## Accelerated Waveform Methods for Parallel Transient Simulation of Semiconductor Devices (1996)

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Venue: | IN PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON COMPUTER-AIDED DESIGN |

Citations: | 4 - 1 self |

### BibTeX

@INPROCEEDINGS{Lumsdaine96acceleratedwaveform,

author = {Andrew Lumsdaine and Mark W. Reichelt and Jeffrey M. Squyres and Jacob K. White},

title = {Accelerated Waveform Methods for Parallel Transient Simulation of Semiconductor Devices},

booktitle = {IN PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON COMPUTER-AIDED DESIGN},

year = {1996},

pages = {270--274},

publisher = {}

}

### OpenURL

### Abstract

Simulating transients in semiconductor devices involves numerically solving the time-dependent drift-diffusion equations, usually in two or three space dimensions. Because of the computation cost of these simulations, methods that perform careful domain decomposition so as to exploit parallel processing have received much recent attention. In this paper, we describe using accelerated waveform relaxation (WR) to perform parallel device transient simulation using both clusters of workstations and the IBM SP-2. The accelerated WR algorithms are compared to pointwise direct and iterative methods, and it is shown that the accelerated WR method is competitive on a single processor. In addition, it is shown that with a domain decomposition chosen for rapid iterative method convergence rather than parallel efficiency, the pointwise methods parallelize poorly but the WR mcthod achieves near linear speedup (with respect to the number of processors) on the IBM SP-2.

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Citation Context ...OR (CSOR) is a generalized waveform extension of the well-known successive overrelaxation (SOR) method used to accelerate the convergence of relaxation methods for solving linear systems of equations =-=[15]-=-. To abbreviate the description of the CSOR algorithm, we will consider the problem of numerically solving the linear initial-value problem, (t) +Az(t) = b(t) with z(0) = 0, (2) where AsII nxn, b(t)sI... |

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Citation Context ...(t), which can be performed in parallel, followed by a global sum of the results. Another Krylov-subspace method that is suitable for accelerating WR is the conjugate gradient squared algorithm (CGS) =-=[23]-=-. The waveform CGS (WCGS) algorithm is a straightforward extension of CGS in which, like WGMRES, the matrix-vector products are replaced by operator-waveform products, and the vector inner products ar... |

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Citation Context ...ates after each timestep computation. As with any iterative scheme, efficiency of WR depends on rapid convergence, and there have been several investigations into accelerating WR, including multigrid =-=[7]-=-, Krylov-subspace [8, 9], and convolution SOR techniques [10, 11]. In this paper we extend the *The MathWorks, Inc., 24 Prime Park Way, Natick, MA 01760 (mwr@mathworks.com). tDept. of Comp. Sci. and E... |

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Citation Context ... variety of MOS devices. 2 Device Transient Simulation Charge transport within a semiconductor device is assumed to be governed by the Poisson equation, and the electron and hole continuity equations =-=[12, 13]-=-. Given a two-dimensional rectangular discretization mesh, the device equation system is typicalIy discretized with a finite-difference formula applied to the Poisson equation, and an exponentially-fi... |

16 |
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Citation Context ...]. As with any iterative scheme, overall computational efficiency of WR depends on rapid convergence, and there have been several investigations into accelerating WR [4, 7], including using multigrid =-=[8, 9]-=-, Krylov-subspace [10, 11], and convolution SOR techniques [12]. In this paper, we extend the results in [10, 11, 12, 13] and provide experimental results using waveform methods on two different paral... |

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Citation Context ...s at single timepoints [5]-[7]. As with any iterative schemc, overall computational efficiency of WR depends on rapid convergence, and there have been several investigations into accelerating WR [5], =-=[8]-=-, including using multigrid [9], [ 10], Krylov-subspace [ 11 ], [ 12], and convolution successive overrelaxation (CSOR) techniques [13]. In this paper, we extend the results in [11]-[14] and provide e... |

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Citation Context ...tep computation. As with any iterative scheme, efficiency of WR depends on rapid convergence, and there have been several investigations into accelerating WR, including multigrid [7], Krylov-subspace =-=[8, 9]-=-, and convolution SOR techniques [10, 11]. In this paper we extend the *The MathWorks, Inc., 24 Prime Park Way, Natick, MA 01760 (mwr@mathworks.com). tDept. of Comp. Sci. and Eng., University of Notre... |

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Citation Context ... can be IBund in [26]. To use the wavefo ylov-subspace methods on the nonlinear device system (1), Newton's method is applied to (1), in a process sometimes refeed to as the wavefo Newton method (WN) =-=[27], to obtai-=-n the following iteration d : JF(X") TM - F(z TM) with x+(0) -- x0. (12) Here, ff is the Jacobtan of F. We note that (15) is a linear time-vying system to bc solved for x "+, which can be ac... |

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Citation Context ...veform Newton/Waveform CGS (WN/WCGS). The program was written in C using a message-passing single-program multiple-data (SPMD) paradigm. The messagepassing was effected using the MPICH implementation =-=[28]-=- (from Argonne National Labs and Mississippi State University) of the Message Passing Interface (MPI) standard [27]. Because the code was written in C and because implementations of MPI exist for many... |

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Citation Context ... have been several investigations into accelerating WR [4, 7], including using multigrid [8, 9], Krylov-subspace [10, 11], and convolution SOR techniques [12]. In this paper, we extend the results in =-=[10, 11, 12, 13]-=- and provide experimental results using waveform methods on two different parallel machines — a cluster of SPARC workstations and an IBM SP-2 — for performing transient simulation on a variety of MOS ... |

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Citation Context ...-order backward difference formula. For an N-node mesh, the resulting sequence of nonlinear algebraic systems in 3N unknowns is typically solved with some variant of Newton’s method and/or relaxation =-=[14, 18]-=-. This approach can be disadvantageous for a parallel implementation, especially for MIMD parallel computers having a high communication latency, since the processors will have 4sto synchronize repeat... |

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Citation Context ...performing senficonductor device transient simulation and the increasing availability of parallel computers suggest that parallel algorithms be developed and used for this problem. Results in [1] and =-=[2]-=- demonstrate that SIMD type parallel machines can be used effectively for device transient simulation. However, special-purpose SIMD machines have not been cost-effective enough (in terms of hardware ... |

6 |
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Citation Context ...scheme, efficiency of WR depends on rapid convergence, and there have been several investigations into accelerating WR, including multigrid [7], Krylov-subspace [8, 9], and convolution SOR techniques =-=[10, 11]-=-. In this paper we extend the *The MathWorks, Inc., 24 Prime Park Way, Natick, MA 01760 (mwr@mathworks.com). tDept. of Comp. Sci. and Eng., University of Notre Dame, Notre Dame, IN 46556 (Andrew. Lums... |

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Citation Context ... as x k+1 (z) =HGJ (z)x k (z), where HGJ (z) = 1; z;1 h I +D ;1 (L +U )� andx k+1 (z) =x k+1 (z);x k (z). Clearly, the spectrum ofHGJ (z) depends on z. In a result reminiscent of classical SOR theory =-=[20, 21]-=-, it can be shown that for a common, but restricted, class of matricesA, the z-transform !opt(z) of the optimal CSOR sequence !opt[m] for solving the linear initial value problem (15) may be expressed... |

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Citation Context ...tep computation. As with any iterative scheme, efficiency of WR depends on rapid convergence, and there have been several investigations into accelerating WR, including multigrid [7], Krylov-subspace =-=[8, 9]-=-, and convolution SOR techniques [10, 11]. In this paper we extend the *The MathWorks, Inc., 24 Prime Park Way, Natick, MA 01760 (mwr@mathworks.com). tDept. of Comp. Sci. and Eng., University of Notre... |

5 |
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Citation Context ...mented in the WR-based device transient simulation program pWORDS, which supports computation on the Intel iPSC/860 as well as workstation clusters running the Parallel Virtual Machine (PVM) software =-=[18]-=-. The pWORDS program uses a manager/worker scheme in which a host program assigns tasks to compute nodes and gathers their results when they are nished. 4.1 Parallel Waveform Methods The waveform solu... |

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Citation Context ...ense of performing senficonductor device transient simulation and the increasing availability of parallel computers suggest that parallel algorithms be developed and used for this problem. Results in =-=[1]-=- and [2] demonstrate that SIMD type parallel machines can be used effectively for device transient simulation. However, special-purpose SIMD machines have not been cost-effective enough (in terms of h... |

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Citation Context ... As with any iterative schemc, overall computational efficiency of WR depends on rapid convergence, and there have been several investigations into accelerating WR [5], [8], including using multigrid =-=[9]-=-, [ 10], Krylov-subspace [ 11 ], [ 12], and convolution successive overrelaxation (CSOR) techniques [13]. In this paper, we extend the results in [11]-[14] and provide experimental results using wavef... |

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Citation Context ...scheme, efficiency of WR depends on rapid convergence, and there have been several investigations into accelerating WR, including multigrid [7], Krylov-subspace [8, 9], and convolution SOR techniques =-=[10, 11]-=-. In this paper we extend the *The MathWorks, Inc., 24 Prime Park Way, Natick, MA 01760 (mwr@mathworks.com). tDept. of Comp. Sci. and Eng., University of Notre Dame, Notre Dame, IN 46556 (Andrew. Lums... |

3 | et al., "A users' guide to PVM parallel virtual machine - Beguilin - 1992 |

2 |
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Citation Context ...cy and limited communication bandwidth. To obtain high parallel performance on a loosely coupled MIMD parallel computer, it is critical that a numerical method avoid frequent parallel synchronization =-=[3]-=-. For the application of semiconductor device transient simulation, this has been achieved by careful domain partitioning, for example see [4]. The waveform relaxation (WR) approach to solving time-de... |

2 |
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Citation Context ... of WR depends on rapid convergence, and there have been several investigations into accelerating WR [4, 7], including using multigrid [8, 9], Krylov-subspace [10, 11], and convolution SOR techniques =-=[12]-=-. In this paper, we extend the results in [10, 11, 12, 13] and provide experimental results using waveform methods on two different parallel machines — a cluster of SPARC workstations and an IBM SP-2 ... |

2 | Accelerating dynamic iteration methods with application to semiconductor device simulation - Lumsdaine, White - 1995 |

1 |
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Citation Context ... numerical method avoid frequent parallel synchronization [3]. For the application of semiconductor device transient simulation, this has been achieved by careful domain partitioning, for example see =-=[4]-=-. The waveform relaxation (WR) approach to solving time-dependent initial-value problems reduces parallel synchonization using an almost orthogonal approach to careful domain partitioning. The iterate... |

1 |
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Citation Context ...ormance of the accelerated waveform methods and the pointwise Newton-CGS algorithms, numerical expcriments were conducted using eight examples. instead of plain WR, the more efficient WR Newton (WRN) =-=[6]-=- variant was used. Although it is more efficient, the WRN method is not an accelerated waveform method. That is, its convergence rate is the same as that of WR but it takes about l/3 the work per iter... |

1 | Parallel Multigrid Waveform Relaxation for Parabolic Problems, Teubner-Skripten zur Numerik - Vandev, alle - 1993 |

1 | Conjugate direction waveform methods br transient two dimensional simulation of MOS de vices - Lumsdaine, Reichelt, et al. - 1991 |

1 |
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Citation Context ... equivalently, to the solution of (6). A more detailed analysis of convergence can be derived by considering (13) on the interval [0, oc) in which case ]C has nonzero spectral radius [5]. As shown in =-=[12]-=-, Krylov-subspace methods can be applied to (13) to accelerate the convergence of WR, but as tC is not self-adjoint, a variant suitable for nonself-adjoint operators must be used. One such method is w... |

1 |
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Citation Context ... there have been several investigations into accelerating WR [5], [8], including using multigrid [9], [ 10], Krylov-subspace [ 11 ], [ 12], and convolution successive overrelaxation (CSOR) techniques =-=[13]-=-. In this paper, we extend the results in [11]-[14] and provide experimental results using waveform methods on two different parallel machines (a cluster of SPARC workstations and an IBM SP-2) for per... |

1 |
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Citation Context ...s, No and Nn are the donor and acceptor concentrations, R is the net generation and recombination rate, q is the .magnitude of electronic charge, andsis the spatially dependent dielecthc pemfittivity =-=[15]-=-, [16]. The cuent densities J, and Jp are given by the driftdiffusion approximations = u + qDVn Jp = q/xppsqsqDpVp - kT/xppu - qDpVp where , and/p e the electron and hole mobilities, d and Dj are the ... |

1 |
iterative Solution of Laje Liqueur Systems
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Citation Context ...k = 0,1,..- until convergedsIterate: For each node i, solve for .}4t,+t , %swavefos: with .1(0) = 0 = LUMSDAINE et d.: ACCELERATED WAVEFORM METHODS 719 In a result reminiscent of classical SOR theory =-=[21]-=-, [22], it can be shown that for a common, but restricted, class of matrices A, the z-transform w(,v,(z ) of the optimal CSOR sequence Wov,[m] for solving the linear IVP (10) with Algorithm 3.2 may be... |

1 |
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Citation Context ...veform GMRES (WGM720 IEEE TRANSACTIONS ON COMPUTER AIDED DESIGN OF IN'IEGRATED CIRCUITS AND SYSTEMS, VOL. 15, NO. 7, JULY 1996 RES), an extension of the generalized minimum residual algorithm (GMRES) =-=[24]-=- to the space H and is shown in Algorithm 3.5. Algorithm 3.5 (Waveform GMRES) 1) Start: Set rs- p- (I-1C)x , vs= r/llr[],s= Ilrll 2) Iterate: For k = 1, 2,- - , until converged:s= :i = 1,2,....ks0 + (... |

1 |
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(Show Context)
Citation Context ...:(t)W(), which can be peffoed in parallel, followed by a global sum of the results. Another ylov-subspace method that is suitable for accelerating WR is the conjugate gradient squared algorithm (CGS) =-=[25]-=-. The waveform CGS (WCGS) algorithm is a straightforwd extension of CGS in which, like WGMRES, the matfixvector products are replaced by operator-wavefo products, and the vector inner products arc rep... |

1 |
Accelm'ating dynamic iteraliou methods with application to seraiconductor device simulation
- Lumsdaine, White
- 1992
(Show Context)
Citation Context ...and the vector inner products arc replaced by waveform inner products. Because this extension is so straighfforwd, the WCGS algorithm will not be lismd here, but a description of WCGS can be IBund in =-=[26]-=-. To use the wavefo ylov-subspace methods on the nonlinear device system (1), Newton's method is applied to (1), in a process sometimes refeed to as the wavefo Newton method (WN) [27], to obtain the f... |

1 |
A transpose-free quasiminimum residual algorithm for nonHermitian linear systems
- Frcund
- 1993
(Show Context)
Citation Context ...e linear systems arising at each Newton iteration of each timestep of an implicit integration fomula applied to (1). Stabilized variants of the conjngate-gradient squared (CGS) algorithm [25], [3 l], =-=[32]-=- are the most popular iterative Inethods for device simulation and they have proved to be the most effective serial algorithm (as well as the most effective parallel pointwise algorithm) for the examp... |

1 |
Message Passing Interl.cc Forum, "MPI-2
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(Show Context)
Citation Context ...ap these communication operations with local computation, a significant amount of interprocessor synchronization is still required. Nonblocking collective operations, such as those proposed for MPI-2 =-=[33]-=-, may prove helpful, however. V. EXPERIMENTAL RESULTS To compare the parallel performance of the accelerated waveform methods and the pointwise Newton-CGS algorithms, numerical expcriments were conduc... |

1 |
CODECS: A mixed-levd device and circuit simulator
- Mayaram, Pederson
- 1988
(Show Context)
Citation Context ...etize the system in time with a low-order implicit integration method. The resulting sequence of nonlinear algebraic systems is typically solved with some variant of Newton's method and/or relaxation =-=[12, 14]-=-. This approach can be disadvantageous for a parallel implementation, especially for MIMD parallel computers having a high communication latency, since the processors will have to synchronize repeated... |

1 |
et al., "A users' guide to PVM parallel virtual machine
- Beguilln
- 1992
(Show Context)
Citation Context ...mented in the WR-based device transient simulation program pWORDS, which supports computation on the Intel iPSC/860 as well as workstation clusters running the Parallel Virtual Machine (PVM) software =-=[18]-=-. The pWORDS program uses a manager /worker scheme in which a host program assigns tasks to compute nodes and gathers their results when they are finished. 4.1 Parallel Waveform Methods The waveform s... |