Tensor Computation: A New Framework for High-Dimensional Problems in EDA

Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit simulation), nonlinearity of devices and circuits, large number of design or optimization parameters (e.g. full-chip routing/placement and circuit sizing), or extensive process variations (e.g. variability/reliability analysis and design for manufacturability). The computational challenges generated by such high dimensional problems are generally hard to handle efficiently with traditional EDA core algorithms that are based on matrix and vector computation. This paper presents "tensor computation" as an alternative general framework for the development of efficient EDA algorithms and tools. A tensor is a high-dimensional generalization of a matrix and a vector, and is a natural choice for both storing and solving efficiently high-dimensional EDA problems. This paper gives a basic tutorial on tensors, demonstrates some recent examples of EDA applications (e.g., nonlinear circuit modeling and high-dimensional uncertainty quantification), and suggests further open EDA problems where the use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and System

Zero-dimensional limit of the two-dimensional Lugiato-Lefever equation

We study effects of tight harmonic-oscillator confinement on the electromagnetic field in a laser cavity by solving the two-dimensional Lugiato-Lefever (2D LL) equation, taking into account self- focusing or defocusing nonlinearity, losses, pump, and the trapping potential. Tightly confined (quasi-zero-dimensional) optical modes (pixels), produced by this model, are analyzed by means of the variational approximation, which provides a qualitative picture of the ensuing phenomena. This is followed by systematic simulations of the time-dependent 2D LL equation, which reveal the shape, stability, and dynamical behavior of the resulting localized patterns. In this way, we produce stability diagrams for the expected pixels. Then, we consider the LL model with the vortical pump, showing that it can produce stable pixels with embedded vorticity (vortex solitons) in remarkably broad sta- bility areas. Alongside confined vortices with the simple single-ring structure, in the latter case the LL model gives rise to stable multi-ring states, with a spiral phase field. In addition to the numeri- cal results, a qualitatively correct description of the vortex solitons is provided by the Thomas-Fermi approximation.Comment: 11 pages, 15 figures, Eur. Phys. Journal D, in press (Topical Issue "Theory and Applications of the Lugiato-Lefever Equation"

Efficient simulation of solution curves and bifurcation loci in injection-locked oscillators

A new method is presented for the two-level harmonic-balance analysis of multivalued synchronized solution curves in injection-locked oscillators. The method is based on the extraction of a nonlinear admittance function, which describes the circuit response from the input source terminals. It does not require any optimization or parameter switching procedures, this constituting a significant advantage compared with previous analysis techniques. With additional mathematical conditions, it enables a straightforward determination of the turning point and Hopf bifurcation loci that delimit the stable injection-locked operation bands. The codimension two bifurcation point at which the turning point and Hopf bifurcation loci merge is analyzed in detail, as well as the saddle-connection locus. As it is shown, a second intersection of the saddle-connection locus with the turning point locus acts as a boundary between synchronization points and points associated with jumps and hysteresis. The likely observation of chaotic solutions in the neighborhood of the saddle-connection locus is discussed too. The techniques have been validated by application to several injection-locked oscillators, obtaining good agreement with the experimental results.This work was supported by the Spanish Ministry of Economy and competitiveness under contract TEC2011-29264-C03-01 and the predoctoral fellowship for researchers in training of the University of Cantabria and the Regional Ministry of Education of the Government of Cantabria

Frequency domain steady-state simulation of oscillators

The focus of this work is on developing algorithms for frequency domain steady-state analysis of oscillators. Convergence problems associated with the frequency domain harmonic balance simulation of oscillators have been examined. Globally convergent homotopy methods have been combined with the harmonic balance method for robust high-Q oscillator simulation. Various homotopy options are evaluated leading to an algorithm that is applicable to a wide variety of oscillator circuits. Two new approaches have also been developed for the simulation of ring oscillators using the harmonic balance method. These include a single-delay cell method and a multiple-probe method. The new methods that have been proposed are robust compared to traditional methods and readily converge for a wide range of single-ended and differential oscillators. They enable harmonic balance simulation of “difficult-to-converge” oscillator circuits

Efficient and Robust Simulation, Modeling and Characterization of IC Power Delivery Circuits

As the Moore’s Law continues to drive IC technology, power delivery has become one of the most difficult design challenges. Two of the major components in power delivery are DC-DC converters and power distribution networks, both of which are time-consuming to simulate and characterize using traditional approaches. In this dissertation, we propose a complete set of solutions to efficiently analyze DC-DC converters and power distribution networks by finding a perfect balance between efficiency and accuracy. To tackle the problem, we first present a novel envelope following method based on a numerically robust time-delayed phase condition to track the envelopes of circuit states under a varying switching frequency. By adopting three fast simulation techniques, our proposed method achieves higher speedup without comprising the accuracy of the results. The robustness and efficiency of the proposed method are demonstrated using several DCDC converter and oscillator circuits modeled using the industrial standard BSIM4 transistor models. A significant runtime speedup of up to 30X with respect to the conventional transient analysis is achieved for several DC-DC converters with strong nonlinear switching characteristics. We then take another approach, average modeling, to enhance the efficiency of analyzing DC-DC converters. We proposed a multi-harmonic model that not only predicts the DC response but also captures the harmonics of arbitrary degrees. The proposed full-order model retains the inductor current as a state variable and accurately captures the circuit dynamics even in the transient state. Furthermore, by continuously monitoring state variables, our model seamlessly transitions between continuous conduction mode and discontinuous conduction mode. The proposed model, when tested with a system decoupling technique, obtains up to 10X runtime speedups over transistor-level simulations with a maximum output voltage error that never exceeds 4%. Based on the multi-harmonic averaged model, we further developed the small-signal model that provides a complete characterization of both DC averages and higher-order harmonic responses. The proposed model captures important high-frequency overshoots and undershoots of the converter response, which are otherwise unaccounted for by the existing techniques. In two converter examples, the proposed model corrects the misleading results of the existing models by providing the truthful characterization of the overall converter AC response and offers important guidance for converter design and closed-loop control. To address the problem of time-consuming simulation of power distribution networks, we present a partition-based iterative method by integrating block-Jacobi method with support graph method. The former enjoys the ease of parallelization, however, lacks a direct control of the numerical properties of the produced partitions. In contrast, the latter operates on the maximum spanning tree of the circuit graph, which is optimized for fast numerical convergence, but is bottlenecked by its difficulty of parallelization. In our proposed method, the circuit partitioning is guided by the maximum spanning tree of the underlying circuit graph, offering essential guidance for achieving fast convergence. The resulting block-Jacobi-like preconditioner maximizes the numerical benefit inherited from support graph theory while lending itself to straightforward parallelization as a partitionbased method. The experimental results on IBM power grid suite and synthetic power grid benchmarks show that our proposed method speeds up the DC simulation by up to 11.5X over a state-of-the-art direct solver

Efficient and Robust Simulation, Modeling and Characterization of IC Power Delivery Circuits

As the Moore’s Law continues to drive IC technology, power delivery has become one of the most difficult design challenges. Two of the major components in power delivery are DC-DC converters and power distribution networks, both of which are time-consuming to simulate and characterize using traditional approaches. In this dissertation, we propose a complete set of solutions to efficiently analyze DC-DC converters and power distribution networks by finding a perfect balance between efficiency and accuracy. To tackle the problem, we first present a novel envelope following method based on a numerically robust time-delayed phase condition to track the envelopes of circuit states under a varying switching frequency. By adopting three fast simulation techniques, our proposed method achieves higher speedup without comprising the accuracy of the results. The robustness and efficiency of the proposed method are demonstrated using several DCDC converter and oscillator circuits modeled using the industrial standard BSIM4 transistor models. A significant runtime speedup of up to 30X with respect to the conventional transient analysis is achieved for several DC-DC converters with strong nonlinear switching characteristics. We then take another approach, average modeling, to enhance the efficiency of analyzing DC-DC converters. We proposed a multi-harmonic model that not only predicts the DC response but also captures the harmonics of arbitrary degrees. The proposed full-order model retains the inductor current as a state variable and accurately captures the circuit dynamics even in the transient state. Furthermore, by continuously monitoring state variables, our model seamlessly transitions between continuous conduction mode and discontinuous conduction mode. The proposed model, when tested with a system decoupling technique, obtains up to 10X runtime speedups over transistor-level simulations with a maximum output voltage error that never exceeds 4%. Based on the multi-harmonic averaged model, we further developed the small-signal model that provides a complete characterization of both DC averages and higher-order harmonic responses. The proposed model captures important high-frequency overshoots and undershoots of the converter response, which are otherwise unaccounted for by the existing techniques. In two converter examples, the proposed model corrects the misleading results of the existing models by providing the truthful characterization of the overall converter AC response and offers important guidance for converter design and closed-loop control. To address the problem of time-consuming simulation of power distribution networks, we present a partition-based iterative method by integrating block-Jacobi method with support graph method. The former enjoys the ease of parallelization, however, lacks a direct control of the numerical properties of the produced partitions. In contrast, the latter operates on the maximum spanning tree of the circuit graph, which is optimized for fast numerical convergence, but is bottlenecked by its difficulty of parallelization. In our proposed method, the circuit partitioning is guided by the maximum spanning tree of the underlying circuit graph, offering essential guidance for achieving fast convergence. The resulting block-Jacobi-like preconditioner maximizes the numerical benefit inherited from support graph theory while lending itself to straightforward parallelization as a partitionbased method. The experimental results on IBM power grid suite and synthetic power grid benchmarks show that our proposed method speeds up the DC simulation by up to 11.5X over a state-of-the-art direct solver

Oscillation modes in symmetrical wireless-locked systems

Time synchronization of multiple elements of a wireless network can be achieved through the wireless coupling of their oscillator circuits. Most previous works on wireless locking of oscillators analyze the system in an idealized manner, representing the oscillator elements with phase models and describing the propagation effects with constant scalar coefficients and time delays. Here, a realistic analysis of the wireless system is presented, which relies on the extraction of the oscillator models from harmonic-balance (HB) simulations and takes into account the antenna gains and propagation effects. The most usual network configurations, corresponding to ring, fully connected, and star topologies, are investigated in detail. In symmetric conditions, the oscillation modes are detected through an eigenvalue/eigenvector calculation of an equivalent coupling matrix. For each particular mode, the system is analyzed in the following manners: by means of an analytical formulation, able to provide all the coexistent solutions, and through a circuit-level HB simulation of an equivalent system with a reduced number of oscillator elements. The stability properties are determined by means of a perturbation system of general application to any coupled structure. A specific formulation is also derived to predict the impact of discrepancies between the oscillator elements. All the results have been validated with independent circuit-level simulations and measurements.This work was supported in part by the Spanish Ministry of Economy and Competitiveness under the research project TEC2017-88242-C3-1-R, in part by the European Regional Development Fund (ERDF/FEDER), in part by Juan de la Cierva Research Program under IJCI-2014-19141, and in part by the Parliament of Cantabria under the project Cantabria Explora 12.JP02.64069

Parallel Algorithms for Time and Frequency Domain Circuit Simulation

As a most critical form of pre-silicon verification, transistor-level circuit simulation is an indispensable step before committing to an expensive manufacturing process. However, considering the nature of circuit simulation, it can be computationally expensive, especially for ever-larger transistor circuits with more complex device models. Therefore, it is becoming increasingly desirable to accelerate circuit simulation. On the other hand, the emergence of multi-core machines offers a promising solution to circuit simulation besides the known application of distributed-memory clustered computing platforms, which provides abundant hardware computing resources. This research addresses the limitations of traditional serial circuit simulations and proposes new techniques for both time-domain and frequency-domain parallel circuit simulations. For time-domain simulation, this dissertation presents a parallel transient simulation methodology. This new approach, called WavePipe, exploits coarse-grained application-level parallelism by simultaneously computing circuit solutions at multiple adjacent time points in a way resembling hardware pipelining. There are two embodiments in WavePipe: backward and forward pipelining schemes. While the former creates independent computing tasks that contribute to a larger future time step, the latter performs predictive computing along the forward direction. Unlike existing relaxation methods, WavePipe facilitates parallel circuit simulation without jeopardizing convergence and accuracy. As a coarse-grained parallel approach, it requires low parallel programming effort, furthermore it creates new avenues to have a full utilization of increasingly parallel hardware by going beyond conventional finer grained parallel device model evaluation and matrix solutions. This dissertation also exploits the recently developed explicit telescopic projective integration method for efficient parallel transient circuit simulation by addressing the stability limitation of explicit numerical integration. The new method allows the effective time step controlled by accuracy requirement instead of stability limitation. Therefore, it not only leads to noticeable efficiency improvement, but also lends itself to straightforward parallelization due to its explicit nature. For frequency-domain simulation, this dissertation presents a parallel harmonic balance approach, applicable to the steady-state and envelope-following analyses of both driven and autonomous circuits. The new approach is centered on a naturally-parallelizable preconditioning technique that speeds up the core computation in harmonic balance based analysis. The proposed method facilitates parallel computing via the use of domain knowledge and simplifies parallel programming compared with fine-grained strategies. As a result, favorable runtime speedups are achieved

Nonlinear microwave simulation techniques

The design of high performance circuits with short manufacturing cycles and low cost demands reliable analysis tools, capable to accurately predict the circuit behaviour prior to manufacturing. In the case of nonlinear circuits, the user must be aware of the possible coexistence of different steady-state solutions for the same element values and the fact that steady-state methods, such as harmonic balance, may converge to unstable solutions that will not be observed experimentally. In this contribution, the main numerical iterative methods for nonlinear analysis, including time-domain integrations, shooting, harmonic balance and envelope transient, are briefly presented and compared. The steady-state methods must be complemented with a stability steady-state analysis to verify the physical existence of the solution. This stability analysis can also be combined with the use of auxiliary generators to simulate the circuit self-oscillation and predict qualitative changes in the solution under the continuous variation of a parameter. The methods will be applied to timely circuit examples that are demanding from the nonlinear analysis point of view.This work has been supported by the Spanish Government under contract TEC2014-60283-C3-1-R and the Parliament of Cantabria (12.JP02.64069)