Haotian Liu

Model Reduction and Simulation of Nonlinear Circuits via Tensor Decomposition

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.

A Constructive Algorithm for Decomposing a Tensor into a Finite Sum of Orthonormal Rank-1 Terms

We propose a constructive algorithm that decomposes an arbitrary real tensor into a finite sum of orthonormal rank-1 outer products. The algorithm, called TTr1SVD, works by converting the tensor into a tensor-train rank-1 (TTr1) series via the singular value decomposition (SVD). TTr1SVD naturally generalizes the SVD to the tensor regime with properties such as uniqueness for a fixed order of indices, orthogonal rank-1 outer product terms, and easy truncation error quantification. Using an outer product column table it also allows, for the first time, a complete characterization of all tensors orthogonal with the original tensor. Incidentally, this leads to a strikingly simple constructive proof showing that the maximum rank of a real

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

2 \times 2 \times 2

Fast nonlinear model order reduction via associated transforms of high-order volterra transfer functions

tensor over the real field is 3. We also derive a conversion of the TTr1 decomposition into a Tucker decomposition with a sparse core tensor. Numerical examples illustrate each of the favorable properties of the TTr1 decomposition.

Autonomous Volterra Algorithm for Steady-State Analysis of Nonlinear Circuits

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.

A tensor-based volterra series black-box nonlinear system identification and simulation framework

We present a new and fast way of computing the projection matrices serving high-order Volterra transfer functions in the context of (weakly and strongly) nonlinear model order reduction. The novelty is to perform, for the first time, the association of multivariate (Laplace) variables in high-order multiple-input multiple-output (MIMO) transfer functions to generate the standard single-s transfer functions. The consequence is obvious: instead of finding projection subspaces about every si, only that about a singles is required. This translates into drastic saving in computation and memory, and much more compact reduced-order nonlinear models, without compromising any accuracy.

A novel linear algebra method for the determination of periodic steady states of nonlinear oscillators

We present a novel algorithm, named autonomous Volterra (AV), that achieves efficient steady-state analysis of nonlinear circuits. With elegant analytic forms and availability of efficient solvers, AV constitutes a competitive steady-state algorithm besides the two mainstreams, namely, shooting Newton (SN) and harmonic balance (HB). Nonlinear systems are first captured in nonlinear differential algebraic equations, followed by expansion into linear Volterra subsystems. A key step of steady-state analysis lies in modeling each Volterra subsystem with autonomous nonlinear inputs. The steady-state solution of these subsystems then proceeds with a series of Sylvester equation solves, completely avoiding the guesses of initial condition and time stepping as in SN, as well as the uncertain length of Fourier series as in HB. Error control in AV is also straightforward by monitoring the norms of the Sylvester equation solutions. We further demonstrate that AV is readily parallelizable with superior scalability toward large-scale problems.

STORM: A nonlinear model order reduction method via symmetric tensor decomposition

Tensors are a multi-linear generalization of matrices to their d-way counterparts, and are receiving intense interest recently due to their natural representation of high-dimensional data and the availability of fast tensor decomposition algorithms. Given the input-output data of a nonlinear system/circuit, this paper presents a non-linear model identification and simulation framework built on top of Volterra series and its seamless integration with tensor arithmetic. By exploiting partially-symmetric polyadic decompositions of sparse Toeplitz tensors, the proposed framework permits a pleasantly scalable way to incorporate high-order Volterra kernels. Such an approach largely eludes the curse of dimensionality and allows computationally fast modeling and simulation beyond weakly non-linear systems. The black-box nature of the model also hides structural information of the system/circuit and encapsulates it in terms of compact tensors. Numerical examples are given to verify the efficacy, efficiency and generality of this tensor-based modeling and simulation framework.

STAVES: Speedy Tensor-Aided Volterra-Based Electronic Simulator

Periodic steady-state (PSS) analysis of nonlinear oscillators has always been a challenging task in circuit simulation. We present a new way that uses numerical linear algebra to identify the PSS(s) of nonlinear circuits. The method works for both autonomous and excited systems. Using the harmonic balancing method, the solution of a nonlinear circuit can be represented by a system of multivariate polynomials. Then, a Macaulay matrix based root-finder is used to compute the Fourier series coefficients. The method avoids the difficult initial guess problem of existing numerical approaches. Numerical examples show the accuracy and feasibility over existing methods.

Frequency-domain transient analysis of multitime partial differential equation systems

Nonlinear model order reduction has always been a challenging but important task in various science and engineering fields. In this paper, a novel symmetric tensor-based order-reduction method (STORM) is presented for simulating large-scale nonlinear systems. The multidimensional data structure of symmetric tensors, as the higher order generalization of symmetric matrices, is utilized for the effective capture of high-order nonlinearities and efficient generation of compact models. Compared to the recent tensor-based nonlinear model order reduction (TNMOR) algorithm [1], STORM shows advantages in two aspects. First, STORM avoids the assumption of the existence of a low-rank tensor approximation. Second, with the use of the symmetric tensor decomposition, STORM allows significantly faster computation and less storage complexity than TNMOR. Numerical experiments demonstrate the superior computational efficiency and accuracy of STORM against existing nonlinear model order reduction methods.