This paper studies how to compute all real eigenvalues, associated to real eigenvectors, of a symmetric tensor. As is well known, the largest or smallest eigenvalue can be found by solving a polynomial optimization problem, while the other middle ones cannot. We propose a new approach for computing all real eigenvalues sequentially, from the largest to the smallest. It uses Jacobian semidefinite relaxations in polynomial optimization. We show that each eigenvalue can be computed by solving a finite hierarchy of semidefinite relaxations. Numerical experiments are presented to show how to do this.
Stochastic spectral methods have achieved great success in the uncertainty quantification of many engineering problems, including electronic and photonic integrated circuits influenced by fabrication process variations. Existing techniques employ a generalized polynomial-chaos expansion, and they almost always assume that all random parameters are mutually independent or Gaussian correlated. However, this assumption is rarely true in real applications. How to handle non-Gaussian correlated random parameters is a long-standing and fundamental challenge. A main bottleneck is the lack of theory and computational methods to perform a projection step in a correlated uncertain parameter space. This paper presents an optimization-based approach to automatically determinate the quadrature nodes and weights required in a projection step, and develops an efficient stochastic collocation algorithm for systems with non-Gaussian correlated parameters. We also provide some theoretical proofs for the complexity and error bound of our proposed method. Numerical experiments on synthetic, electronic and photonic integrated circuit examples show the nearly exponential convergence rate and excellent efficiency of our proposed approach. Many other challenging uncertainty-related problems can be further solved based on this work.
This paper generalizes stochastic collocation methods to handle correlated non-Gaussian random parameters. The key challenge is to perform a multivariate numerical integration in a correlated parameter space when computing the coefficient of each basis function via a projection step. We propose an optimization model and a block coordinate descent solver to compute the required quadrature samples. Our method is verified with a CMOS ring oscillator and an optical ring resonator, showing 3000× speedup over Monte Carlo.
Uncertainty quantification has become an efficient tool for yield prediction, but its power in yield-aware optimization has not been well explored from either theoretical or application perspectives. Yield optimization is a much more challenging task. On one side, optimizing the generally non-convex probability measure of performance metrics is difficult. On the other side, evaluating the probability measure in each optimization iteration requires massive simulation data, especially when the process variations are non-Gaussian correlated. This paper proposes a data-efficient framework for the yield-aware optimization of the photonic IC. This framework optimizes design performance with a yield guarantee, and it consists of two modules: a modeling module that builds stochastic surrogate models for design objectives and chance constraints with a few simulation samples, and a novel yield optimization module that handles probabilistic objectives and chance constraints in an efficient deterministic way. This deterministic treatment avoids repeatedly evaluating probability measures at each iteration, thus it only requires a few simulations in the whole optimization flow. We validate the accuracy and efficiency of the whole framework by a synthetic example and two photonic ICs. Our optimization method can achieve more than 30× reduction of simulation cost and better design performance on the test cases compared with a recent Bayesian yield optimization approach.
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