Heterogeneous photonic-electronic systems, such as co-packaged optics and photonic-electronic artificial intelligence (AI) accelerators, are rapidly gaining traction but also pose significant design challenges due to distinct design methodologies. Digital and analog electronics are typically described using hardware description languages and SPICE, respectively, whereas photonic devices and systems are represented using permittivity tensors on the Yee grid and the Scattering matrix formulation. This disparity necessitates an end-to-end photonic-electronic cosimulation tool to streamline co-design. Most preliminary cosimulation approaches rely on translating photonic compact models into Verilog-A or SPICE models to simulate everything there, which not only introduces the additional complexity of model conversion but also has potential numerical stability problems. Additionally, another critical functionality missing from the current implementation is enabling gradient calculation in these co-simulators, which will be crucial for end-to-end gradient-based electronic-photonic system optimization. To address these challenges, we introduce SPIPE, a differentiable SPICE-level co-simulation framework for integrated photonic-electronic systems. SPIPE is the first co-simulator to overcome model conversion issues and to provide differentiability. Numerical experiments on several circuits confirm the accuracy of SPIPE when compared to analytical solutions and real-world experimental data. Furthermore, in cases where existing simulators are applicable, SPIPE achieves a runtime reduction of 2!+85!A compared to an industry-standard simulator. SPIPE features an integrated simulation interface with a low usage barrier, opening avenues for more accessible and effective photonic-electronic co-design. SPIPE is open sourced: https://github.com/zhengqigao/spipe.

The state-of-the-art (SOTA) deep learning based time series models are inspired by convolutional neural networks (CNN), recurrent neural networks (RNN) or transformers which are successful architectures for domains like vision, text, etc. However, the gold standard architecture for time series modeling is not yet established. In this paper, we propose a new neural network structure that can be used as a strong baseline for time series problems, leveraging dilated kernels with fully convolutional networks (FCNs). The proposed model, called the dilated multi-kernel fully convolutional network (DM-FCN), is a composite model that leverages a vast receptive field and is designed to capture the long-distance interaction in multivariate time series data. We evaluate the performance of the DM-FCN model on a variety of time series benchmarks. Our results show that the baseline DM-FCN model outperforms state-of-the-art models on many of the benchmarks by a large margin. By integrating statistical insights, we also evaluated different variations of DM-FCN and deliberated on model selections across diverse time series data.

Reducing the likelihood of generating harmful and toxic output is an essential task when aligning large language models (LLMs). Existing methods mainly rely on training an external reward model (i.e., another language model) or fine-tuning the LLM using self-generated data to influence the outcome. In this paper, we show that LLMs have the capability of self-detoxification without external reward model learning or retraining of the LM. We propose \textit{Self-disciplined Autoregressive Sampling (SASA)}, a lightweight controlled decoding algorithm for toxicity reduction of LLMs. SASA leverages the contextual representations from an LLM to learn linear subspaces from labeled data characterizing toxic v.s. non-toxic output in analytical forms. When auto-completing a response token-by-token, SASA dynamically tracks the margin of the current output to steer the generation away from the toxic subspace, by adjusting the autoregressive sampling strategy. Evaluated on LLMs of different scale and nature, namely Llama-3.1-Instruct (8B), Llama-2 (7B), and GPT2-L models with the RealToxicityPrompts, BOLD, and AttaQ benchmarks, SASA markedly enhances the quality of the generated sentences relative to the original models and attains comparable performance to state-of-the-art detoxification techniques, significantly reducing the toxicity level by only using the LLM's internal representations.

Accurately predicting the energy consumption of Battery Electric Vehicles (BEVs) is essential for addressing range anxiety, optimizing route planning, and governing infrastructure investments in a rapidly electrifying transportation sector. This paper presents a generalized, flexible, and probably the most comprehensive modeling framework designed to estimate BEV energy consumption under different driving conditions, vehicle configurations, and environmental influences. The model is structured in mechanical, electrical, and auxiliary sub-models. The model incorporates detailed input parameters, such as aerodynamic coefficients, transmission and motor characteristics, regenerative braking constraints, battery capacity, climate control demands, and ambient conditions. Validation results demonstrate a strong alignment between measured and predicted power profiles, with a high coefficient of determination, low RMSE, and low MAE confirming the model’s reliability and adaptability. The introduced framework can be extended to various BEV segments, driving cycles, and environmental conditions, providing valuable information for vehicle manufacturers, fleet managers, and policymakers aiming to improve EV performance, route efficiency, and charging infrastructure deployment.

Neural networks are powerful tools in various applications, and quantifying their uncertainty is crucial for reliable decision-making. In the deep learning field, the uncertainties are usually categorized into aleatoric (data) and epistemic (model) uncertainty. In this paper, we point out that the existing popular variance attenuation method highly overestimates aleatoric uncertainty. To address this issue, we proposed a new estimation method by actively de-noising the observed data. By conducting a broad range of experiments, we demonstrate that our proposed approach provides a much closer approximation to the actual data uncertainty than the standard method.

Accurate estimation of rare failure occurrence probability is crucial for ensuring the proper and reliable functioning of integrated circuits (ICs). Conventional Monte Carlo methods are inefficient, demanding an exorbitant number of samples to achieve reliable estimates. Inspired by the exact sampling capabilities of normalizing flows, we revisit this problem and propose normalizing flow assisted importance sampling, termed NOFIS. NOFIS first learns a sequence of proposal distributions associated with predefined nested subset events by minimizing KL divergence losses. Next, it estimates the rare event probability by utilizing importance sampling in conjunction with the last proposal. The efficacy of our NOFIS method is substantiated through comprehensive qualitative visualizations, affirming the optimality of the learned proposal distribution, as well as 10 quantitative experiments, which highlight NOFIS's superior accuracy over baseline approaches.

Recent years have witnessed a paradigm shift in deep learning from task-centric model design to task-agnostic representation learning and task-specific fine-tuning. Pretrained model representations are commonly evaluated extensively across various real-world tasks and used as a foundation for different downstream tasks. This paper proposes a solution for assessing the quality of representations in a task-agnostic way. To circumvent the need for real-world data in evaluation, we explore the use of synthetic binary classification tasks with Gaussian mixtures to probe pretrained models and compare the robustness-accuracy performance on pretrained representations with an idealized reference. Our approach offers a holistic evaluation, revealing intrinsic model capabilities and reducing the dependency on real-life data for model evaluation. Evaluated with various pretrained image models, the experimental results confirm that our task-agnostic evaluation correlates with actual linear probing performance on downstream tasks and can also guide parameter choice in robust linear probing to achieve a better robustness-accuracy trade-off.

Abstract. This work aims to accelerate the convergence of proximal gradient methods used to solve regularized linear inverse problems. This is achieved by designing a polynomial-based preconditioner that targets the eigenvalue spectrum of the normal operator derived from the linear operator. The preconditioner does not assume any explicit structure on the linear function and thus can be deployed in diverse applications of interest. The efficacy of the preconditioner is validated on three different Magnetic Resonance Imaging applications, where it is seen to achieve faster iterative convergence (around [Formula: see text] faster, depending on the application of interest) while achieving similar reconstruction quality.

Deep neural networks (DNN) have shown great capacity of modeling a dynamical system; nevertheless, they usually do not obey physics constraints such as conservation laws. This paper proposes a new learning framework named $\textbf{ConCerNet}$ to improve the trustworthiness of the DNN based dynamics modeling to endow the invariant properties. $\textbf{ConCerNet}$ consists of two steps: (i) a contrastive learning method to automatically capture the system invariants (i.e. conservation properties) along the trajectory observations; (ii) a neural projection layer to guarantee that the learned dynamics models preserve the learned invariants. We theoretically prove the functional relationship between the learned latent representation and the unknown system invariant function. Experiments show that our method consistently outperforms the baseline neural networks in both coordinate error and conservation metrics by a large margin. With neural network based parameterization and no dependence on prior knowledge, our method can be extended to complex and large-scale dynamics by leveraging an autoencoder.

As a seminal tool in self-supervised representation learning, contrastive learning has gained unprecedented attention in recent years. In essence, contrastive learning aims to leverage pairs of positive and negative samples for representation learning, which relates to exploiting neighborhood information in a feature space. By investigating the connection between contrastive learning and neighborhood component analysis (NCA), we provide a novel stochastic nearest neighbor viewpoint of contrastive learning and subsequently propose a series of contrastive losses that outperform the existing ones. Under our proposed framework, we show a new methodology to design integrated contrastive losses that could simultaneously achieve good accuracy and robustness on downstream tasks. With the integrated framework, we achieve up to 6% improvement on the standard accuracy and 17% improvement on the robust accuracy.

Recent works have developed several methods of defending neural networks against adversarial attacks with certified guarantees. However, these techniques can be computationally costly due to the use of certification during training. We develop a new regularizer that is both more efficient than existing certified defenses, requiring only one additional forward propagation through a network, and can be used to train networks with similar certified accuracy. Through experiments on MNIST and CIFAR-10 we demonstrate improvements in training speed and comparable certified accuracy compared to state-of-the-art certified defenses.

The fragility of modern machine learning models has drawn a considerable amount of attention from both academia and the public. While immense interests were in either crafting adversarial attacks as a way to measure the robustness of neural networks or devising worst-case analytical robustness verification with guarantees, few methods could enjoy both scalability and robustness guarantees at the same time. As an alternative to these attempts, randomized smoothing adopts a different prediction rule that enables statistical robustness arguments which easily scale to large networks. However, in this paper, we point out the side effects of current randomized smoothing workflows. Specifically, we articulate and prove two major points: 1) the decision boundaries of smoothed classifiers will shrink, resulting in disparity in class-wise accuracy; 2) applying noise augmentation in the training process does not necessarily resolve the shrinking issue due to the inconsistent learning objectives.

Recent studies have shown that deep reinforcement learning agents are vulnerable to small adversarial perturbations on the agent's inputs, which raises concerns about deploying such agents in the real world. To address this issue, we propose RADIAL-RL, a principled framework to train reinforcement learning agents with improved robustness against $l_p$-norm bounded adversarial attacks. Our framework is compatible with popular deep reinforcement learning algorithms and we demonstrate its performance with deep Q-learning, A3C and PPO. We experiment on three deep RL benchmarks (Atari, MuJoCo and ProcGen) to show the effectiveness of our robust training algorithm. Our RADIAL-RL agents consistently outperform prior methods when tested against attacks of varying strength and are more computationally efficient to train. In addition, we propose a new evaluation method called Greedy-Worst-Case Reward (GWC) to measure attack agnostic robustness of deep RL agents. We show that GWC can be evaluated efficiently and is a good estimate of the reward under the worst possible sequence of adversarial attacks. All code used for our experiments is available at https://github.com/tuomaso/radial_rl_v2.

The rapid growth of deep learning applications in real life is accompanied by severe safety concerns. To mitigate this uneasy phenomenon, much research has been done providing reliable evaluations of the fragility level in different deep neural networks. Apart from devising adversarial attacks, quantifiers that certify safeguarded regions have also been designed in the past five years. The summarizing work in (Salman et al. 2019) unifies a family of existing verifiers under a convex relaxation framework. We draw inspiration from such work and further demonstrate the optimality of deterministic CROWN (Zhang et al. 2018) solutions in a given linear programming problem under mild constraints. Given this theoretical result, the computationally expensive linear programming based method is shown to be unnecessary. We then propose an optimization-based approach FROWN (Fastened CROWN): a general algorithm to tighten robustness certificates for neural networks. Extensive experiments on various networks trained individually verify the effectiveness of FROWN in safeguarding larger robust regions.

This work studies the sensitivity of neural networks to weight perturbations, firstly corresponding to a newly developed threat model that perturbs the neural network parameters. We propose an efficient approach to compute a certified robustness bound of weight perturbations, within which neural networks will not make erroneous outputs as desired by the adversary. In addition, we identify a useful connection between our developed certification method and the problem of weight quantization, a popular model compression technique in deep neural networks (DNNs) and a ‘must-try’ step in the design of DNN inference engines on resource constrained computing platforms, such as mobiles, FPGA, and ASIC. Specifically, we study the problem of weight quantization – weight perturbations in the non-adversarial setting – through the lens of certificated robustness, and we demonstrate significant improvements on the generalization ability of quantized networks through our robustness-aware quantization scheme.

Verifying robustness of neural networks given a specified threat model is a fundamental yet challenging task. While current verification methods mainly focus on the $\ell_p$-norm threat model of the input instances, robustness verification against semantic adversarial attacks inducing large $\ell_p$-norm perturbations, such as color shifting and lighting adjustment, are beyond their capacity. To bridge this gap, we propose \textit{Semantify-NN}, a model-agnostic and generic robustness verification approach against semantic perturbations for neural networks. By simply inserting our proposed \textit{semantic perturbation layers} (SP-layers) to the input layer of any given model, \textit{Semantify-NN} is model-agnostic, and any $\ell_p$-norm based verification tools can be used to verify the model robustness against semantic perturbations. We illustrate the principles of designing the SP-layers and provide examples including semantic perturbations to image classification in the space of hue, saturation, lightness, brightness, contrast and rotation, respectively. In addition, an efficient refinement technique is proposed to further significantly improve the semantic certificate. Experiments on various network architectures and different datasets demonstrate the superior verification performance of \textit{Semantify-NN} over $\ell_p$-norm-based verification frameworks that naively convert semantic perturbation to $\ell_p$-norm. The results show that \textit{Semantify-NN} can support robustness verification against a wide range of semantic perturbations.

Recent works have empirically shown that there exist adversarial examples that can be hidden from neural network interpretability (namely, making network interpretation maps visually similar), or interpretability is itself susceptible to adversarial attacks. In this paper, we theoretically show that with a proper measurement of interpretation, it is actually difficult to prevent prediction-evasion adversarial attacks from causing interpretation discrepancy, as confirmed by experiments on MNIST, CIFAR-10 and Restricted ImageNet. Spurred by that, we develop an interpretability-aware defensive scheme built only on promoting robust interpretation (without the need for resorting to adversarial loss minimization). We show that our defense achieves both robust classification and robust interpretation, outperforming state-of-the-art adversarial training methods against attacks of large perturbation in particular.

Deep neural networks are known to be fragile to small adversarial perturbations, which raises serious concerns when a neural network policy is interconnected with a physical system in a closed loop. In this paper, we show how to combine recent works on static neural network certification tools with robust control theory to certify a neural network policy in a control loop. We give a sufficient condition and an algorithm to ensure that the closed loop state and control constraints are satisfied when the persistent adversarial perturbation is l-infinity norm bounded. Our method is based on finding a positively invariant set of the closed loop dynamical system, and thus we do not require the continuity of the neural network policy. Along with the verification result, we also develop an effective attack strategy for neural network control systems that outperforms exhaustive Monte-Carlo search significantly. We show that our certification algorithm works well on learned models and could achieve 5 times better result than the traditional Lipschitz-based method to certify the robustness of a neural network policy on the cart-pole balance control problem.

Randomized smoothing is a recently proposed defense against adversarial attacks that has achieved state-of-the-art provable robustness against $\ell_2$ perturbations. A number of works have extended the guarantees to other metrics, such as $\ell_1$ or $\ell_\infty$, by using different smoothing measures. Although the current framework has been shown to yield near-optimal $\ell_p$ radii, the total safety region certified by the current framework can be arbitrarily small compared to the optimal. In this work, we propose a framework to improve the certified safety region for these smoothed classifiers without changing the underlying smoothing scheme. The theoretical contributions are as follows: 1) We generalize the certification for randomized smoothing by reformulating certified radius calculation as a nested optimization problem over a class of functions. 2) We provide a method to calculate the certified safety region using zeroth-order and first-order information for Gaussian-smoothed classifiers. We also provide a framework that generalizes the calculation for certification using higher-order information. 3) We design efficient, high-confidence estimators for the relevant statistics of the first-order information. Combining the theoretical contribution 2) and 3) allows us to certify safety region that are significantly larger than ones provided by the current methods. On CIFAR and Imagenet, the new regions achieve significant improvements on general $\ell_1$ certified radii and on the $\ell_2$ certified radii for color-space attacks ($\ell_2$ perturbation restricted to only one color/channel) while also achieving smaller improvements on the general $\ell_2$ certified radii. As discussed in the future works section, our framework can also provide a way to circumvent the current impossibility results on achieving higher magnitudes of certified radii without requiring the use of data-dependent smoothing techniques.

We propose a new tensor completion method based on tensor trains. The to-be-completed tensor is modeled as a low-rank tensor train, where we use the known tensor entries and their coordinates to update the tensor train. A novel tensor train initialization procedure is proposed specifically for image and video completion, which is demonstrated to ensure fast convergence of the completion algorithm. The tensor train framework is also shown to easily accommodate Total Variation and Tikhonov regularization due to their low-rank tensor train representations. Image and video inpainting experiments verify the superiority of the proposed scheme in terms of both speed and scalability, where a speedup of up to 155× is observed compared to state-of-the-art tensor completion methods at a similar accuracy. Moreover, we demonstrate the proposed scheme is especially advantageous over existing algorithms when only tiny portions (say, 1%) of the to-be-completed images/videos are known.

Verifying robustness of neural network classifiers has attracted great interests and attention due to the success of deep neural networks and their unexpected vulnerability to adversarial perturbations. Although finding minimum adversarial distortion of neural networks (with ReLU activations) has been shown to be an NP-complete problem, obtaining a non-trivial lower bound of minimum distortion as a provable robustness guarantee is possible. However, most previous works only focused on simple fully-connected layers (multilayer perceptrons) and were limited to ReLU activations. This motivates us to propose a general and efficient framework, CNN-Cert, that is capable of certifying robustness on general convolutional neural networks. Our framework is general – we can handle various architectures including convolutional layers, max-pooling layers, batch normalization layer, residual blocks, as well as general activation functions; our approach is efficient – by exploiting the special structure of convolutional layers, we achieve up to 17 and 11 times of speed-up compared to the state-of-the-art certification algorithms (e.g. Fast-Lin, CROWN) and 366 times of speed-up compared to the dual-LP approach while our algorithm obtains similar or even better verification bounds. In addition, CNN-Cert generalizes state-of-the-art algorithms e.g. Fast-Lin and CROWN. We demonstrate by extensive experiments that our method outperforms state-of-the-art lowerbound-based certification algorithms in terms of both bound quality and speed.

The vulnerability to adversarial attacks has been a critical issue for deep neural networks. Addressing this issue requires a reliable way to evaluate the robustness of a network. Recently, several methods have been developed to compute robustness quantification for neural networks, namely, certified lower bounds of the minimum adversarial perturbation. Such methods, however, were devised for feed-forward networks, e.g. multi-layer perceptron or convolutional networks. It remains an open problem to quantify robustness for recurrent networks, especially LSTM and GRU. For such networks, there exist additional challenges in computing the robustness quantification, such as handling the inputs at multiple steps and the interaction between gates and states. In this work, we propose POPQORN (Propagated-output Quantified Robustness for RNNs), a general algorithm to quantify robustness of RNNs, including vanilla RNNs, LSTMs, and GRUs. We demonstrate its effectiveness on different network architectures and show that the robustness quantification on individual steps can lead to new insights.

We propose a novel framework PROVEN to \textbf{PRO}babilistically \textbf{VE}rify \textbf{N}eural network’s robustness with statistical guarantees. PROVEN provides probability certificates of neural network robustness when the input perturbation follow distributional characterization. Notably, PROVEN is derived from current state-of-the-art worst-case neural network robustness verification frameworks, and therefore it can provide probability certificates with little computational overhead on top of existing methods such as Fast-Lin, CROWN and CNN-Cert. Experiments on small and large MNIST and CIFAR neural network models demonstrate our probabilistic approach can tighten up robustness certificate to around $1.8 \times$ and $3.5 \times$ with at least a $99.99%$ confidence compared with the worst-case robustness certificate by CROWN and CNN-Cert.

Verifying the robustness property of a general Rectified Linear Unit (ReLU) network is an NP-complete problem. Although finding the exact minimum adversarial distortion is hard, giving a certified lower bound of the minimum distortion is possible. Current available methods of computing such a bound are either time-consuming or deliver low quality bounds that are too loose to be useful. In this paper, we exploit the special structure of ReLU networks and provide two computationally efficient algorithms (Fast-Lin, Fast-Lip) that are able to certify non-trivial lower bounds of minimum adversarial distortions. Experiments show that (1) our methods deliver bounds close to (the gap is 2-3X) exact minimum distortions found by Reluplex in small networks while our algorithms are more than 10,000 times faster; (2) our methods deliver similar quality of bounds (the gap is within 35% and usually around 10%; sometimes our bounds are even better) for larger networks compared to the methods based on solving linear programming problems but our algorithms are 33-14,000 times faster; (3) our method is capable of solving large MNIST and CIFAR networks up to 7 layers with more than 10,000 neurons within tens of seconds on a single CPU core. In addition, we show that there is no polynomial time algorithm that can approximately find the minimum $\ell_1$ adversarial distortion of a ReLU network with a $0.99\ln n$ approximation ratio unless NP=P, where $n$ is the number of neurons in the network.

This research investigates the effects that electrical mismatches and partial shading can have on the performance of photovoltaic arrays. The analysis adopts a probabilistic point of view where the most relevant parameters of the solar units are seen as random variables. The analysis relies on an efficient and robust simulation technique, based on Wave Digital principles, that is tailored to the modular topology of solar arrays. It is shown how electrical mismatch, solar shading and array topology can interact among them in a quite complex way.

Finding minimum distortion of adversarial examples and thus certifying robustness in neural networks classifiers is known to be a challenging problem. Nevertheless, recently it has been shown to be possible to give a non-trivial certified lower bound of minimum distortion, and some recent progress has been made towards this direction by exploiting the piece-wise linear nature of ReLU activations. However, a generic robustness certification for \textit{general} activation functions still remains largely unexplored. To address this issue, in this paper we introduce CROWN, a general framework to certify robustness of neural networks with general activation functions. The novelty in our algorithm consists of bounding a given activation function with linear and quadratic functions, hence allowing it to tackle general activation functions including but not limited to the four popular choices: ReLU, tanh, sigmoid and arctan. In addition, we facilitate the search for a tighter certified lower bound by \textit{adaptively} selecting appropriate surrogates for each neuron activation. Experimental results show that CROWN on ReLU networks can notably improve the certified lower bounds compared to the current state-of-the-art algorithm Fast-Lin, while having comparable computational efficiency. Furthermore, CROWN also demonstrates its effectiveness and flexibility on networks with general activation functions, including tanh, sigmoid and arctan.

This paper shows how arrays of coupled resonant oscillators can provide wildly disordered phase responses corresponding to nonsynchronized regimes. The unpredictability of their time response makes oscillator arrays attractive randomness sources for cryptographic applications. We describe a possible implementation of a random number generator (RNG) combining the proposed randomness source with a few low cost elements needed for bit stream generation. Several bit stream sequences are simulated with a numerically efficient phase-domain model able to include oscillator phase noise and parameters variability. The randomness of the generated bit streams are checked with the tests provided by the National Institute of Standards and Technology. Our results show that the statistical properties of the proposed RNG are indeed resilient to temperature-induced thermal noise variations as well as to the statistical uncertainty of oscillating frequencies and coupling strengths.

This paper presents an effective approach to variability analysis of integrated capacitors due to manufacturing process uncertainty. The proposed approach combines the generalized polynomial chaos method for uncertainty quantification with the efficient floating random walk algorithm for capacitance extraction. For applications where detailed statistical descriptions are required, the method allows achieving a 1000× acceleration compared to standard Monte Carlo analysis. Application to variability analysis in digital to analog converters is illustrated.

Many critical electronic design automation (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 multirate 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.

Parallelization of traditional accelerated techniques for integral equation solvers has been shown to be inefficient and to scale poorly with the number of parallel computing nodes. This is because traditional methods typically represent the solution using piecewise constant (PWC) basis functions, resulting in gigantic systems of linear equations to solve. In this paper, we propose instantiable basis functions, which generate smaller systems than PWC basis functions for the same accuracy. Furthermore, they redistribute computation from the system solving step to the embarrassingly parallelizable system setup step, hence enabling highly scalable and efficient parallelization. In the examples, we tested, our new solver is six to ten times faster than FASTCAP in serial execution and achieves 90% parallel efficiency in a ten-core distributed-memory system. We developed a toolkit that automates a complete extraction flow from GDSII layout files to capacitance matrices. Our code has been released in the public domain.

Volterra series is a powerful tool for black-box macro-modeling of nonlinear devices. However, the exponential complexity growth in storing and evaluating higher order Volterra kernels has limited so far its employment on complex practical applications. On the other hand, tensors are a higher order generalization of matrices that can naturally and efficiently capture multi-dimensional data. Significant computational savings can often be achieved when the appropriate low-rank tensor decomposition is available. In this paper we exploit a strong link between tensors and frequency-domain Volterra kernels in modeling nonlinear systems. Based on such link we have developed a technique called speedy tensor-aided Volterra-based electronic simulator (STAVES) utilizing high-order Volterra transfer functions for highly accurate time-domain simulation of nonlinear systems. The main computational tools in our approach are the canonical tensor decomposition and the inverse discrete Fourier transform. Examples demonstrate the efficiency of the proposed method in simulating some practical nonlinear circuit structures.

Model order reduction of nonlinear circuits (especially highly nonlinear circuits) has always been a theoretically and numerically challenging task. In this paper, we utilize tensors (namely, a higher order generalization of matrices) to develop a tensor-based nonlinear model order reduction algorithm we named TNMOR for the efficient simulation of nonlinear circuits. Unlike existing nonlinear model order reduction methods, in TNMOR high-order nonlinearities are captured using tensors, followed by decomposition and reduction to a compact tensor-based reduced-order model. Therefore, TNMOR completely avoids the dense reduced-order system matrices, which in turn allows faster simulation and a smaller memory requirement if relatively low-rank approximations of these tensors exist. Numerical experiments on transient and periodic steady-state analyses confirm the superior accuracy and efficiency of TNMOR, particularly in highly nonlinear scenarios.

An array of weakly coupled oscillators can generate multiphase signals, i.e., multiple sinusoidal signals with specific phase separations. Multiphase oscillators are attractive solutions in many electronic applications such as the synchronization of multiple processing units in digital electronics and the frequency synthesis in mixed-signal radio frequency circuits. Due to the complexity of multiphase oscillators and the large number of design parameters, novel simulation techniques are highly desired to efficiently handle such large-scale problems. In this paper, an efficient phase-domain simulation technique is proposed to calculate the phase response of inductance capacitance oscillator array. By some practical examples, it is shown how the proposed method can be exploited to identify the array topologies and parameter settings that guarantee stable phase separations. It is also shown how the proposed technique can be used to evaluate phase-noise performance.

Hierarchical uncertainty quantification can reduce the computational cost of stochastic circuit simulation by employing spectral methods at different levels. This paper presents an efficient framework to simulate hierarchically some challenging stochastic circuits/systems that include high-dimensional subsystems. Due to the high parameter dimensionality, it is challenging to both extract surrogate models at the low level of the design hierarchy and to handle them in the high-level simulation. In this paper, we develop an efficient analysis of variance-based stochastic circuit/microelectromechanical systems simulator to efficiently extract the surrogate models at the low level. In order to avoid the curse of dimensionality, we employ tensor-train decomposition at the high level to construct the basis functions and Gauss quadrature points. As a demonstration, we verify our algorithm on a stochastic oscillator with four MEMS capacitors and 184 random parameters. This challenging example is efficiently simulated by our simulator at the cost of only 10min in MATLAB on a regular personal computer.

This paper describes a novel approach for passivity enforcement of compact dynamical models of electrical interconnects. The proposed approach is based on a parameterization of general state-space scattering models with fixed poles. We formulate the passivity constraints as a unitary boundedness condition on the H∞norm of the system transfer function. When this condition is not verified, we use it as an explicit constraint within an iterative perturbation loop of the system state-space matrices. Since the resulting optimization framework is convex but nonsmooth, we solve it via localization based algorithms, such as the ellipsoid and the cutting plane methods. The proposed technique solves two critical bottleneck issues of the existing approaches for passivity enforcement of linear macromodels. Compared to quasi-optimal schemes based on singular value or Hamiltonian eigenvalue perturbation, we are able to guarantee convergence to the optimal solution. Compared to convex formulations based on direct Bounded Real Lemma constraints, we are able to reduce both memory and time requirements by orders of magnitude. We demonstrate the effectiveness of our approach on a number of cases for which existing algorithms either fail or exhibit very slow convergence.

Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellent performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine a set of orthonormal polynomials and a proper numerical quadrature rule. The former are used as the basis functions in a generalized polynomial chaos expansion. The latter is used to compute the integrals involved in stochastic spectral methods. Obtaining such information requires knowing the density function of the random input a-priori. However, individual system components are often described by surrogate models rather than density functions. In order to apply stochastic spectral methods in hierarchical uncertainty quantification, we first propose to construct physically consistent closed-form density functions by two monotone interpolation schemes. Then, by exploiting the special forms of the obtained density functions, we determine the generalized polynomial-chaos basis functions and the Gauss quadrature rules that are required by a stochastic spectral simulator. The effectiveness of our proposed algorithm is verified by both synthetic and practical circuit examples.

In this paper, we present the first validation of the virtual source (VS) charge-based compact model for standard cell libraries and large-scale digital circuits. With only a modest number of physically meaningful parameters, the VS model accounts for the main short-channel effects in nanometer technologies. Using a novel DC and transient parameter extraction methodology, the model is verified with simulated data from a well-characterized, industrial 40-nm bulk silicon model. The VS model is used to fully characterize a standard cell library with timing comparisons showing less than 2.7% error with respect to the industrial design kit. Furthermore, a 1001-stage inverter chain and a 32-bit ripple-carry adder are employed as test cases in a vendor CAD environment to validate the use of the VS model for large-scale digital circuit applications. Parametric Vdd sweeps show that the VS model is also ready for usage in low-power design methodologies. Finally, runtime comparisons have shown that the use of the VS model results in a speedup of about 7.6×.

Due to significant manufacturing process variations, the performance of integrated circuits (ICs) has become increasingly uncertain. Such uncertainties must be carefully quantified with efficient stochastic circuit simulators. This paper discusses the recent advances of stochastic spectral circuit simulators based on generalized polynomial chaos (gPC). Such techniques can handle both Gaussian and non-Gaussian random parameters, showing remarkable speedup over Monte Carlo for circuits with a small or medium number of parameters. We focus on the recently developed stochastic testing and the application of conventional stochastic Galerkin and stochastic collocation schemes to nonlinear circuit problems. The uncertainty quantification algorithms for static, transient and periodic steady-state simulations are presented along with some practical simulation results. Some open problems in this field are discussed.

Uncertainties have become a major concern in integrated circuit design. In order to avoid the huge number of repeated simulations in conventional Monte Carlo flows, this paper presents an intrusive spectral simulator for statistical circuit analysis. Our simulator employs the recently developed generalized polynomial chaos expansion to perform uncertainty quantification of nonlinear transistor circuits with both Gaussian and non-Gaussian random parameters. We modify the nonintrusive stochastic collocation (SC) method and develop an intrusive variant called stochastic testing (ST) method. Compared with the popular intrusive stochastic Galerkin (SG) method, the coupled deterministic equations resulting from our proposed ST method can be solved in a decoupled manner at each time point. At the same time, ST requires fewer samples and allows more flexible time step size controls than directly using a nonintrusive SC solver. These two properties make ST more efficient than SG and than existing SC methods, and more suitable for time-domain circuit simulation. Simulation results of several digital, analog and RF circuits are reported. Since our algorithm is based on generic mathematical models, the proposed ST algorithm can be applied to many other engineering problems.

We present an efficient and scalable framework for the generation of guaranteed passive compact dynamical models for multiport structures. The proposed algorithm enforces passivity using frequency independent linear matrix inequalities, as opposed to the existing optimization based algorithms which enforce passivity using computationally expensive frequency dependent constraints. We have tested our algorithm for various multiport structures. An excellent match between the given samples and our passive model was achieved.

Passivity-preserving model order reduction (MOR) of descriptor systems (DSs) is highly desired in the simulation of VLSI interconnects and on-chip passives. One popular method is PRIMA, a Krylov-subspace projection approach which preserves the passivity of positive semidefinite (PSD) structured DSs. However, system passivity is not guaranteed by PRIMA when the system is indefinite. Furthermore, the possible polynomial parts of singular systems are normally not captured. For indefinite DSs, positive-real balanced truncation (PRBT) can generate passive reduced-order models (ROMs), whose main bottleneck lies in solving the dual expensive generalized algebraic Riccati equations (GAREs). This paper presents a novel moment-matching MOR for indefinite DSs, which preserves both the system passivity and, if present, also the improper polynomial part. This method only requires solving one GARE, therefore it is cheaper than existing PRBT schemes. On the other hand, the proposed algorithm is capable of preserving the passivity of indefinite DSs, which is not guaranteed by traditional moment-matching MORs. Examples are finally presented showing that our method is superior to PRIMA in terms of accuracy.

Traditional parallel boundary element methods suffer from low parallel efficiency and poor scalability due to the long system solving time bottleneck. In this paper, we demonstrate how to avoid such a bottleneck by using an instantiable basis function approach. In our demonstrated examples, we achieve 90% parallel efficiency and scalability both in shared memory and distributed memory parallel systems.

This paper presents an approach for the model order reduction of fully parameterized linear dynamic systems. In a fully parameterized system, not only the state matrices, but also can the input/output matrices be parameterized. The algorithm presented in this paper is based on neither conventional moment-matching nor balanced-truncation ideas. Instead, it uses “optimal (block) vectors” to construct the projection matrix, such that the system errors in the whole parameter space are minimized. This minimization problem is formulated as a recursive least square (RLS) optimization and then solved at a low cost. Our algorithm is tested by a set of multi-port multi-parameter cases with both intermediate and large parameter variations. The numerical results show that high accuracy is guaranteed, and that very compact models can be obtained for multi-parameter models due to the fact that the ROM size is independent of the number of parameters in our approach.

In this paper we summarize recent developments in compact dynamical modeling for both linear and nonlinear systems arising in analog applications. These techniques include methods based on the projection framework, rational fitting of frequency response samples, and nonlinear system identification from time domain data. By combining traditional projection and fitting methods with recently developed convex optimization techniques, it is possible to obtain guaranteed stable and passive parameterized models that are usable in time domain simulators and may serve as a valuable tool for analog designers in both top-down and bottom-up design flows.

In this paper we present an efficient algorithm for variation-aware interconnect extraction. The problem we are addressing can be formulated mathematically as the solution of linear systems with matrix coefficients that are dependent on a set of random variables. Our algorithm is based on representing the solution vector as a summation of terms. Each term is a product of an unknown vector in the deterministic space and an unknown direction in the stochastic space. We then formulate a simple nonlinear optimization problem which uncovers sequentially the most relevant directions in the combined deterministic-stochastic space. The complexity of our algorithm scales with the sum (rather than the product) of the sizes of the deterministic and stochastic spaces, hence it is orders of magnitude more efficient than many of the available state of the art techniques. Finally, we validate our algorithm on a variety of onchip and off-chip capacitance and inductance extraction problems, ranging from moderate to very large size, not feasible using any of the available state of the art techniques.

In this paper we present a stochastic model order reduction technique for interconnect extraction in the presence of process variabilities, i.e. variation-aware extraction. It is becoming increasingly evident that sampling based methods for variation-aware extraction are more efficient than more computationally complex techniques such as stochastic Galerkin method or the Neumann expansion. However, one of the remaining computational challenges of sampling based methods is how to simultaneously and efficiently solve the large number of linear systems corresponding to each different sample point. In this paper, we present a stochastic model reduction technique that exploits the similarity among the different solves to reduce the computational complexity of subsequent solves. We first suggest how to build a projection matrix such that the statistical moments and/or the coefficients of the projection of the stochastic vector on some orthogonal polynomials are preserved.We further introduce a proximity measure, which we use to determine apriori if a given system needs to be solved, or if it is instead properly represented using the currently available basis. Finally, in order to reduce the time required for the system assembly, we use the multivariate Hermite expansion to represent the system matrix. We verify our method by solving a variety of variation-aware capacitance extraction problems ranging from on-chip capacitance extraction in the presence of width and thickness variations, to off-chip capacitance extraction in the presence of surface roughness. We further solve very large scale problems that cannot be handled by any other state of the art technique.

In this paper we present a passive reduced order modeling algorithm for linear multiport interconnect structures. The proposed technique uses rational fitting via semidefinite programming to identify a passive transfer matrix from given frequency domain data samples. Numerical results are presented for a power distribution grid and an array of inductors, and the proposed approach is compared to two existing rational fitting techniques.

This paper presents a system identification technique for generating stable compact models of typical analog circuit blocks in radio frequency systems. The identification procedure is based on minimizing the model error over a given training data set subject to an incremental stability constraint, which is formulated as a semidefinite optimization problem. Numerical results are presented for several analog circuits, including a distributed power amplifier, as well as a MEM device. It is also shown that our dynamical models can accurately predict important circuit performance metrics, and may thus, be useful for design optimization of analog systems.

With the adoption of ultra regular fabric paradigms for controlling design printability at the 22nm node and beyond, there is an emerging need for a layout-driven, pattern-based parasitic extraction of alternative fabric layouts. In this paper, we propose a hierarchical floating random walk (HFRW) algorithm for computing the 3D capacitances of a large number of topologically different layout configurations that are all composed of the same layout motifs. Our algorithm is not a standard hierarchical domain decomposition extension of the well established floating random walk technique, but rather a novel algorithm that employs Markov Transition Matrices. Specifically, unlike the fast-multipole boundary element method and hierarchical domain decomposition (which use a far-field approximation to gain computational efficiency), our proposed algorithm is exact and does not rely on any tradeoff between accuracy and computational efficiency. Instead, it relies on a tradeoff between memory and computational efficiency. Since floating random walk type of algorithms have generally minimal memory requirements, such a tradeoff does not result in any practical limitations. The main practical advantage of the proposed algorithm is its ability to handle a set of layout configurations in a complexity that is basically independent of the set size. For instance, in a large 3D layout example, the capacitance calculation of 120 different configurations made of similar motifs is accomplished in the time required to solve independently just 2 configurations, i.e. a 60x speedup.

This paper presents theoretical and practical results concerning the stability of piecewise-linear (PWL) reduced models for the purposes of analog macromodeling. Results include proofs of input-output (I/O) stability for PWL approximations to certain classes of nonlinear descriptor systems, along with projection techniques that are guaranteed to preserve I/O stability in reduced-order PWL models. We also derive a new PWL formulation and introduce a new nonlinear projection, allowing us to extend our stability results to a broader class of nonlinear systems described by models containing nonlinear descriptor functions. Lastly, we present algorithms to compute efficiently the required stabilizing nonlinear left-projection matrix operators.

In this paper we present an efficient algorithm for extracting the complete statistical distribution of the input impedance of interconnect structures in the presence of a large number of random geometrical variations. The main contribution in this paper is the development of a new algorithm, which combines both Neumann expansion and Hermite expansion, to accurately and efficiently solve stochastic linear system of equations. The second contribution is a new theorem to efficiently obtain the coefficients of the Hermite expansion while computing only low order integrals. We establish the accuracy of the proposed algorithm by solving stochastic linear systems resulting from the discretization of the stochastic volume integral equation and comparing our results to those obtained from other techniques available in the literature, such as Monte Carlo and stochastic finite element analysis. We further prove the computational efficiency of our algorithm by solving large problems that are not solvable using the current state of the art.

In this work we present a stability-preserving projection framework for model reduction of linear systems. Specifically, given one projection matrix (e.g. a right-projection matrix), we derive a set of linear constraints for the other projection matrix (e.g. the left-projection matrix) resulting in a projection framework that is guaranteed to generate a stable reduced model. Several efficient techniques for solving the proposed system of constraints are presented, including an optimization problem formulation for finding the optimal stabilizing projection, and a formulation with computational complexity independent of the size of the original system. The resulting algorithms can create accurate stable and passive models of arbitrary indefinite systems at a significantly cheaper cost than existing methods such as balanced truncation. Nevertheless, our algorithms integrate fully and effortlessly with most of the available standard model order reduction approaches for very large systems generated in VLSI applications (such as moment-matching methods, POD, or poor manpsilas TBR), which can guarantee stability and passivity only in very specialized cases. Our algorithms have been tested on a large variety of typical VLSI applications, including field-solver-extracted models of RF inductors for analog applications, power distribution grids for large VLSI digital integrated circuits, and MEMS devices for sensing and actuation applications. The results have been successfully compared to those from existing and much more expensive stabilizing reduction techniques.

Lithographic limitations and manufacturing uncertainties are resulting in fabricated shapes on wafer that are topologically equivalent, but geometrically different from the corresponding drawn shapes. While first-order sensitivity information can measure the change in pattern parasitics when the shape variations are small, there is still a need for a high-order algorithm that can extract parasitic variations incrementally in the presence of a large number of simultaneous shape variations. This paper proposes such an algorithm based on the wellknown method of floating random walk (FRW). Specifically, we formalize the notion of random path sharing between several conductors undergoing shape perturbations and use it as a basis of a fast capacitance sensitivity extraction algorithm and a fast incremental variational capacitance extraction algorithm. The efficiency of these algorithms is further improved with a novel FRW method for dealing with layered media. Our numerical examples show a 10X speed up with respect to the boundary-element method adjoint or finite-difference sensitivity extraction, and more than 560X speed up with respect to a non-incremental FRW method for a high-order variational extraction.

In this paper, an optimization-based model order reduction (MOR) framework is proposed. The method involves setting up a quasi-convex program that solves a relaxation of the optimal${\cal H}_{\infty}$norm MOR problem. The method can generate guaranteed stable and passive reduced models and is very flexible in imposing additional constraints such as exact matching of specific frequency response samples. The proposed optimization-based approach is also extended to solve the parameterized model-reduction problem (PMOR). The proposed method is compared to existing moment matching and optimization-based MOR methods in several examples. PMOR models for large RF inductors over substrate and power-distribution grid are also constructed.

In this paper we present several results concerning the stabilization of piecewise-linear reduced order models. We include proofs of internal and external stability for models whose system matrices possess special structures. We then introduce a new projection scheme, and a new set of weighting functions which allow us to extend some of these results to piecewise-linear systems comprised of arbitrary matrices, at least one of which is Hurwitz. Included are an algorithm for creating switching piecewise-linear reduced models comprised of globally exponentially stable systems, and stable simulation results for a system which produces unstable results when using the standard TPWL method.

Typical nonlinear model order reduction ap- proaches need to address two issues: reducing the order of the model, and approximating the vector field. In this paper we focus exclusively on the second issue, and present results characterizing the repercussions at the system level of vector field approximations. The error assessment problem is formulated as the L2 gain upper bounding problem of a scaled feedback interconnection. Applying the small gain theorem in the proposed setup, we prove that the L2 gain of the error system is upper bounded by the L2 gain of the vector field approximation error, provided it is small. In addition, the paper also presents a numerical procedure, based on the IQC/LMI approach, to perform the error estimation task with less conservatism. A numerical example is given in this paper to demonstrate the practical implications of the presented results.

This paper presents a parameterized reduction technique for highly nonlinear systems. In our approach, we first approximate the nonlinear system with a convex combination of parameterized linear models created by linearizing the nonlinear system at points along training trajectories. Each of these linear models is then projected using a moment-matching scheme into a low-order subspace, resulting in a parameterized reduced-order nonlinear system. Several options for selecting the linear models and constructing the projection matrix are presented and analyzed. In addition, we propose a training scheme which automatically selects parameter-space training points by approximating parameter sensitivities. Results and comparisons are presented for three examples which contain distributed strong nonlinearities: a diode transmission line, a microelectromechanical switch, and a pulse-narrowing nonlinear transmission line. In most cases, we are able to accurately capture the parameter dependence over the parameter ranges of plusmn50% from the nominal values and to achieve an average simulation speedup of about 10x.

An efficient approach to full-wave impedance extraction is developed that accounts for substrate effects through the use of two-layer media Green's functions in a mixed-potential-integral-equation (MPIE) solver. Particularly, the choice of implementation for the layered media Green's functions motivates the development of accelerated techniques for both volume and surface integrations in the solver. Solver accuracy is validated against measurements taken on fabricated devices; solver efficiency is demonstrated by its 9.8X reduction in cost in comparison to the traditional integration approach.

Existing Full-wave Model Order Reduction (FMOR) approaches are based on Expanded Taylor Series Approximations (ETAS) of the oscillatory full-wave system matrix. The accuracy of such approaches hinges on the worst case interaction distances, producing accurate models over a very narrow band of frequencies. In this paper we present Segregation by Primary Phase Factors (SPPF), a novel algorithm for FMOR enabling wideband interconnect impedance analysis. SPPF extracts exponential terms (primary phase factors) and then approximates the smoother remainder with an ETAS, thus resulting in good accuracies over a very wide band of frequencies. We also present a technique to improve conditioning for the required computation. Example results are given for simple interconnect structures modeled using a discretized mixed potential integral equation formulation.

In this paper an optimization based model order reduction (MOR) framework is proposed. The method involves setting up a quasiconvex program that explicitly minimizes a relaxation of the optimal H∞ norm MOR problem. The method generates guaranteed stable and passive reduced models and it is very flexible in imposing additional constraints. The proposed optimization approach is also extended to parameterized model reduction problem (PMOR). The proposed method is compared to existing moment matching and optimization based MOR methods in several examples. A PMOR model for a large RF inductor is also constructed.

In this paper we present a parameterized reduction technique for non-linear systems. Our approach combines an existing non-parameterized trajectory piecewise linear method for non-linear systems, with an existing moment matching parameterized technique for linear systems. Results and comparisons are presented for two examples: an analog non-linear circuit, and a MEM switch.

In this paper, we describe an approach for generating accurate geometrically parameterized integrated circuit interconnect models that are efficient enough for use in interconnect synthesis. The model-generation approach presented is automatic, and is based on a multiparameter moment matching model-reduction algorithm. A moment-matching theorem proof for the algorithm is derived, as well as a complexity analysis for the model-order growth. The effectiveness of the technique is tested using a capacitance extraction example, where the plate spacing is considered as the geometric parameter, and a multiline bus example, where both wire spacing and wire width are considered as geometric parameters. Experimental results demonstrate that the generated models accurately predict capacitance values for the capacitor example, and both delay and cross-talk effects over a reasonably wide range of spacing and width variation for the multiline bus example.

The major concerns in state-of-the-art model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms, such as PRIMA, generate guaranteed-passive models for systems with special internal structure, using numerically stable and efficient Krylov-subspace iterations. Truncated balanced realization (TBR) algorithms, as used to date in the design automation community, can achieve smaller models with better error control, but do not necessarily preserve passivity. In this paper, we show how to construct TBR-like methods that generate guaranteed passive reduced models and in addition are applicable to state-space systems with arbitrary internal structure.

This paper presents a class of algorithms suitable for model reduction of distributed systems. Distributed systems are not suitable for treatment by standard model-reduction algorithms such as PRIMA, PVL, and the Arnoldi schemes because they generate matrices that are dependent on frequency (or other parameters) and cannot be put in a lumped or state-space form. Our algorithms build on well-known projection-based reduction techniques, and so require only matrix-vector product operations and are thus suitable for operation in conjunction with electromagnetic analysis codes that use iterative solution methods and fast-multipole acceleration techniques. Under the condition that the starting systems satisfy system-theoretic properties required of physical systems, the reduced systems can be guaranteed to be passive. For distributed systems, we argue that causality of the underlying representation is as important a consideration.

The major concerns in state-of-the-art model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms such as PRIMA generate guaranteed-passive models, for systems with special internal structure, using numerically stable and efficient Krylov-subspace iterations. Truncated Balanced Realization (TBR) algorithms, as used to date in the design automation community, can achieve smaller models with better error control, but do not necessarily preserve passivity. In this paper we show how to construct TBR-like methods that guarantee passive reduced models and in addition are applicable to state-space systems with arbitrary internal structure.

Modeling the exponentially varying current distributions in conductor interiors associated with high frequency interconnect behavior causes a rapid increase in the computation time and memory required even by recently developed fast electromagnetic analysis programs. In this paper we describe a procedure to generate numerically a set of basis functions which efficiently represent conductor current variation, and thus improving solver efficiency. The method is based on solving a sequence of template problems, and is easily generalized to arbitrary conductor cross-sections. Results are presented to demonstrate that the numerically computed basis functions are seven to twenty times more efficient than the commonly used piece-wise constant basis functions.

In this paper we describe an approach for generating geometrically-parameterized integrated-circuit interconnect models that are efficient enough for use in interconnect synthesis. The model generation approach presented is automatic, and is based on a multi-parameter model-reduction algorithm. The effectiveness of the technique is tested using a multi-line bus example, where both wire spacing and wire width are considered as geometric parameters. Experimental results demonstrate that the generated models accurately predict both delay and cross-talk effects over a wide range of spacing and width variation.

In this paper, we present an efficient method to model the interior of the conductors in a quasi-static or full-wave integral equation solver. We show how interconnect cross-sectional current distributions can be modeled using a small number of conduction modes as basis functions for the discretization of the Mixed Potential Integral Equation (MPIE). Two examples are presented to demonstrate the computational attractiveness of our method. In particular, we show how our new approach can successfully and efficiently capture skin effects, proximity effects and transmission line resonances.

Interconnect structures including dielectrics can be modeled by an integral equation method using volume currents and surface charges for the conductors, and volume polarization currents and surface charges for the dielectrics. In this paper we describe a mesh analysis approach for computing the discretized currents in both the conductors and the dielectrics. We then show that this fully mesh-based formulation can be cast into a form using provably positive semidefinite matrices, making for easy application of Krylov-subspace based model-reduction schemes to generate accurate guaranteed passive reduced-order models. Several printed circuit board examples are given to demonstrate the effectiveness of the strategy.