In post–layout circuit simulation, efficient model order reduction (MOR) for many–port resistor–capacitor (RC) circuits remains a crucial issue. The current mainstream MOR methods for such circuits include high–order moment matching methods and elimination methods. High-order moment matching methods–characterized by high accuracy, such as PRIMA and TurboMOR–tend to generate large dense reduced-order systems when the number of ports is large, which impairs the efficiency of MOR. Another common type of MOR method for many–port circuits is based on Gaussian elimination, with the SIP method as a representative. The main limitation of this method lies in the inadequate matching of high–order moments. In this paper, we propose a sparse multipoint moment matching method and present comprehensive theoretical analysis results regarding the multi–frequency high–order moment matching property. Meanwhile, to enhance the algorithm's efficiency, sparse control and deflation techniques are introduced to further optimize the algorithm. Numerical experiments demonstrated that, compared to SIP, the accuracy is improved by more than two orders of magnitude at high frequency points without adding many extra linear components. Compared to TurboMOR methods, our method achieves a speed improvement of more than twice while maintaining the same level of precision.

Efficient model order reduction for many-port resistor-capacitor (RC) networks is essential in post-layout circuit simulation. Existing high-accuracy elimination-based methods have certain limitations, such as fixed frequency points, large reduced-order models, or high reduction cost. This paper proposes FlexRC, a flexible multi-point model order reduction method for many-port RC networks. FlexRC starts from the same elimination step as previous methods, and then constructs a nonorthogonal projection basis by a modified block rational Arnoldi process to generate a sparse banded reduced model. FlexRC features three adjustable components: user-specified frequency points, a tolerance-controlled port-reduction technique for the internal subsystem, and an optional sparsity-control strategy. We discuss passivity under port-reduction perturbations, analyze moment matching, and provide a conservative error estimate for port reduction. Numerical experiments on industrial RC examples and IBM power-grid examples demonstrate the effectiveness of FlexRC in terms of reduction time and transient simulation time.

Optimizing multiple competitive black-box objectives with tight constraints poses a common challenge in analog circuit design. Multiobjective Bayesian optimization (MOBO) is a sample-efficient approach to identify the optimal tradeoffs, namely, the Pareto front (PF). However, existing MOBO methods exhibit limitations in handling high-dimensional design space, large sample budgets, many objectives and tight constraints. This article introduces VTSMOC, a sample-efficient and computationally lightweight approach for addressing high-dimensional constrained multiobjective optimization problems. VTSMOC decomposes the design space into Voronoi cells, dynamically constructing a hierarchical Voronoi tree through clustering observations with dominance relationships. Promising leaf nodes in the Voronoi tree are pinpointed by traversing the tree with gradient bandit. The diversity of PF is ensured by parallel sampling within different promising cells, selected using a diffusive strategy. We also propose the expected PF improvement (EPFI) and probability of PF improvement (PPFI) acquisition functions to facilitate the PF efficiently along the radial direction of PF surface. Compared to state-of-the-art methods, VTSMOC achieves significant improvements in both sample and computational efficiency.

Harmonic balance (HB) method is a powerful frequency-domain method used in RF circuit simulations. The key point of HB method is efficiently solving the Jacobian system in Newton’s method. In this paper, we first introduce a new time-domain preconditioner for HB Jacobian. Unlike existing time-domain preconditioners, which cannot balance the efficiency of solving the linear system corresponding to the preconditioner with the reduction in iteration step for strongly nonlinear circuit, the proposed preconditioner successfully addresses both aspects. We also present a new preconditioning method that extends time-domain preconditioners to circuit with distributed devices, which was previously unattainable. Finally, a matrix norm-based metric is proposed to measure the strength of circuit nonlinearity, which can help us a priori choose the appropriate preconditioner.

Periodic small-signal analysis is crucial but time-consuming in RF simulation, since it may deal with many frequency points. While the Krylov subspace recycling method has greatly accelerated the simulation, the increasing memory cost in large-scale RF simulation has become a new bottleneck. A remedy for memory shortage is to restart the recycling algorithm, but may cause excessive extra iterations. To address this issue, this paper outlines a framework of recycling subspace truncation method for periodic small-signal analysis, provided with an efficient initial guess choice method and a Floquet-based subspace truncation strategy. Numerical results show that compared to the existing methods, the proposed method achieves up to 2.5x speedup in the same memory cost.

Accurate and efficient computation of Floquet multipliers and subspaces is essential for analyzing limit cycle in dynamical systems and periodic steady state in Radio Frequency simulation. This problem is typically addressed by solving a periodic linear eigenvalue problem, which is discretized from the linear time-periodic system using one-step collocation methods. Collocation methods become costly for large-scale cases. Our alternative approach is to use multistep methods. The multistep method leads to a periodic polynomial eigenvalue problem (pPEP), and introduces additional parasitic periodic eigenvalues. We prove that as the stepsize decreases, the computed Floquet multipliers and their associated invariant subspace converge with higher order, while the parasitic periodic eigenvalues converge to zero geometrically and therefore Floquet multipliers are not affected by those parasitic ones. A memory-efficient algorithm pTOAR is designed to solve the large-scale pPEP. Its computational and memory costs are almost independent of the choice of multistep methods. Numerical results coincide with our convergence analysis, and also demonstrate the efficiency of pTOAR.

This paper studies the convex isotonic regression with generalized order restrictions induced by a directed tree. The proposed model covers various intriguing optimization problems with shape or order restrictions, including the generalized nearly isotonic optimization and the total variation on a tree. Inspired by the success of the pool-adjacent-violator algorithm and its active-set interpretation, we propose an active-set-based recursive approach for solving the underlying model. Unlike the brute-force approach that traverses an exponential number of possible active-set combinations, our algorithm has a polynomial time computational complexity under mild assumptions.

This paper presents an aggregation-based two-grid method for solving a multilevel block Toeplitz system. Different from the existing multigrid methods for multilevel block Toeplitz systems, we aggregate a given multilevel block Toeplitz matrix to a new multilevel Toeplitz matrix in such a way that a very sparse coarse grid matrix is constructed in practice. Then, we give an asymptotically tight bound of the convergence rate and provide an algorithm for selecting the optimal prolongation vector and the relaxation factor for our method. Numerical experiments on artificial examples are provided for visualizing the correctness of our analysis, while experiments associated with practical examples show the efficiency of our method in terms of computing time.

Given sample data points {(xj, fj)}Nj =1, in [Brubeck, Nakatsukasa, and Trefethen, SIAM Review, 63 (2021), pp. 405-415], an Arnoldi-based pro-cedure is proposed to accurately evaluate the best fitting polynomial, in the least squares sense, at new nodes {sj}Mj=1, based on the Vandermonde ba-sis. Numerical tests indicated that this procedure can in general achieve high accuracy. The main purpose of this paper is to perform a forward rounding error analysis in finite precision. Our result establishes sensitivity factors re-garding the accuracy of the algorithm, and provides a theoretical justification for why the algorithm works. For least-squares approximation on an interval, we propose a variant of this Arnoldi-based evaluation by using the Chebyshev polynomial basis. Numerical tests are reported to demonstrate our forward rounding error analysis.

A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair (A, C) is introduced in this paper. The 2DEVP can be regarded as a linear algebra formulation of the well-known eigenvalue optimization problem of the parameter matrix A - \mu C. We first present fundamental properties of the 2DEVP, such as the existence and variational characterizations of 2Deigenvalues, and then devise a Rayleigh quotient iteration (RQI)-like algorithm, 2DRQI in short, for computing a 2D-eigentriplet of the 2DEVP. The efficacy of the 2DRQI is demonstrated by large scale eigenvalue optimization problems arising from the minmax of Rayleigh quotients and the distance to instability of a stable matrix.

As the resolution continues to increase, the scale of the power grid of the flat panel display (FPD) becomes huge. This imposes a severe computational challenge for analysis and storage. In this paper, based on the highly periodic FPD structure, we present an efficient simulator which contains a novel stamping scheme and a fast structural solver. Experiments on real industrial cases show that compared to the conventional stamping scheme, our proposed stamping scheme achieves up to 118× speedup in 74× less memory; compared to the state-of-art direct solver, Cholmod, our proposed structural solver achieves up to 17× speedup in 5× less memory.

Implicit determinant method is an effective method for some linear eigenvalue optimization problems since it solves linear systems of equations rather than eigenpairs. In this paper, we generalize the implicit determinant method to solve an Hermitian eigenvalue optimization problem for smooth case and non-smooth case. We prove that the implicit determinant method converges locally and quadratically. Numerical experiments confirm our theoretical results and illustrate the efficiency of implicit determinant method.

Efficient high-dimensional performance modeling of analog/RF circuits over multiple corners is an important-yet-challenging task. In this article, we propose a novel performance modeling approach for analog/RF circuits, referred to as correlated Bayesian model fusion (C-BMF). The key idea is to encode the correlation information for both model template and coefficient magnitude among different corners by using a unified prior distribution. Next, the prior distribution is combined with a few simulation samples via Bayesian inference to efficiently determine the unknown model coefficients. Two circuit examples designed in a commercial 40-nm CMOS process demonstrate that C-BMF achieves about $2\times $ cost reduction over the traditional state-of-the-art modeling technique without surrendering any accuracy.

Graph spectral sparsification-based preconditioning technique has shown promising results for power grid analysis. However, the conventional methods converge slowly for high-accuracy requirement. In this work, we propose an efficient approach to address this issue. Instead of using the Cholesky factorization, we employ the incomplete Cholesky factorization to factorize the spectral sparsifier. We also propose a concept of graph spectral pattern, which can further reduce the preconditioned conjugate gradient (PCG) iterations using less number of nonzeros. Experiments show that under 10−6 relative tolerance, our proposed preconditioning technique achieves $1.17\times $ speedup compared to AMGPCG in average; compared to the conventional spectral sparsification-based preconditioning techniques, our proposed approach achieves up to $8.53\times $ speedup of the factorization, $8.74\times $ speedup of the PCG iteration, and $5.6\times $ speedup of the total time. Moreover, the speedup of the total time continues to enlarge for higher-accuracy requirement, e.g., 10−12. Finally, but not least, our method is compatible with existing graph spectral sparsification algorithms for power grid analysis.

In Part I of this paper, we introduced a two dimensional eigenvalue problem (2DEVP) of a matrix pair and investigated its fundamental theory such as existence, variational characterization and number of 2D-eigenvalues. In Part II, we proposed a Rayleigh quotient iteration (RQI)-like algorithm (2DRQI) for computing a 2D-eigentriplet of the 2DEVP near a prescribed point, and discussed applications of 2DEVP and 2DRQI for solving the minimax problem of Rayleigh quotients, and computing the distance to instability. In this third part, we present convergence analysis of the 2DRQI. We show that under some mild conditions, the 2DRQI is locally quadratically convergent for computing a nonsingular 2D-eigentriplet.

In Part I of this paper, we introduced a 2D eigenvalue problem (2DEVP) and presented theoretical results of the 2DEVP and its intrinsic connetion with the eigenvalue optimizations. In this part, we devise a Rayleigh quotient iteration (RQI)-like algorithm, 2DRQI in short, for computing a 2D-eigentriplet of the 2DEVP. The 2DRQI performs $2\times$ to $8\times$ faster than the existing algorithms for large scale eigenvalue optimizations arising from the minmax of Rayleigh quotients and the distance to instability of a stable matrix.

In this article, to accurately estimate the rare failure rates for large-scale circuits (e.g., SRAM) where process variations are modeled as truncated normal distributions in high-dimensional space, we propose a novel truncated scaled-sigma sampling (T-SSS) method. Similar to scaled-sigma sampling (SSS), T-SSS distorts the truncated normal distributions by a scaling factor, resulting in an analytical model for failure rate estimation. By drawing random samples from the distorted distribution and estimating a sequence of scaled failure rates, we can solve all unknown model coefficients and predict the original failure rate by extrapolation. The accuracy of T-SSS is further assessed by estimating its confidence interval (CI) based on resampling. Our numerical results demonstrate that the proposed T-SSS method can achieve superior accuracy over the state-of-the-art method without increasing the computational cost.

Two‐dimensional eigenvalue problems (2DEVP) equations in complex Hermitian case contain complex quadratic forms which is non‐holomorphic. In this case, standard Newton method fails to apply. An existing strategy to solve this problem is to transform them into real problems (TRN). However, this method doubles the size of equations and thus is time consuming. On the other hand, the non‐isolation of the solution set in 2DEVP also complicates the analysis. In this article, we propose a Newton type method which solves the problem caused by non‐holomophism and non‐isolation. It has locally quadratic convergence rate and is about at least twice as much efficient as TRN. We hope our ideas can provide insights for solving other problems including non‐holomorphism and non‐isolation. We apply our algorithm to calculate the distance to instability. Numerical experiments show its advantages in efficiency while keeping good convergence compared with the current state of art algorithms.

In this paper, we propose an efficient parametric yield estimation method based on Bayesian Inference. By observing that nowadays analog and mixed-signal circuit is designed via a multi-stage flow, and that the circuit performance correlation of early stage and late stage is naturally symmetrical, we introduce a general region to capture the common features of the early and late stage. Meanwhile, two private regions are also incorporated to represent the unique features of these two stages respectively. Afterwards, we introduce classifiers one for each region to explicitly encode the correlation information. Next, we set up a graphical model, and consequently adopt Bayesian Inference to calculate the model parameters. Finally, based on the obtained optimal model parameters, we can accurately and efficiently estimate the parametric yield with a simple sampling method. Our numerical experiments demonstrate that compared to the state-of-the-art algorithms, our proposed method can better estimate the yield while significantly reducing the number of circuit simulations.

Pareto Front (PF) modeling is essential in decision making problems across all domains such as economics, medicine or engineering. In Operation Research literature, this task has been addressed based on multi-objective optimization algorithms. However, without learning models for PF, these methods cannot examine whether a new provided point locates on PF or not. In this paper, we reconsider the task from Data Mining perspective. A novel projection based active Gaussian process regression (P- aGPR) method is proposed for efficient PF modeling. First, P- aGPR chooses a series of projection spaces with dimensionalities ranking from low to high. Next, in each projection space, a Gaussian process regression (GPR) model is trained to represent the constraint that PF should satisfy in that space. Moreover, in order to improve modeling efficacy and stability, an active learning framework has been developed by exploiting the uncertainty information obtained in the GPR models. Different from all existing methods, our proposed P-aGPR method can not only provide a generative PF model, but also fast examine whether a provided point locates on PF or not. The numerical results demonstrate that compared to state-of-the-art passive learning methods the proposed P-aGPR method can achieve higher modeling accuracy and stability.