R. Saleh and J. White, "Accelerating relaxation algorithms for circuit simulation using waveform-newton and stepsize refinement," IEEE Trans. CAD, vol. 9, pp. 951-958, September 1990.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... interesting applications are not necessarily described by a linear system of ODEs, but rather by a nonlinear system of ODEs: 8) t) r( t) t) 0 To solve (8) we apply Newton s method directly to the nonlinear ODE system (in a process sometimes referred to as the waveform Newton method (WN) [19]) to obtain the following iteration: 9) t L JF(3m) Em l JF( Em) E rn F(m) Here, J is the Jacobtan of F. We note that (9) is a linear time varying IVP to be solved for , which can be accomplished with a waveform conjugate direction method. The resulting operator ....

R. SALEH AND J. WHITE, Accelerating relaxation algorithms for circuit simulation using waveform-newton and step-size refinement, IEEE Trans. CAD, 9 (1990), pp. 951-958.

....are imposed by ohmic contacts at the source and along the bottom of the substrate and Neumann reflecting boundary conditions are imposed along the left and right sides. The drain driven karD example is described in [16] We have compared our approach with parallelized pointwise Newton GMRES, WRN [18], WN WGMRES, and WRN with CSOR acceleration. The backward Euler method with 256 fixed timesteps is used for all experiments, on a simulation interval of 51.2 or 512 picoseconds. Although the use of global uniform timesteps precludes multirate integration, it also simplifies the problem of ....

R. Saleh and J. White. Accelerating relaxation algorithms for circuit simulation using waveformnewton and step size refinement. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 9:951--958, September 1990.

....an accurate solution of each nonlinear equation will require multiple Newton iterations at each timestep. Instead, by only taking one Newton iteration using an initial guess obtained from the previous waveform iterate at each timestep, one obtains the waveform relaxation Newton (WRN) algorithm [13]. In order to use the waveform Krylov subspace methods, Newton s method is applied to (4) in a process sometimes referred to as the waveform Newton method (WN) 13] to obtain the following iteration: i d dt J F (x m ) j x m 1 = J F (x m )x m Gamma F (x m ) x m 1 (0) ....

.... guess obtained from the previous waveform iterate at each timestep, one obtains the waveform relaxation Newton (WRN) algorithm [13] In order to use the waveform Krylov subspace methods, Newton s method is applied to (4) in a process sometimes referred to as the waveform Newton method (WN) [13] to obtain the following iteration: i d dt J F (x m ) j x m 1 = J F (x m )x m Gamma F (x m ) x m 1 (0) x 0 : 5) Here, J F is the Jacobian of F . We note that (5) is a linear time varying system to be solved for x m 1 , which can be accomplished with WGMRES. The ....

R. Saleh and J. White, Accelerating relaxation algorithms for circuit simulation using waveform-Newton and step-size refinement, IEEE Trans. CAD, 9 (1990), pp. 951--958.

....these methods have persisted even in more recent commercially available simulators, 6, 7, 15] Due in part to the problems with Newton methods, other solve techniques have been pursued in the past fifteen years. Much work has been done on relaxation techniques and parallel circuit simulation, [7, 12, 13, 16, 18, 25, 27, 31, 35, 38, 45, 43, 42, 49, 50, 48, 47]. In general, these techniques were found to work well for particular types of circuits, but were not robust enough for a general simulator. Another solution technique that has been used consistently is continuation or homotopy methods. These techniques have been used from the early stages of ....

Jacob White, R. A. Saleh, Alberto L. Sangiovanni-Vincentelli, and A. Richard Newton. Accelerating relaxation algorithms for circuit simulation using waveform Newton, iterative step size refinement and parallel techniques. In Proceedings of the 1985 International Conference on Computer-Aided Design, pages 5--7. IEEE, 1985.

.... accepted as the next step to be adopted for hardware algorithm acceleration in order to satisfy the increasing demand for more computer power [2] 4] 3] Numerous implementations of CAD algorithms on parallel processing machines, including simulators at various levels, have been reported [5] [6] [7] Most implementations are of logic simulators [8] due to the high degree of parallelism inherent to the eventdriven approach exploited in those simulators. Traditional circuit simulators do not share this characteristic and therefore do not show such a high degree of parallelism. Behind any ....

- Jacob White et al., "Accelerating Relaxation Algorithms for Circuit Simulation using Waveform Newton, Iterative step size refinement and Parallel Techniques", International Conference on CAD, 1985

.... (x(t) t) 0 x(0) x 0 : 6) In order to use the previously developed methods, which only apply to linear systems, we must first linearize (6) To linearize (6) we apply Newton s method directly to the nonlinear ODE system (in a process sometimes referred to as the waveform Newton method (WN) [40]) to obtain the following iteration: i d dt J F (x m ) j x m 1 = J F (x m )x m Gamma F (x m ) x m 1 (0) x 0 : 7) Here, J F is the Jacobian of F . We note that (7) is a linear time varying IVP to be solved for x m 1 , which can be accomplished with a waveform ....

....of frequency and to use a power method to estimate an optimal opt [m] 37] There are a variety of alternative approaches to extending the CSOR algorithm to problems with nonlinearities. We used a waveform extension of relaxation Newton methods (WRN) for solving nonlinear algebraic problems [35, 40]. For the nonlinear problem of the form of (6) the iteration update equation for the i th component of x in a CSOR Newton algorithm is given by d dt x k 1 i (t) F i (x k (t) x i i x k 1 i (t) Gamma x k i (t) j F i (x k (t) t) 0; followed by x k 1 i (t) x k i (t) ....

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R. Saleh and J. White, Accelerating relaxation algorithms for circuit simulation using waveform-Newton and step-size refinement, IEEE Trans. CAD, 9 (1990), pp. 951--958.

....reflecting boundary conditions are imposed along the left and right sides. The description of device and illustration of the drain driven karD example are described in Figure 1 and Table I which are taken from [23] We have compared our approach with parallelized pointwise Newton GMRES, WRN [25], WN WGMRES, and WRN with CSOR acceleration. The backward Euler method with 256 fixed timesteps is used for all experiments, 6 Algorithm 2 Improved Quasi Minimal Residual Method 1: v 1 = w 1 = r 0 = b Gamma Ax 0 ; 1 = 1; 0 = 1; 2: p 0 = q 0 = u 0 = d 0 = f 0 = 0; fl 1 = v 1 ; v 1 ) ....

R. Saleh and J. White. Accelerating relaxation algorithms for circuit simulation using waveform-newton and step size refinement. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 9:951--958, September 1990.

....For k = 1,2, until satisfied do: hLk ( I )vk,vJ) j 1,2, k 9 (I g)v = i=x hi,v , v = 9 i h x, 3. Form approximate solution: o V y, where y minimizes ] fie To apply WGMRES to the nonlinear device sys tem (1) the system is linearized with waveform Newton (WN) [17] and WGMRES is used to solve the resulting sequence of time varying linear IVPs [9] 4 Parallel Implementation Various parallel solution methods have been implemented in the WR based device transient simulation program pWORDS, which supports computation on the Intel iPSC 860 as well as ....

....were imposed by ohmic contacts at the source and along the bottom of the substrate and Neumann reflecting boundary conditions were imposed along the left and right sides. The drain driven karD example is shown in Figure 1. The experiments compared parallelized pointwise Newton GMRES, WRN [17], WN WGMRES, and WRN with CSOR acceleration. The backward Euler methoe[ with 256 fixed timesteps was used for all experiment , on a simulation interval of 51.2 or 512 picoseconds. Although the use of global uniform timesteps precludes multirate integration (one of the primary computational ....

R. Saleh and J. White, "Accelerating relaxation algorithms for circuit simulation using waveform-newton and stepsize refinement," IEEE Trans. CAD, vol. 9, pp. 951-958, September 1990.

....That is, use a single iteration of some kind of waveform Newton algorithm to approximately solve the WR iteration equations. This approach is the main focus of this paper. It is reasonably straightforward to derive the waveform Newton (WN) and waveform relaxation Newton (WRN) algorithms [12] [14], and show that WRN has similar convergence properties to standard WR. In addition, the iteration equations for WRN are time varying linear differential equations and are easier to solve than the nonlinear differential iteration equations of WR. However, WRN does not prove to be much more ....

.... WRN with Timestep Refinement The WRN algorithm combined with the timestep refinement strategy also has several advantages that make it appropriate for use on parallel processors [20] For any decomposition method, the decomposed subsystems can be solved independently on parallel processors [14]. Also, as with any waveform relaxation method, solving the decomposed subsystems is a significant computation involving numerically integrating the independent differential equations over some interval of time. The WRN algorithm has an additional advantage tbr parallel computation. Since the ....

[Article contains additional citation context not shown here]

J. White. R. Saleh, A Sangiovanni-Vincentelli, and A. R. Newton. 'Accelerating relaxation algorithms for circuit simulation using waveform Newton, step-size refinement and parallel techniques," in Proc. lnt. Conf on CAD, Santa Clara, CA, Nov. 1985. pp. 5-7.

....solution is a complicated computation, any parallel onmmunioation overhead is likely to be an insignificant portion of the computation time. Performing the subcircuit computations in parallel, which will be referred to as block level parallelism, has been the focus of most previous work[11,12,13,15] However, it is only of the parallelism that can be exploited in these methods. There is substantial parallelism inside the block computation, but it is more dif ficult to exploit because the computation is harder to decompese into independent pieces. In this paper, a two level approach ....

....too much scheduling overhead to be efficiently implemented on a loosely coupled parallel processor. The algorithm was implemented on a Sequent Balance 8000 system, a single bus con nected multiprocessor and the following table of results indicate that the algorithm is, in fact, effective[12] TABLE 1 TIMEPOINT PIPELINING WR CPU TIME VS # OF PROCESSORS Circuit FET n I 3 6 9 uP Control 116 704 247 159 149 Eprom 348 745 265 185 182 Cmos Ram 428 3379 1217 642 496 3. INSIDE BLOCK PARALLELISM If a block relaxation method like WR is used, the problem of computing the time domain ....

[Article contains additional citation context not shown here]

J. White, R. Saleh, A. Sangiovanni-Vincentelli, and A. R. Newton, "Accelerating Relaxation Algorithms for Circuit Simulation using Waveform Newton, Iterative Step Size Refinement, and Parallel Techniques" Int. conf on computer-Aided Design, Santa Clara, California, November 1985.

....P k j=1 h j;k v j ffl h k 1;k = k v k 1 k ffl v k 1 = v k 1 =h k 1;k 3. Form approximate solution: x k = x 0 V k y k , where y k minimizes kfie 1 Gamma H k y k k, To apply WGMRES to the nonlinear device system (1) the system is linearized with waveform Newton (WN) [17] and WGMRES is used to solve the resulting sequence of time varying linear IVPs [9] 4 Parallel Implementation Various parallel solution methods have been implemented in the WR based device transient simulation program pWORDS, which supports computation on the Intel iPSC 860 as well as ....

....were imposed by ohmic contacts at the source and along the bottom of the substrate and Neumann reflecting boundary conditions were imposed along the left and right sides. The drain driven karD example is shown in Figure 1. The experiments compared parallelized pointwise Newton GMRES, WRN [17], WN WGMRES, and WRN with CSOR acceleration. The backward Euler method with 256 fixed timesteps was used for all experiments, on a simulation interval of 51.2 or 512 picoseconds. Although the use of global uniform timesteps precludes multirate integration (one of the primary computational ....

R. Saleh and J. White, "Accelerating relaxation algorithms for circuit simulation using waveform-newton and stepsize refinement," IEEE Trans. CAD, vol. 9, pp. 951--958, September 1990.

....Methods for Nonlinear Systems. Consider the problem of numerically solving the nonlinear IVP: d dt x(t) F (x(t) t) 0 x(0) x 0 : 3.1) To solve (3. 1) we apply Newton s method directly to the nonlinear ODE system (in a process sometimes referred to as the waveform Newton method (WN) [29]) to obtain the following iteration: i d dt J F (x m ) j x m 1 = J F (x m )x m Gamma F (x m ) x m 1 (0) x 0 : 3.2) Here, J F is the Jacobian of F . We note that (3.2) is a linear time varying IVP to be solved for x m 1 , which can be accomplished with a waveform ....

R. SALEH AND J. WHITE, Accelerating relaxation algorithms for circuit simulation using waveform-Newton and step-size refinement, IEEE Trans. CAD, 9 (1990), pp. 951--958.

.... straightforward, the WCGS algorithm will not be listed here, but a description of WCGS can be found in [24] To use the waveform Krylov subspace methods on the nonlinear device system (6) Newton s method is applied to (6) in a process sometimes referred to as the waveform Newton method (WN) [25] to obtain the following iteration: Gamma d dt J F (x m ) Delta x m 1 = J F (x m )x m Gamma F (x m ) with x m 1 (0) x 0 : 20) Here, J F is the Jacobian of F . We note that (20) is a linear time varying system to be solved for x m 1 , which can be accomplished with ....

R. Saleh and J. White, "Accelerating relaxation algorithms for circuit simulation using waveform-Newton and step-size refinement," IEEE Trans. CAD, vol. 9, no. 9, pp. 951--958, 1990.

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