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Mohammad Alkhaleel - One of the best experts on this subject based on the ideXlab platform.
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parameter optimization in waveform relaxation for fractional order RC ciRCuits
IEEE Transactions on Circuits and Systems I-regular Papers, 2017Co-Authors: Shulin Wu, Mohammad AlkhaleelAbstract:The longitudinal waveform relaxation (WR) proposed by Gander and Ruehli converges faster than the classical WR method. For the former, a free parameter $\alpha$ is contained, which has a significant effect on the convergence rate. The optimization of this parameter is thus an important issue in practice. Here, we apply this new WR method to the fractional-order RC ciRCuits, and optimize such a parameter at the continuous and discrete levels (this gives two parameters $\alpha ^{c}_{\mathrm{ opt}}$ and $\alpha ^{d}_{\mathrm{ opt}}$ ). We consider three simple but widely used convolution quadrature for discretization, based on the implicit-Euler method, the two-step backward difference formula, and the trapezoidal rule, and we derive the parameter $\alpha ^{d}_{\mathrm{ opt}}$ for each quadrature. Interestingly, it is found that for the former two quadratures, the optimized parameter $\alpha ^{d}_{\mathrm{ opt}}$ results in a much better convergence rate than $\alpha ^{c}_{\mathrm{ opt}}$ , while for the quadrature based on the trapezoidal rule, $\alpha ^{d}_{\mathrm{ opt}}$ and $\alpha ^{c}_{\mathrm{ opt}}$ result in the same convergence rate.
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convergence analysis of the neumann neumann waveform relaxation method for time fractional RC ciRCuits
Simulation Modelling Practice and Theory, 2016Co-Authors: Mohammad AlkhaleelAbstract:Abstract The classical waveform relaxation (WR) methods rely on decoupling the large-scale ODEs system into small-scale subsystems and then solving these subsystems in a Jacobi or Gauss–Seidel pattern. However, in general it is hard to find a clever partition and for strongly coupled systems the classical WR methods usually converge slowly and non-uniformly. On the contrary, the WR methods of longitudinal type, such as the Robin-WR method and the Neumann–Neumann waveform relaxation (NN-WR) method, possess the advantages of simple partitioning procedure and uniform convergence rate. The Robin-WR method has been extensively studied in the past few years, while the NN-WR method is just proposed very recently and does not get much attention. It was shown in our previous work that the NN-WR method converges much faster than the Robin-WR method, provided the involved parameter, namely β, is chosen properly. In this paper, we perform a convergence analysis of the NN-WR method for time-fractional RC ciRCuits, with special attention to the optimization of the parameter β. For time-fractional PDEs, this work corresponds to the study of the NN-WR method at the semi-discrete level. We present a detailed numerical test of this method, with respect to convergence rate, CPU time and asymptotic dependence on the problem/discretization parameters, in the case of two- and multi-subciRCuits.
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optimization of transmission conditions in waveform relaxation techniques for RC ciRCuits
SIAM Journal on Numerical Analysis, 2014Co-Authors: Mohammad Alkhaleel, Martin J Gander, Albert E. RuehliAbstract:Waveform relaxation techniques have become increasingly important with the wide availability of parallel computers with a large number of processors. A limiting factor for classical waveform relaxation, however, is the convergence speed for an important class of problems, especially if long time windows are considered. In contrast, the optimized waveform relaxation algorithm discussed in this paper is well suited to address this problem. Today several numerical analyses have shown that optimized waveform relaxation algorithms can oveRCome slow convergence over long time windows. However, the optimized waveform relaxation techniques require the determination of optimized parameters. In this paper, we present a theoretical foundation for the determination of the optimized parameters for an important class of RC ciRCuits.
Mohammad Al-khaleel - One of the best experts on this subject based on the ideXlab platform.
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Optimized waveform relaxation methods for RC ciRCuits: discrete case
ESAIM: Mathematical Modelling and Numerical Analysis, 2016Co-Authors: Mohammad Al-khaleelAbstract:The optimized waveform relaxation (OWR) methods, benefiting from intelligent information exchange between subsystems – the so-called transmission conditions (TCs), are recognized as efficient solvers for large scale ciRCuits and get a lot of attention in recent years. The TCs contain a free parameter, namely α , which has a significant influence on the convergence rates. So far, the analysis of finding the best parameter is merely performed at the continuous level and such an analysis does not take into account the influence of temporal discretizations. In this paper, we show that the temporal discretizations do have an important effect on the OWR methods. Precisely, for the Backward–Euler method, compared to the parameter α c opt from the continuous analysis, we show that the convergence rates can be further improved by using the one α d opt analyzed at the discrete level, while for the Trapezoidal rule, it is better to use α c opt . This conclusion is confirmed by numerical results.
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Neumann-Neumann Waveform Relaxation methods for fractional RC ciRCuits
2015 6th International Conference on Information and Communication Systems (ICICS), 2015Co-Authors: Mohammad Al-khaleelAbstract:The Waveform Relaxation (WR) methods are recognized as efficient solvers for large scale ciRCuits and attract a lot of attention in recent years due to their favorable advantages where they are ideally suited for the use of multiple parallel processors for problems with multiple time scales. However, applying classical WR techniques to strongly coupled systems leads to non-uniform convergence. Therefore, more uniform WR methods have been developed. This paper is concerned to generalize the Neumann-Neumann waveform relaxation (NN-WR) method invented recently for time-dependent PDEs to time-fractional ciRCuits which seems to be a promising method in ciRCuit simulations. By choosing the RC ciRCuit in infinite size as the model, we perform a convergence analysis for the NN-WR method and this corresponds to the analysis of this method for PDEs at the semi-discrete level. The NN-WR method contains a free parameter, namely β, which has a significant effect on the convergence rate. For PDEs, the analysis at the space-time continuous level shows β = 1 over 4, while the analysis in this paper shows that, at the semi-discrete level, i.e., for the ciRCuit problem, we can have a better choice which leads to much faster convergence in practical computing. A comparison with the so-called Robin WR is also included.
Bernard N Sheehan - One of the best experts on this subject based on the ideXlab platform.
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predicting coupled noise in RC ciRCuits by matching 1 2 and 3 moments
Design Automation Conference, 2000Co-Authors: Bernard N Sheehan, M GraphicsAbstract:This paper develops the noise-counterparts to familiar delay formulas like Elmore or PRIMO. By matching the first few moments of the network's transfer impedance, we obtain efficient and accurate predictions for maximum noise between two capacitively coupled RC networks. Unlike many crosstalk equations in the literature, the method applies to general topologies and models transition-time dependence as well. Efficient enough for large ciRCuits, the moment-matching noise formulas developed here can serve as a key ingredient in CAD methodologies that ensure a layout is free of noise problems.
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predicting coupled noise in RC ciRCuits
Design Automation and Test in Europe, 2000Co-Authors: Bernard N SheehanAbstract:A novel method which can be regarded as the noise-counterpart of the celebrated Elmore's delay formula-both being based on the first two moments of the network's transfer function-efficiently and accurately predicts maximum noise between two capacitively coupled RC networks, without simulation. The method applies to general topologies (with significant simplification for coupled trees), accurately models how coupling varies with driver transition time, and quantifies the uncertainty in the calculated noise values. Efficient enough for large ciRCuits, the new method can serve as a key ingredient in CAD methodologies to ensure that a layout is noise-problem free.
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ticer realizable reduction of extracted RC ciRCuits
International Conference on Computer Aided Design, 1999Co-Authors: Bernard N SheehanAbstract:Time Constant Equilibration Reduction (TICER) is a novel RC reduction method tailored for extract/reduce CAD tools. Geometry-minded extraction tools fracture nets into parasitics based on local changes in geometry. The resulting RC ciRCuits can have a huge dynamic range of time-constants; by eliminating the extreme time-constants, TICER produces smaller, less-stiff RC networks. It produces realizable RC ciRCuits; can retain original network topology; scales well to large networks (/spl sim/10/sup 7/ nodes); preserves dc and ac behavior; handles resistor loops and floating capacitors; has controllable accuracy; operates in linear time on most nets.
Shiori Omokawa - One of the best experts on this subject based on the ideXlab platform.
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an l band sige hbt active differential equalizer with tunable positive negative gain slopes using transistor loaded RC ciRCuits
Asia-Pacific Microwave Conference, 2018Co-Authors: Yasushi Itoh, Weng Xiaole, Shiori OmokawaAbstract:An L-band SiGe HBT active differential equalizer with tunable positive/negative gain slopes has been realized. the active equalizer employs transistor-loaded parallel RC ciRCuits in the load and feedback paths of the differential amplifier in order to achieve tunable positive/negative gain slopes simultaneously. The implemented active equalizer has achieved positive gain slopes up to + 13.2dB/GHz and negative gain slopes down to − 6.7dB/GHz across 0.1 to 0.8GHz. The active differential equalizer provides positive and negative gain slopes coincidently by making a gain variation larger at lower frequencies and smaller at higher frequencies to compensate for various types of the frequency-and temperature-dependent gain slopes.
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an l band sige hbt active differential equalizer with variable positive negative gain slopes using parallel RC ciRCuits
Asia-Pacific Microwave Conference, 2017Co-Authors: Yasushi Itoh, Hiroaki Takagi, Weng Xiaole, Shiori OmokawaAbstract:An L-band SiGe HBT active differential equalizer with variable positive/negative gain slopes has been designed, fabricated and performed. The active equalizer employs parallel RC ciRCuits in the feedback and load ciRCuits of the differential amplifier to achieve positive/negative gain slopes simultaneously. The implemented active equalizer has achieved positive gain slopes of 0 to +18dB/GHz as well as negative gain slopes of −13 to 0dB/GHz across 0.1 to 1GHz. The active differential equalizer presented in this paper has an outstanding feature of providing both positive and negative gain slopes coincidently from a single ciRCuit, which can be easily employed in the microwave and optical systems with various requirements for frequency and temperature compensations.
Albert E. Ruehli - One of the best experts on this subject based on the ideXlab platform.
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asymptotic analysis for overlap in waveform relaxation methods for RC type ciRCuits
Journal of Scientific Computing, 2020Co-Authors: Martin J Gander, Pratik M Kumbhar, Albert E. RuehliAbstract:Waveform relaxation (WR) methods are based on partitioning large ciRCuits into sub-ciRCuits which then are solved separately for multiple time steps in so called time windows, and an iteration is used to converge to the global ciRCuit solution in each time window. Classical WR converges quite slowly, especially when long time windows are used. To oveRCome this issue, optimized WR (OWR) was introduced which is based on optimized transmission conditions that transfer information between the sub-ciRCuits more efficiently than classical WR. We study here for the first time the influence of overlapping sub-ciRCuits in both WR and OWR applied to RC ciRCuits. We give a ciRCuit interpretation of the new transmission conditions in OWR, and derive closed form asymptotic expressions for the ciRCuit elements representing the optimization parameter in OWR. Our analysis shows that the parameter is quite different in the overlapping case, compared to the non-overlapping one. We then show numerically that our optimized choice performs well, also for cases not covered by our analysis. This paper provides a general methodology to derive optimized parameters and can be extended to other ciRCuits or system of differential equations or space–time PDEs.
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analysis of overlap in waveform relaxation methods for RC ciRCuits
International Conference on Domain Decomposition Methods, 2017Co-Authors: Martin J Gander, Pratik M Kumbhar, Albert E. RuehliAbstract:Waveform relaxation (WR) methods are based on partitioning large ciRCuits into sub-ciRCuits which can be solved separately, and an iteration using transmission conditions then leads to better and better approximations of the entire ciRCuit. Optimized waveform relaxation (OWR) methods work similarly, but they use more effective transmission conditions between sub-ciRCuits. We study here for the first time the influence of overlap on WR and OWR applied to RC ciRCuits. We derive an optimization problem which characterizes the best choice of certain resistance parameters in the transmission conditions for convergence, and give an asymptotic solution of this optimization problem. We also illustrate our results with numerical experiments.
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optimization of transmission conditions in waveform relaxation techniques for RC ciRCuits
SIAM Journal on Numerical Analysis, 2014Co-Authors: Mohammad Alkhaleel, Martin J Gander, Albert E. RuehliAbstract:Waveform relaxation techniques have become increasingly important with the wide availability of parallel computers with a large number of processors. A limiting factor for classical waveform relaxation, however, is the convergence speed for an important class of problems, especially if long time windows are considered. In contrast, the optimized waveform relaxation algorithm discussed in this paper is well suited to address this problem. Today several numerical analyses have shown that optimized waveform relaxation algorithms can oveRCome slow convergence over long time windows. However, the optimized waveform relaxation techniques require the determination of optimized parameters. In this paper, we present a theoretical foundation for the determination of the optimized parameters for an important class of RC ciRCuits.
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convergence of waveform relaxation for RC ciRCuits
Institute for Mathematics and Its Applications, 1994Co-Authors: Albert E. Ruehli, Charles A ZukowskiAbstract:The waveform relaxation [WR] method of ciRCuit simulation has demonstrated the ability to handle large digital VLSI ciRCuits without sacrificing accuracy. Existing programs use reasonable heuristics for ciRCuit partitioning of present day ciRCuits. As ciRCuit models begin to include more and more parasitic elements, due to shrinking geometries, decreasing signal rise times and increasing operating frequencies, the partitioning will become even more important. This paper addresses the question of how partitioning should be done within complex inteRConnect models. Specifically, we consider the partitioning of a limiting case RC ciRCuit example, and investigate its convergence properties and optimal time window size.
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rapid convergence of waveform relaxation
Applied Numerical Mathematics, 1993Co-Authors: Benedict Leimkuhler, Albert E. RuehliAbstract:Abstract Many VLSI ciRCuits to which WR might be applied exhibit a regular, symmetric structure. For such ciRCuits, general spectral estimates of the convergence rate are not necessarily very accurate. By relying on a detailed analysis of the system structure, estimates for the convergence of the waveform relaxation method are given for RC ciRCuits arising as simplified models of a VLSI inteRConnect. These examples suggest a new approach to WR convergence estimation.
