Trilinos/NOX Nonlinear Solver . Capabilities: • Newton-Based Nonlinear Solver - Linked to Trilinos linear solvers for scalability - Matrix-Free option • Anderson Acceleration for Fixed-Point iterations • Globalizations for improved robustness - Line Searches, Trust Region, Homotopy methods • Customizable: C++ abstractions at every level • Extended by LOCA package - Parameter continuation, Stability analysis, Bifurcation tracking . Download: Part of Trilinos (trilinos.sandia.gov) . Further information: Andy Salinger [[email protected]] ‹#› 1 NOX and ML are part of larger Trilinos solver stack: Linear solvers, Equations solvers, Analysis tools EquationSolvers" Analysis Tools" " Linear Solver (embedded)" ! Interface Analysis Tools" Nonlinear" Solver!NOX Analysis Tools" " Time Integration" " (black-box)" ! Continuation" " Optimization" " " LinearSolvers" Sensitivity" Analysis" UQ (sampling)" " Stability Analysis" Parameter Studies" Linear Algebra" " " " " Optimization" " Iterative Solvers" UQ Solver" " " Direct Solvers" " Solved Problem Interface Algebraic" " Nonlinear Model Preconditioners" " Interface Multilevel" ! Preconditioners" ! ML Your Model Here FASTMath SciDAC Institute" ‹#› 2 Trilinos/NOX: Robustness for Ice Sheet Simulation: PISCEES SciDAC Application project (BER-ASCR) . Ice Sheets modeled by nonlinear Stokes’s equation • Initial solve is fragile: Full Newton fails • Homotopy continuation on regularization parameter “γ” saves the day γ=10-10" γ=10-10" γ=10-1.0" γ=10-6.0" γ=10-2.5" γ=10-10" Greenland Ice Sheet" Surface Velocities" (constant friction model)" ‹#› 3 Multigrid in Trilinos .! ML: algebraic multigrid (AMG) •! Aggregation-based AMG for -! Poisson, elasticity, convection-diffusion, eddy current (E&M) •! Various coarsening and data rebalancing options •! Smoothers: SOR, polynomial, ILU, block variants, user-provided .! MueLu: templated multigrid framework •! Same main algorithms as above, plus energy minimizing AMG •! Leverages Trilinos templated sparse linear algebra stack -! Can use kernels optimized for various compute node types -! Templated scalar type allows for mixed precision, UQ, … •! Advanced data reuse possibilities, extensible by design .! Download/further information: www.trilinos.org ‹#› 4 ML and MueLu: Application highlights ML scales up to 512K BG/Q & 128K Titan cores in Drekar simulations SARANS turbulence*" Tokamak*" MHD generator*" *courtesy E. Cyr/SNL" PPE&Solve&Weak&Scaling&& 1.0# 0.9# 0.8# MueLu scales up to 524K BG/Q 0.7# 0.6# cores in Sierra/Nalu simulations 0.5# 0.4# 0.3# 0.2# 0.1# Turbulent OpenJet (Re=6600)" 0.0# 128# 1,024# 8,192# 65,536# 524,288# (courtesy P. Lin/SNL) Parallel&Efficiency:&Time/(Krylov&Itera7on)& #&MPI&Processes& Edge2based# Element2based# ‹#› 5 PARPACK . Capabilities: • Compute a few eigenpairs of a Hermitian and non-Hermitian matrix • Both standard and generalized eigenvalues • Extremal and interior eigenvalues • Reverse communication allows easy integration with application • MPI/BLACS communication . Download: http://www.caam.rice.edu/software/ARPACK/ . Further information: beyond PARPACK • EIGPEN (based on penalty trace minimization for computing many eigenpairs) • Parallel multiple shift-invert interface for computing many eigenpairs ‹#› 6 PARPACK . Hybrid MPI/OpenMP implementation . MATLAB interface available (eigs) ‹#› 7 MFDn: Nuclear Configuration Interaction Calculation . MFDn (Many-body Fermion Dynamics for nuclear physics) is the most robust and efficient configuration interaction code for studying nuclear structure . Solve the nuclear quantum many-body problem . PARPACK allows low excitation states to be extracted from a low dimensional Krylov subspace 10 10 9 10 8 10 7 10 6 10 4He 5 6Li 10 8Be 4 10B 10 12C 3 16O 10 19F 2 10 23Na M-scheme basis space dimension 1 27Al 10 0 10 0 2 4 6 8 10 12 14 Nmax ‹#› 8 .