetric Cap Pseudo-arclength Turning Point Continuation

  20. Summary • QR approach provides a convenient way to solve bordered systems • Nonsingular • Only involves one linear solve • Only requires simple vector operations • Doesn’t change dimension of the linear system • Become a workhorse tool in LOCA • Highly encouraged by minimally augmented turning point formulation • No singular matrix solves • Improves robustness, scalability and accuracy • Requires linear algebra-specific implementation • Future work • Minimally augmented pitchfork, Hopf bifurcations • Preconditioners for

  21. Points of Contact • LOCA: Trilinos continuation and bifurcation package • Sub-package of NOX • Andy Salinger (agsalin@sandia.gov) • Eric Phipps (etphipp@sandia.gov) • www.software.sandia.gov/nox • NOX: Trilinos nonlinear solver package • Roger Pawlowski (rppawlo@sandia.gov) • Tammy Kolda (tgkolda@sandia.gov) • www.software.sandia.gov/nox • Trilinos: Collection of large-scale linear/nonlinear solvers • Epetra, Ifpack, AztecOO, ML, NOX, LOCA, … • Mike Heroux (maherou@sandia.gov) • www.software.sandia.gov/Trilinos • Trilinos Release 6.0 currently available, 7.0 this fall • 7.0 will include all LOCA algorithms presented here • Salinas: Massively Parallel Structural Dynamics Finite Element Code • Garth Reese (gmreese@sandia.gov)

  22. Modified Minimally Augmented Turning Point Formulation for Large-Scale Problems With iterative solvers, solves for u and v are in-exact. Does this impact nonlinear convergence? Instead solve for updates to u and v every nonlinear iteration:

  23. RILU fill factor: 6 • RILU overlap: 6 • Krylov space: 2000 Snap-through Buckling of a Shallow CapModified Minimally Augmented Formulation

  24. Snap-through Buckling of a Shallow CapMinimally Augmented Formulation • RILU fill factor: 6 • RILU overlap: 6 • Krylov space: 2000 • Bordering • Minimally augmented

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