Zhiqiang Cai

Zhiqiang Cai, Rob Falgout, Shun Zhang

The first-order system LL* (FOSLL*) approach for general second-order elliptic partial differential equations was proposed and analyzed in [10], in order to retain the full efficiency of the L2 norm first-order system least-squares (FOSLS) ap- proach while exhibiting the generality of the inverse-norm FOSLS approach. The FOSLL* approach in [10] was applied to the div-curl system with added slack vari- ables, and hence it is quite complicated. In this paper, we apply the FOSLL* approach to the div system and establish its well-posedness. For the corresponding finite ele- ment approximation, we obtain a quasi-optimal a priori error bound under the same regularity assumption as the standard Galerkin method, but without the restriction to sufficiently small mesh size. Unlike the FOSLS approach, the FOSLL* approach does not have a free a posteriori error estimator, we then propose an explicit residual error estimator and establish its reliability and efficiency bounds

Zhiqiang Cai, Cuiyu He, Shun Zhang

In \cite{CaZh:09}, we introduced and analyzed an improved Zienkiewicz-Zhu (ZZ) estimator for the conforming linear finite element approximation to elliptic interface problems. The estimator is based on the piecewise "constant" flux recovery in the $H(div;Ω)$ conforming finite element space. This paper extends the results of \cite{CaZh:09} to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear $H(div)$ space.

Zhiqiang Cai, Cuiyu He, Shun Zhang

For elliptic interface problems, this paper studies residual-based a posteriori error estimations for various finite element approximations. For the conforming and the Raviart-Thomas mixed elements in two-dimension and for the Crouzeix-Raviart nonconforming and the discontinuous Galerkin elements in both two- and three-dimensions, the global reliability bounds are established with constants independent of the jump of the diffusion coefficient. Moreover, we obtain these estimates with no assumption on the distribution of the diffusion coefficient.

Zhiqiang Cai, Shun Zhang

We introduced and analyzed robust recovery-based a posteriori error estimators for various lower order finite element approximations to interface problems in [9, 10], where the recoveries of the flux and/or gradient are implicit (i.e., requiring solutions of global problems with mass matrices). In this paper, we develop fully explicit recovery-based error estimators for lower order conforming, mixed, and non- conforming finite element approximations to diffusion problems with full coefficient tensor. When the diffusion coefficient is piecewise constant scalar and its distribution is local quasi-monotone, it is shown theoretically that the estimators developed in this paper are robust with respect to the size of jumps. Numerical experiments are also performed to support the theoretical results.

Zhiqiang Cai, Anastassia Doktorova, Robert D. Falgout et al.

This paper studies the shallow Ritz method for solving one-dimensional diffusion-reaction problems. The method is capable of improving the order of approximation for non-smooth problems. By following a similar approach to the one presented in [9], we present a damped block Newton (dBN) method to achieve nearly optimal order of approximation. The dBN method optimizes the Ritz functional by alternating between the linear and non-linear parameters of the shallow ReLU neural network (NN). For diffusion-reaction problems, new difficulties arise: (1) for the linear parameters, the mass matrix is dense and even more ill-conditioned than the stiffness matrix, and (2) for the non-linear parameters, the Hessian matrix is dense and may be singular. This paper addresses these challenges, resulting in a dBN method with computational cost of ${\cal O}(n)$. The ideas presented for diffusion-reaction problems can also be applied to least-squares approximation problems. For both applications, starting with the non-linear parameters as a uniform partition, numerical experiments show that the dBN method moves the mesh points to nearly optimal locations.

Zhiqiang Cai, Jingshuang Chen, Min Liu

Designing an optimal deep neural network for a given task is important and challenging in many machine learning applications. To address this issue, we introduce a self-adaptive algorithm: the adaptive network enhancement (ANE) method, written as loops of the form train, estimate and enhance. Starting with a small two-layer neural network (NN), the step train is to solve the optimization problem at the current NN; the step estimate is to compute a posteriori estimator/indicators using the solution at the current NN; the step enhance is to add new neurons to the current NN. Novel network enhancement strategies based on the computed estimator/indicators are developed in this paper to determine how many new neurons and when a new layer should be added to the current NN. The ANE method provides a natural process for obtaining a good initialization in training the current NN; in addition, we introduce an advanced procedure on how to initialize newly added neurons for a better approximation. We demonstrate that the ANE method can automatically design a nearly minimal NN for learning functions exhibiting sharp transitional layers as well as discontinuous solutions of hyperbolic partial differential equations.

Zhiqiang Cai, Jingshuang Chen, Min Liu

This paper studies least-squares ReLU neural network method for solving the linear advection-reaction problem with discontinuous solution. The method is a discretization of an equivalent least-squares formulation in the set of neural network functions with the ReLU activation function. The method is capable of approximating the discontinuous interface of the underlying problem automatically through the free hyper-planes of the ReLU neural network and, hence, outperforms mesh-based numerical methods in terms of the number of degrees of freedom. Numerical results of some benchmark test problems show that the method can not only approximate the solution with the least number of parameters, but also avoid the common Gibbs phenomena along the discontinuous interface. Moreover, a three-layer ReLU neural network is necessary and sufficient in order to well approximate a discontinuous solution with an interface in $\mathbb{R}^2$ that is not a straight line.

Zhiqiang Cai, Jingshuang Chen, Min Liu et al.

This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional construction and employs least-squares functionals as loss functions to determine parameters of the deep neural network. There are various least-squares functionals for a partial differential equation. This paper focuses on the so-called first-order system least-squares (FOSLS) functional studied in [3], which is based on a first-order system of scalar second-order elliptic PDEs. Numerical results for second-order elliptic PDEs in one dimension are presented.

Zhiqiang Cai, Shun Zhang

For elliptic interface problems in two- and three-dimensions, this paper establishes a priori error estimates for Crouzeix-Raviart nonconforming, Raviart-Thomas mixed, and discontinuous Galerkin finite element approximations. These estimates are robust with respect to the diffusion coefficient and optimal with respect to local regularity of the solution. Moreover, we obtain these estimates with no assumption on the distribution of the diffusion coefficient.