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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. - PowerPoint PPT Presentation
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for more information ... http://www.tops-scidac.org
unscalable
scalable
Problem Size (increasing with number of processors)
Tim
e to
So
luti
on
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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
Keyword, key challenge: “Optimal”
Scalable SolversIn support of SciDAC fusion, astophysical, combustion, and other simulations, TOPS is creating a new generation of solvers for PDE field problems.
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
Data Layout
structured composite block-struc unstruc CSR
Linear Solvers
GMG, ... FAC, ... Hybrid, ... AMGe, ... ILU, ...
Linear System Interfaces
PETSc codeUser code
ApplicationInitialization
FunctionEvaluation
JacobianEvaluation
Post-Processing
PC KSP
Main Routine
Linear Solvers (SLES)
Nonlinear Solvers (SNES)
Timestepping Solvers (TS)
ADIC code
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.
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3 12 27 48 75
ASM-GMRESAMG-FMGRES
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3 12 27 48 75
ASM-GMRESAMG-FMGRESAMG inner
size, procs
size, procs
iters
time
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
PDE solver software is strategic to SciDAC Applications
PERC, CCA
TSTTAPDEC
TOPS
SS
SDM
How SciDAC apps engage TOPS solvers 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
Early TOPS partnersTOPS 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.
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