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1. Main Routine Linear System Interfaces Applications 200 Timestepping Solvers (TS) unscalable 150 APDEC TSTT SDM 100 Nonlinear Solvers (SNES) Time to Solution TOPS Linear Solvers 50 Linear Solvers (SLES) scalable GMG, ... FAC, ... Hybrid, ... AMGe, ... ILU, ... 0 1000 1 10 100 Problem Size (increasing with number of processors) PERC, CCA PC KSP Data Layout structured composite block-struc unstruc CSR Application Initialization Function Evaluation Jacobian Evaluation Post- Processing SS User code PETSc code ADIC code Scalable Solvers In support of SciDAC fusion, astophysical, combustion, and other simulations, TOPS is creating a new generation of solvers for PDE field problems. How SciDAC apps engage TOPS solvers PDE solver software is strategic to SciDAC • Directly (now) • Apps code sets up own discretization, possibly built on a grid made up of distributed objects from PETSc (like M3D) • Apps code calls a TOPS solver, possibly with explicit matrix elements, or in a Jacobian-free mode • Through APDEC or TSTT (coming in 2003) • Apps code calls on Chombo, Overture, Trellis, etc., to express its PDEs with an automatically adapted discretization • Through componentization (coming later) • Apps code, discretization frameworks, TOPS solvers are all peer components interacting in a Common Component Architecture framework • Benefits to apps • With solver, get stability analysis and sensitivity analysis functionality • Many DOE mission-critical systems are modeled by PDEs • Finite-dimensional models of PDEs must be large for accuracy • Qualitative insight is not enough (Hamming notwithstanding) • Simulations must resolve policy controversies • Advances in algorithms are at least as important as advances in hardware, in supporting simulation • Easily demonstrated for PDEs in the period 1945–2000 • Continuous problems provide exploitable hierarchy of approximation models, creating hope for “optimal” algorithms • Software lags both hardware and algorithms Keyword, key challenge: “Optimal” Early TOPS partners TOPS has many application partners, including the Center for Extended Magnetohydrodynamic Modeling (CEMM, left), the Center for Magnetic Reconnection Studies (CMRS, below left), and the Terascale Supernovae Initiative (TSI, below right). For CEMM, TOPS’s scalable linear solvers power linear solvers inside an operator-split time integration of tokamak dynamics. For CMRS, TOPS has developed a fully implicit nonlinear capability, permitting accurate implicit time stepping that exceeds the Courant stability limit for an explicit method. For TSI, TOPS is extending TSI’s 1D operator-split solvers to 2D and 3D operator-split and nonlinearly implicit, both. • Convergence rate nearly independent of discretization parameters • Multilevel schemes for linear and nonlinear problems • Newton-like schemes for quadratic convergence of nonlinear problems • Convergence rate as independent as possible of physical parameters • Continuation schemes • Asymptotics-induced, operator-split preconditioning time size, procs Algebraic multigrid (AMG) above, shows perfect iteration scaling, above, in contrast to additive Schwarz (ASM), but still needs performance work to achieve temporal scaling, below, on CEMM fusion code, M3D iters size, procs Interoperability TOPS brings together and will make interoperable some of the most popular solver software toolkits in the DOE, such as Hypre, PETSc, and SUNDIALS. TOPS solvers will also interoperate with APDEC and TSTT codes. Multiple interfaces TOPS’s conceptual interfaces (from Hypre, below) allow users to access its multilevel solvers from data structures close to the applications. TOPS’s interface to automatic differentiation tools (through PETSc, below right) provides rapid nonlinear solution and optimization, all matrix-free. for more information ... http://www.tops-scidac.org