Abstract. An implementation of GMRES with multiple preconditioners (MPGMRES) is proposed for solving shifted linear systems with shift-and-invert preconditioners. With this type of preconditioner, the Krylov subspace can be built without requiring the matrix-vector product with the shifted matrix. Furthermore, the multipreconditioned search space is shown to grow only linearly with the number of preconditioners. This allows for a more efficient implementation of the algorithm. The proposed implementation is tested on shifted systems that arise in computational hydrology and the evaluation of different matrix functions. The numerical results indicate the effectiveness of the proposed approach.

The use of a constraint (indefinite) preconditioner for the iterative solution of the linear system arising from the finite element discretization of coupled Stokes-Darcy flow is studied. Spectral and field-of-values bounds for the preconditioned system which are independent of the underlying mesh size are presented. Numerical experiments on several two-and three-dimensional problems illustrate the effectiveness of our approach.

Fast and accurate thermal analysis is crucial for determining the propagation of heat and tracking the formation of hot spots in integrated circuits (ICs). Existing academic thermal analysis tools primarily use compact models to accelerate thermal simulations but are limited to linear problems on relatively simple circuit geometries. The Manchester Thermal Analyzer (MTA) is a comprehensive tool that allows for fast and highly accurate linear and nonlinear thermal simulations of complex physical structures including the IC, the package, and the heatsink. The MTA is targeted for 2.5/3-D IC designs but also handles standard planar ICs. The MTA discretizes the heat equation in space using the finite element method and performs the time integration with unconditionally stable implicit time stepping methods. To improve the computational efficiency without sacrificing accuracy, the MTA features adaptive spatiotemporal refinement. The largescale linear systems that arise during the simulation are solved with fast preconditioned Krylov subspace methods. The MTA supports the thermal analysis of realistic integrated systems and surpasses the computational abilities and performance of existing academic thermal simulators. For example, the simulation of a processor in a package attached to a heat sink, modeled by a computational grid consisting of over 3 million nodes, takes less than 3 minutes. The MTA is fully parallel and publicly available. *

Thermal analysis is crucial for determining the propagation of heat and to track the formation of hot spots in advanced integrated circuit technologies. At the core of the thermal analysis for the integrated circuits is the numerical solution of the heat equation. Prior academic thermal analysis tools typically compute temperature by applying finite difference methods on uniform grids with time integration methods having fixed time step size. Additionally, the linear systems arising from the discretized heat equation are solved using direct methods based on matrix factorizations. Direct methods, however, do not scale well as the problem size increases. Moreover, most of the tools support only 2-D or a limited number of 3-D technologies. To address these issues, this paper presents a novel thermal analyzer with the ability to model both 2-D and 3-D circuit technologies. The analyzer solves the heat equation using the finite element method for the spatial discretization coupled with implicit time integration methods for advancing the solution in time. It also offers fully adaptive spatio-temporal refinement features for improved accuracy and computational efficiency. The resulting linear systems are solved by a multigrid preconditioned Krylov subspace iterative method, which gives superior performance for 3-D transient analyses. The analyzer is shown to accurately capture the propagation of heat in both the horizontal and vertical directions of integrated systems.

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Thermal analysis is a high performance computing problem because the microscale spatial and time discretization of the heat equation translates into repeatedly solving large linear systems of equations. In previous works, compact models of integrated circuits (IC) were introduced to speed up this process. However, such methods are limited in their accuracy as they approximate the underlying physics. The solution methodologies for such models are also ill-suited to simulate the thermal characteristics of an IC at cell-level. The finite element method (FEM) is an appropriate computational technique for providing both fast and accurate thermal analyses. Considering that the number of cells in modern ICs is in the order of millions, thermal analysis at this abstraction level is a formidable task. Consequently, handling the computational meshes and computing thermal profiles of an IC at the cell-level requires intensive computing power. In order to provide accurate cell-level thermal simulations at a lower computational cost, this work introduces an advanced mesh generator for cell-level floorplans. This mesh generator applies a cell-homogenization technique based on the initial cell-level floorplan and the power trace to create reduced order meshes for e cient cell-level thermal simulations. The e↵ects of using di↵erent homogenization algorithms are explored to illustrate the tradeo↵ between the simulation speed and the solution accuracy. Results show that the proposed technique can achieve a 90% reduction in the number of nodes in the mesh with less than 5% error to the temperature of the full scale mesh. In addition, the simulation time is reduced by an order of magnitude.