Lei Zhang

Jizeng Wang, Lei Zhang, You-He Zhou

A high precision, and space time fully decoupled, wavelet formulation numerical method is developed for a class of nonlinear initial boundary value problems. This method is established based on a proposed Coiflet based approximation scheme with an adjustable high order for a square integrable function over a bounded interval, which allows expansion coefficients to be explicitly expressed by function values at a series of single points. In applying the solution method, the nonlinear initial boundary value problems are first spatially discretized into a nonlinear initial value problem by combining the proposed wavelet approximation scheme and the conventional Galerkin method. A novel high order step by step time integrating approach is then developed for the resulting nonlinear initial value problem using the same function approximation scheme based on wavelet theory. The solution method is shown to have Nth-order accuracy, as long as the Coiflet with [0, 3N-1] compact support is adopted, where N can be any positive even number. In addition, the stability property of the method is analyzed, and the stable domain is determined. Numerical examples are considered to justify both the accuracy and efficiency of the method. Results show that the proposed solution method has better accuracy and efficiency than most other methods.

Lei Zhang, Mukesh Ghimire, Wenlong Zhang et al.

Solving Hamilton-Jacobi-Isaacs (HJI) PDEs numerically enables equilibrial feedback control in two-player differential games, yet faces the curse of dimensionality (CoD). While physics-informed neural networks (PINNs) have shown promise in alleviating CoD in solving PDEs, vanilla PINNs fall short in learning discontinuous solutions due to their sampling nature, leading to poor safety performance of the resulting policies when values are discontinuous due to state or temporal logic constraints. In this study, we explore three potential solutions to this challenge: (1) a hybrid learning method that is guided by both supervisory equilibria and the HJI PDE, (2) a value-hardening method where a sequence of HJIs are solved with increasing Lipschitz constant on the constraint violation penalty, and (3) the epigraphical technique that lifts the value to a higher dimensional state space where it becomes continuous. Evaluations through 5D and 9D vehicle and 13D drone simulations reveal that the hybrid method outperforms others in terms of generalization and safety performance by taking advantage of both the supervisory equilibrium values and costates, and the low cost of PINN loss gradients.

Xinghua Xue, Haitong Huang, Cheng Liu et al.

Winograd convolution is originally proposed to reduce the computing overhead by converting multiplication in neural network (NN) with addition via linear transformation. Other than the computing efficiency, we observe its great potential in improving NN fault tolerance and evaluate its fault tolerance comprehensively for the first time. Then, we explore the use of fault tolerance of winograd convolution for either fault-tolerant or energy-efficient NN processing. According to our experiments, winograd convolution can be utilized to reduce fault-tolerant design overhead by 27.49\% or energy consumption by 7.19\% without any accuracy loss compared to that without being aware of the fault tolerance

Houman Owhadi, Lei Zhang

In this paper, we consider numerical homogenization of acoustic wave equations with heterogeneous coefficients, namely, when the bulk modulus and the density of the medium are only bounded. We show that under a Cordes type condition the second order derivatives of the solution with respect to harmonic coordinates are $L^2$ (instead $H^{-1}$ with respect to Euclidean coordinates) and the solution itself is in $L^{\infty}(0,T,H^2(Ω))$ (instead of $L^{\infty}(0,T,H^1(Ω))$ with respect to Euclidean coordinates). Then, we propose an implicit time stepping method to solve the resulted linear system on coarse spatial scales, and present error estimates of the method. It follows that by pre-computing the associated harmonic coordinates, it is possible to numerically homogenize the wave equation without assumptions of scale separation or ergodicity.

Houman Owhadi, Lei Zhang

This paper addresses the issue of homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available. Namely, we consider divergence form linear parabolic operators in $Ω\subset \R^n$ with $L^\infty(Ω\times (0,T))$-coefficients. It appears that the inverse operator maps the unit ball of $L^2(Ω\times (0,T))$ into a space of functions which at small (time and space) scales are close in $H^1$-norm to a functional space of dimension $n$. It follows that once one has solved these equations at least $n$-times it is possible to homogenize them both in space and in time, reducing the number of operations counts necessary to obtain further solutions. In practice we show that under a Cordes type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to harmonic coordinates are in $L^2$ (instead of $H^{-1}$ with Euclidean coordinates). If the medium is time independent then it is sufficient to solve $n$ times the associated elliptic equation in order to homogenize the parabolic equation.

Houman Owhadi, Lei Zhang

We consider divergence form elliptic operators in dimension $n\geq 2$ with $L^\infty$ coefficients. Although solutions of these operators are only Hölder continuous, we show that they are differentiable ($C^{1,α}$) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators.