An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization

Abstract

We present a scalable approach and implementation for solving stochastic optimization problems on high-performance computers. In this work we revisit the sparse linear algebra computations of the parallel solver PIPS with the goal of improving the shared-memory performance and decreasing the time to solution. These computations consist of solving sparse linear systems with multiple sparse right-hand sides and are needed in our Schur-complement decomposition approach to compute the contribution of each scenario to the Schur matrix. Our novel approach uses an incomplete augmented factorization implemented within the PARDISO linear solver and an outer BiCGStab iteration to efficiently absorb pivot perturbations occurring during factorization. This approach is capable of both efficiently using the cores inside a computational node and exploiting sparsity of the right-hand sides. We report on the performance of the approach on highperformance computers when solving stochastic unit commitment problems of unprecedented size (billions of variables and constraints) that arise in the optimization and control of electrical power grids. Our numerical experiments suggest that supercomputers can be efficiently used to solve power grid stochastic optimization problems with thousands of scenarios under the strict "real-time" requirements of power grid operators. To our knowledge, this has not been possible prior to the present work.

Real-Time Stochastic Optimization of Complex Energy Systems on High-Performance Computers

“…For this we propose a novel algorithmic solution for the sparse scenario computations based on an incomplete, augmented factorization approach. 6 We achieve good peak performance despite the problem's complicating features. This approach decreases execution time by approximately a factor of 10 compared to previous implementations.…”

“…6 The solver uses a static pivoting strategy 14 and perturbs the matrix whenever numerically acceptable pivots can't be found within a diagonal supernode block. This means that only a pertur-…”

“…We use the biconjugate gradient stabilized (BiCGStab) Krylov space iterative method, in which the perturbed factorization of C acts as an efficient preconditioner 6 and keeps the number of absorption BiCGStab steps low.…”

Determinación Computacional del Coeficiente de Transferencia de Calor en Calentadores Eléctricos de Flujo Continuo, mediante Dinámica de Fluidos Computacional

“…En las ecuaciones de movimiento fue empleada una discretización tipo P1+P1 y de segundo orden para la temperatura, como método de solución fue empleado un resolvedor directo tipo PARDISO (PARallel DIrect SOlver), para la solución del sistema en estado estacionario de modo segregado con una tolerancia relativa de 0.001 (Petra et al, 2014).…”

On parallelization of the loop over elements in FEAP

“…So we check how the solution time varies with the number of threads for the problem size fixed per thread. It is the so-called week efficiency, see [21]. In an ideal situation, the execution time does not change with the change of number of threads and size of the task.…”