Study of parallel numerical methods for semiconductor device simulation

Implicit schemes are a popular approach to the discretization of linear partial differential equations by finite differences. They require to solve a linear set of equations in each time step. Since finite difference discretizations lead to a local coupling, these systems of equations are sparse and can be effectively solved by iterative procedures. Numerical procedures of this type are known in control theory as repetitive processes. They have mostly been used to describe control algorithms for processes where passes of finite length are repeated over and over. This contribution shows how the implicit discretization of partial differential equations can be cast into the framework of repetitive processes. Thus it establishes a link between yet unrelated results in numerical mathematics and control theory.

The main purpose of this work is to present a new parallel direct solver: Dissection solver. It is based on LU factorization of the sparse matrix of the linear system and allows to detect automatically and handle properly the zero‐energy modes, which are important when dealing with DDM. A performance evaluation and comparisons with other direct solvers (MUMPS, DSCPACK) are also given for both sequential and parallel computations. Results of numerical experiments with a two‐level parallelization of large‐scale structural analysis problems are also presented: FETI is used for the global problem parallelization and Dissection for the local multithreading. In this framework, the largest problem we have solved is of an elastic solid composed of 400 subdomains running on 400 computation nodes (3200 cores) and containing about 165 millions dof. The computation of one single iteration consumes less than 20 min of CPU time. Several comparisons to MUMPS are given for the numerical computation of large‐scale linear systems on a massively parallel cluster: performances and weaknesses of this new solver are highlighted. Copyright © 2011 John Wiley & Sons, Ltd.

SUMMARY Solving the frequency-domain elastic wave equations relies on an efficient linear solver for the large, sparse, indefinite and ill-conditioned linear system derived from the discretization of the elastic wave equation. Direct solvers, which are mostly based on LU decomposition, are efficient for multiple right-hand sides problems, but the memory requirement is huge due to the fill-in effects. On the contrary, iterative solvers fully benefit from the sparsity of the system, but they require problem-specific preconditioners to ensure the convergence because of the ill-conditioning of the system. In this study, we investigate the performance of a robust iterative method named CARP-CG for frequency-domain elastic wave modeling. CARP-CG method turns the original system into a symmetric positive semi-definite system by Kaczmarz row-projections. Such a system can be efficiently solved by the conjugate gradient (CG) method. The row-projections can be seen as a purely algebraic preconditioning technique which is general and is easy to implement. The parallelization is straightforward through a row-block decomposition combined with a component-averaging method. We discretize the 2D frequency-domain elastic wave equation through a 4th order finite difference scheme. Numerical experiments on the Marmousi2 model exhibit a good scalability of CARP-CG. Comparisons between CARP-CG and standard Krylov iterative solvers (GMRES and CGNR) further emphasize the robustness and the fast convergence of CARP-CG method.

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations. This thesis is devoted to the development of new algorithms to solve this kind of problems. In particular, when the matrices involved are sparse and non-symmetric. The new algorithms are based on the Induced Dimension Reduction method [IDR(s)]. IDR(s) is a Krylov subspace method originally proposed in 2008 to solve systems of linear equations. IDR(s) has received considerable attention due to its stable and fast convergence. It is, therefore, natural to ask if it is possible to extend IDR(s) to solve other matrix problems, and if so, to compare those extensions with other well-established methods. This work aims to answer these questions. The main matrix problems considered in this dissertation are: the standard eigenvalue problem, the quadratic eigenvalue problem, the solution of systems of linear equations, the solution of sequences of systems of linear equations, and linear matrix equations. We focus on examples that arise from the discretization of partial differential equations.

We propose a new approximate factorization for solving linear systems with symmetric positive definite sparse matrices. In a nutshell the algorithm is to apply hierarchically block Gaussian elimination and additionally compress the fill-in. The systems that have efficient compression of the fill-in mostly arise from discretization of partial differential equations. We show that the resulting factorization can be used as an efficient preconditioner and compare the proposed approach with state-of-art direct and iterative solvers.

We present a stable embedded discontinuous Galerkin (EDG) method for scalable electrothermal simulation of 3D semiconductor devices. The framework extends the robust hybridized discontinuous Galerkin method developed for purely electric paradigm to resolve fully coupled electrothermal drift-diffusion equations, addressing a critical gap in existing harmonic-averaging formulations that lack non-isothermal current modeling. It is believed that for the first time, we derive a consistent discretization of thermally coupled current terms within the EDG scheme, ensuring stability under strong convection and Joule heating. A parallel solver is further developed and implemented via the JAUMIN framework using domain decomposition, enabling extreme-scale 3D simulations of semiconductor devices. The validations on p-n junctions, NLDMOSFET and multi-finger FinFET demonstrate superior Newton convergence and comparable accuracy in comparison with that of commercial software COMSOL. It can achieve 79.2% strong scaling efficiency (20×cores) and 59.0% weak scaling efficiency (8× problem size) in the 8-Finger FinFET electrothermal simulations.

Frequency-domain elastic wave modeling relies on an efficient linear solver for the large, sparse, ill-conditioned linear system derived from the discretization of the elastic wave equation. Direct solvers are mostly based on LU decomposition. These methods are efficient for multiple right-hand sides problems, but they require significant memory resources. Conversely, iterative solvers benefit from the sparsity of the system, but they require sophisticated preconditioners to converge due to the system ill-conditioning. In this study, we investigate the performance of an iterative method named CARP-CG for frequency-domain elastic wave modeling. The CARP-CG method transforms the original system into a symmetric positive semi-definite system by cyclic row-projections. This system is efficiently solved with the conjugate gradient (CG) method. The cyclic row-projection transformation can be seen as a purely algebraic preconditioning technique which is easy to implement. The algorithm can be parallelized through a row-block decomposition combined with component-averaging operations. Numerical experiments on the 2D frequency-domain elastic problem with high Poisson's ratio exhibit a good scalability of CARP-CG. Comparisons for different frequencies between CARP-CG and standard Krylov iterative solvers (GMRES and CG on the normal equations) emphasize the robustness and the fast convergence of the method.

In many scientific computing problems, the overall execution time is dominated by the time to solve very large linear systems. Quite often, the matrices are unsymmetric and ill conditioned, with an irregular sparsity structure reflecting the irregular refinement in the discretization grid. With increasing problem size and problem dimension, direct solvers cannot be used because of the huge memory requirements. The performance of preconditioned iterative solvers is largely dominated by memory-related aspects like size, bandwidth, and indirect addressing speed. This article summarizes the experience of the authors on the relationship between memory aspects and performance in real applications in the domain of very large scale integration (VLSI) device simulation. The authors analyze storage requirements of direct and iterative solvers on a statistical data set, and demonstrate performance variations due to memory-related architectural features on a number of computers ranging from workstations to Cray, NEC, and Fujitsu supercomputers for typical and ill-conditioned linear systems, using different iterative methods and preconditioners. The experiments are done using PILS, a package of iterative linear solvers. PILS implements a large number of iterative methods and preconditioners and allows them to be combined in a flexible way.

Sylvester equations AX − XB = C play an important role in numerical linear algebra. For example, they arise in the computation of invariant subspaces, in control problems, as linearizations of algebraic Riccati equations, and in the discretization of partial differential equations. For small systems, direct methods are feasible. For large systems, iterative solution methods are available, like Krylov subspace methods. It can be observed that there are essentially two types of subspace methods for Sylvester equations: one in which block matrices are treated as rigid objects (functions on a grid), and one in which the blocks are seen as a basis of a subspace.

In this paper, we present a fully parallel transistor level full-chip circuit simulation tool with SPICE-accuracy for general circuit designs. The proposed overlapping domain decomposition approach partitions the circuit into a linear subdomain and multiple non-linear subdomains based on circuit non-linearity and connectivity. Parallel iterative matrix solver is used to solve the linear domain while non-linear subdomains are parallelly distributed into different processors topologically and solved by direct solver. To achieve maximum parallelism, device model evaluation is done parallelly. Parallel domain decomposition technique is used to iteratively solve the different partitions of the circuit and ensure convergence. Orders of magnitude speedup over SPICE is observed for sets of large-scale circuit designs on up to 64 processors.

A package of three parallel solvers with a unified user interface is developed for sparse symmetric complex linear systems arising from the discretization of partial differential equations on unstructured meshes. Once the data interface is set up, the package constructs the sparse symmetric complex matrix, and solves the linear system using one of the following methods: a bi-conjugate gradient solver, a two-stage Cholesky LDL/sup T/ factorization, or a hybrid solver combining the above two methods. The user deals only with local matrices on local mesh partitions in each processor. Scaling problem size N with the number of processors P with N/P fixed, test runs on Intel Delta up to 28 processors show that the bi-conjugate gradient method scales linearly with N whereas the hybrid method scales as /spl radic/N. >

The solution of large sparse unsymmetric linear systems is a critical and challenging component of semiconductor device and circuit simulations. The time for a simulation is often dominated by this part. The sparse solver is expected to balance different, and often conflicting requirements. Reliability, a low memory-footprint, and a short solution time are a few of these demands. Currently, no black-box solver exists that can satisfy all criteria. The linear systems from both simulations can be highly ill-conditioned and are, therefore, quite challenging for direct and iterative methods. In this paper, it is shown that algorithms to place large entries on the diagonal using unsymmetric permutations and scalings greatly enhance the reliability of both direct and preconditioned iterative solvers for unsymmetric linear systems arising in semiconductor device and circuit simulations. The numerical experiments indicate that the overall solution strategy is both reliable and cost effective.

Multicolor Fourier analysis of the multigrid method for quadratic FEM discretizations

Scalable line and plane relaxation in a parallel structured multigrid solver

The transient stability analysis of power systems can be described by differential and algebraic equations, and due to the increased speed with parallel computing techniques, use of the waveform relaxation method to solve these equations has attracted much attention. In order to enhance the calculation speed of the waveform relaxation method, it is of prime importance to reduce the required number of iterations for each time window. To this end, an initialization and preconditioning techniques have been proposed to improve convergence of the iterative procedure. In this paper, in order to enhance convergence, we developed a new global initialization technique that consists of linear analysis and a preconditioning technique. Specifically, global initialization is the result of the repeated application of conventional initialization and preconditioning techniques before each of the nonlinear iterative procedures, and then using linear analysis to determine the interaction between the CPUs. By calculating the initial waveforms more accurately, it is expected that additional speedup can be achieved by reducing the required number of iterations necessary for convergence of the nonlinear analysis.