In this paper, we first study the projections onto the set of unit dual quaternions, and the set of dual quaternion vectors with unit norms. Then we propose a power method for computing the dominant eigenvalue of a dual quaternion Hermitian matrix. For a strict dominant eigenvalue, we show the sequence generated by the power method converges to the dominant eigenvalue and its corresponding eigenvector linearly. For a general dominant eigenvalue, we establish linear convergence of the standard part of the dominant eigenvalue. Based upon these, we reformulate the simultaneous localization and mapping problem as a rank-one dual quaternion completion problem. A two-block coordinate descent method is proposed to solve this problem. One block has a closed-form solution and the other block is the best rank-one approximation problem of a dual quaternion Hermitian matrix, which can be computed by the power method. Numerical experiments are presented to show the efficiency of our proposed power method.

Stochastic spectral methods have achieved a great success in the uncertainty quantification of many engineering problems, including variation-aware electronic and photonic design automation. State-of-the-art techniques employ generalized polynomial-chaos expansions and assume that all random parameters are independent or Gaussian correlated. This assumption is rarely true in real applications. How to handle non-Gaussian correlated random parameters is a long-standing and fundamental challenge: It is not clear how to choose basis functions and to perform a projection step in a correlated uncertain parameter space. This paper first presents a new set of basis functions to well capture the impact of non-Gaussian correlated parameters and then proposes an automatic and optimization-based quadrature method to perform projection-based stochastic collocation with a few simulation samples in the correlated parameter space. We further provide some theoretical proofs for the complexity and error bound of our proposed method. The numerical experiments on several synthetic, electronic, and photonic integrated circuit examples show the nearly exponential convergence rate of our approach and its significant ( $700 \times $ – $6000 \times $ ) speedup than Monte Carlo. Many other open problems with non-Gaussian correlated uncertainties can be further solved based on this paper.

We propose a supplement matrix method for computing eigenvalues of a dual Hermitian matrix and discuss its application in multiagent formation control. Suppose we have a ring, which can be the real field, the complex field, or the quaternion ring. We study dual number symmetric matrices, dual complex Hermitian matrices, and dual quaternion Hermitian matrices in a unified frame of dual Hermitian matrices. An [Formula: see text] dual Hermitian matrix has [Formula: see text] dual number eigenvalues. We define determinant, characteristic polynomial, and supplement matrices for a dual Hermitian matrix. Supplement matrices are Hermitian matrices in the original ring. The standard parts of the eigenvalues of that dual Hermitian matrix are the eigenvalues of the standard part of the dual Hermitian matrix in the original ring, while the dual parts of the eigenvalues of that dual Hermitian matrix are the eigenvalues of those supplement matrices. Note that the standard part of the dual Hermitian matrix is still a Hermitian matrix in the original ring. Hence, by applying any practical method for computing eigenvalues of Hermitian matrices in the original ring, we have a practical method for computing eigenvalues of a dual Hermitian matrix. We call this method the supplement matrix method. In multiagent formation control, a desired relative configuration scheme may be given. People need to know if this scheme is reasonable such that a feasible solution of configurations of these multiagents exists. By exploring the eigenvalue problem of dual Hermitian matrices and its link with the unit gain graph theory, we open a cross-disciplinary approach to solve the relative configuration problem. Numerical experiments are reported to show the efficiency of our proposed methods.

Active subspace is a model reduction method widely used in the uncertainty quantification community. In this paper, we propose analyzing the internal structure and vulnerability of deep neural networks using active subspace. Firstly, we employ the active subspace to measure the number of “active neurons” at each intermediate layer, which indicates that the number of neurons can be reduced from several thousands to several dozens. This motivates us to change the network structure and to develop a new and more compact network, referred to as ASNet, that has significantly fewer model parameters. Secondly, we propose analyzing the vulnerability of a neural network using active subspace by finding an additive universal adversarial attack vector that can misclassify a dataset with a high probability. Our experiments on CIFAR-10 show that ASNet can achieve 23.98x parameter and 7.30x flops reduction. The universal active subspace attack vector can achieve around 20% higher attack ratio compared with the existing approaches in our numerical experiments. The PyTorch codes for this paper are available online.

Uncertainty quantification based on generalized polynomial chaos has been used in many applications. It has also achieved great success in variation-aware design automation. However, almost all existing techniques assume that the parameters are mutually independent or Gaussian correlated, which is rarely true in real applications. For instance, in chip manufacturing, many process variations are actually correlated. Recently, some techniques have been developed to handle non-Gaussian correlated random parameters, but they are time-consuming for high-dimensional problems. We present a new framework to solve uncertainty quantification problems with many non-Gaussian correlated uncertainties. Firstly, we propose a set of smooth basis functions to well capture the impact of non-Gaussian correlated process variations. We develop a tensor approach to compute these basis functions in a high-dimension setting. Secondly, we investigate the theoretical aspect and practical implementation of a sparse solver to compute the coefficients of all basis functions. We provide some theoretical analysis for the exact recovery condition and error bound of this sparse solver in the context of uncertainty quantification. We present three adaptive sampling approaches to improve the performance of the sparse solver. Finally, we validate our methods by synthetic and practical electronic/photonic ICs with 19 to 57 non-Gaussian correlated variation parameters. Our approach outperforms Monte Carlo by thousands of times in terms of efficiency. It can also accurately predict the output density functions with multiple peaks caused by non-Gaussian correlations, which are hard to capture by existing methods.