Publications

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Preprints

• Z. Chen, S. Lee, and L. Mu, Automated Detection and Characterization of Singularities in Functions using Neural Networks - from FFT Signals, submitted (2024).

     Abstract: To be added.

Peer-reviewed Publications (Appeared or Accepted) (Click each one for abstract)

• R. Kalelkar, H. Xu, D. Nguyen, and Z. Chen, Generalist CEOs and the Readability of 10-K Reports, Advances in Accounting (2023). Full Text(HTML)

     Abstract: In this paper, we investigate the association between the general managerial ability of CEOs and the readability of 10-K reports. We find that the readability of 10-K reports is lower for firms managed by CEOs with general managerial ability. Our result is robust to change analysis, alternate readability measures, various fixed effects, instrumental variables, and the propensity score approach. Our additional analysis reveals that general managerial ability is negatively associated with the readability of management discussion and analysis (MD&A). Moreover, the disclosure tone of the 10-K and MD&A is conservative when the firms are managed by generalist CEOs. Our findings also reveal that CEO tenure moderates the positive association between the general ability index and the Gunning Fog index of 10-K reports. Finally, we find that overinvestment and misstatement strengthen the association between the general ability index and the readability of 10-K reports, thus supporting the obfuscation hypothesis. We thus conclude that firms incur costs in the form of lower disclosure quality when they opt for a generalist CEO.
• Z. Chen and L. Mu, High Order IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings, Communications on Applied Mathematics and Computation, accepted (2022). DOI, full-text (view only).

     Abstract: In this paper, we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs. To tackle the randomness in the problem, the stochastic Galerkin method of the generalized polynomial chaos approach has been employed. Besides, the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed. We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.
• Z. Chen, L. Liu, and L. Mu, Solving the linear transport equation by a deep neural network approach, Discrete and Continuous Dynamical Systems Series S, 2022, 15(4): 669-686. DOI, Full Text(HTML), arXiv

     Abstract: In this paper, we study the linear transport model by adopting the deep learning method, in particular the deep neural network (DNN) approach. While the interest in using DNN to study partial differential equations is arising, here we adapt it to study kinetic models, in particular the linear transport model. Moreover, theoretical analysis of the convergence of neural networks and its approximated solution towards the analytic solution is shown. We demonstrate the accuracy and effectiveness of the proposed DNN method in numerical experiments.
• L. Lyu and Z. Chen, Local discontinuous Galerkin methods with novel basis for fractional diffusion equations with non-smooth solutions, Commun. Appl. Math. Comput. 4, 227–249 (2022). DOI, Preprint

     Abstract: In this paper, we develop novel local discontinuous Galerkin (LDG) methods for fractional diffusion equations with non-smooth solutions. We consider such problems, for which the solutions are not smooth at the boundary, and therefore the traditional LDG methods with piecewise polynomial solutions suffer accuracy degeneracy. The novel LDG methods utilize a solution information-enriched basis, simulate the problem on a paired special mesh, and achieve optimal order of accuracy. We analyze the $L^2$ stability and optimal error estimate in the $L^2$-norm. Finally, numerical examples are presented to validate the theoretical conclusions.
• L. Mu and Z. Chen, A New WENO Weak Galerkin Finite Element Method for Time Dependent Hyperbolic Equations, Applied Numerical Mathematics (2020), 159: 106-124. DOI, Preprint

     Abstract: In this paper, we develop a new WENO weak Galerkin finite element scheme for solving the time dependent hyperbolic equations. The upwind-type stabilizer is imposed to enforce the flux direction in the scheme. For the linear convection equations, we analyze the $L^2$-stability and error estimate for the $L^2$-norm. We also investigate a simple limiter using weighted essentially non-oscillatory (WENO) methodology for obtaining a robust procedure to achieve high order accuracy and capture the sharp, non-oscillatory shock transitions. The approach applies to linear convection equations and Burgers equations. Finally, numerical examples are presented to validate the theoretical conclusions.
• M. P. Laiu, Z. Chen and C. D. Hauck, A fast implicit solver for semiconductor models in one space dimension, Journal of Computational Physics (2020): 109567. DOI, Preprint

     Abstract: Several different approaches are proposed for solving fully implicit discretizations of a simplified Boltzmann-Poisson system with a linear relaxation-type collision kernel. This system models the evolution of free electrons in semiconductor devices under a low-density assumption. At each implicit time step, the discretized system is formulated as a fixed-point problem, which can then be solved with a variety of methods. A key algorithmic component in all the approaches considered here is a recently developed sweeping algorithm for Vlasov-Poisson systems. A synthetic acceleration scheme has been implemented to accelerate the convergence of iterative solvers by using the solution to a drift-diffusion equation as a preconditioner. The performance of four iterative solvers and their accelerated variants has been compared on problems modeling semiconductor devices with various electron mean-free-path.
• Z. Chen and C. D. Hauck, Multiscale convergence properties for spectral approximations of a model kinetic equation, Mathematics of Computation 88.319 (2019): 2257-2293. DOI, Preprint

     Abstract: In this work, we prove rigorous convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by standard algebraic estimates of the form $N^{−q}$, where $N$ is the number of modes and $q > 0$ depends on the regularity of the solution. However, in the multiscale setting, the error estimate can be expressed in terms of the scaling parameter $\epsilon$, which measures the ratio of the mean-free-path to the characteristic domain length. We show that, for isotropic initial conditions, the error in the spectral approximation is $\mathcal{O}(\epsilon^{N+1})$. More surprisingly, the coefficients of the expansion satisfy super convergence properties. In particular, the error of the lth coefficient of the expansion scales like $\mathcal{O}(\epsilon^{2N})$ when $l = 0$ and $\mathcal{O}(\epsilon^{2N+2-l})$ for all $1 \leq l \leq N$. This result is significant, because the low-order coefficients correspond to physically relevant quantities of the underlying system. All the above estimates involve constants depending on $N$, the time $t$, and the initial condition. We investigate specifically the dependence on $N$, in order to assess whether increasing $N$ actually yields an additional factor of $\epsilon$ in the error. Numerical tests will also be presented to support the theoretical results.
• Z. Chen, L. Liu and L. Mu, DG-IMEX Stochastic Galerkin schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings, Journal of Scientific Computing (2017), 73(2), 566-592. DOI, Preprint

     Abstract: In this paper, we consider the linear transport equation under diffusive scaling and with random inputs. The method is based on the generalized polynomial chaos approach in the stochastic Galerkin framework. Several theoretical aspects will be addressed. A uniform numerical stability with respect to the Knudsen number $\epsilon$, and a uniform in ε error estimate is given. For temporal and spatial discretizations, we apply the implicit–explicit scheme under the micro–macro decomposition framework and the discontinuous Galerkin method, as proposed in Jang et al. (SIAM J Numer Anal 52:2048–2072, 2014) for the deterministic problem. We provide rigorous proof of the stochastic asymptotic-preserving (sAP) property. Extensive numerical experiments that validate the accuracy and sAP of the method are conducted.
• H. Huang, Z. Chen, J. Li and J. Yan, Direct discontinuous Galerkin method and its variations for second order elliptic equations, Journal of Scientific Computing (2017), 70(2), 744-765. DOI, Preprint

     Abstract: In this paper, we study the direct discontinuous Galerkin method and its variations for 2nd order elliptic problems. A priori error estimate under the energy norm is established for all four methods. Optimal error estimate under $L^2$ norm is obtained for DDG method with interface correction and symmetric DDG method. A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal $(k + 1)$-th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal $(k + 1)$-th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of the Helmholtz equation are well resolved.
• Z. Chen, H. Huang and J. Yan, Third order Maximum-Principle-Satisfying Direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes, Journal of Computational Physics, v308(2016), pp.198-217. DOI, Preprint

     Abstract: We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods for convection-diffusion equations on unstructured triangular meshes. We carefully calculate the normal derivative numerical flux across element edges and prove that, with the proper choice of parameter pair $(\beta_0,\beta_1)$ in the numerical flux formula, the quadratic polynomial solution satisfies the strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. A sequence of numerical examples is carried out to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter.
• Y. Chen, Z. Chen, Y. Cheng, A. Gillman and F. Li, Study of Discrete Scattering Operators for Some Linear Kinetic Models, S. C. Brenner (Editor), In Topics in Numerical Partial Differential Equations and Scientific Computing, Springer-Verlag New York, IMA Volumes in Mathematics and its Applications, Vol. 160(2016), pp.99-136. DOI, Preprint

     Abstract: In this paper, we consider spatially homogeneous linear kinetic models arising from semiconductor device simulations and investigate how various deterministic numerical methods approximate their scattering operators. In particular, methods including first and second order discontinuous Galerkin methods, a first order collocation method, a Fourier-collocation spectral method, and a Nyström method are examined when they are applied to one-dimensional models with singular or continuous scattering kernels. Mathematical properties are discussed for the corresponding discrete scattering operators. We also present numerical experiments to demonstrate the performance of these methods. Understanding how the scattering operators are approximated can provide insights into designing efficient algorithms for simulating kinetic models and for the implicit discretizations of the problems in the presence of multiple scales.
• Z. Chen and C.-W. Shu, Recovering exponential accuracy in Fourier spectral methods involving piecewise smooth functions with unbounded derivative singularities, Journal of Scientific Computing, v65(2015), pp.1145-1165. DOI

     Abstract: Fourier spectral methods achieve exponential accuracy both on the approximation level and for solving partial differential equations, if the solution is analytic. If the solution is discontinuous but piecewise analytic up to the discontinuities, Fourier spectral methods produce poor pointwise accuracy, but still maintain exponential accuracy after post-processing (Gottlieb and Shu in SIAM Rev 30:644–668, 1997). In Chen and Shu (J Comput Appl Math 265:83–95, 2014), an extended technique is provided to recover exponential accuracy for functions which have end-point singularities, from the knowledge of point values on standard collocation points. In this paper, we develop a technique to recover exponential accuracy from the first N Fourier coefficients of functions which are analytic in the open interval but have unbounded derivative singularities at endpoints. With this post-processing method, we are able to obtain exponential accuracy of spectral methods applied to linear transport equations involving such functions.
• Z. Chen and C.-W. Shu, Recovering exponential accuracy from collocation point values of smooth functions with end-point singularities, Journal of Computational and Applied Mathematics, v265 (2014), pp.83-95. DOI

     Abstract: Gibbs phenomenon is the particular manner how a global spectral approximation of a piecewise analytic function behaves at the jump discontinuity. The truncated spectral series has large oscillations near the jump, and the overshoot does not decay as the number of terms in the truncated series increases. There is therefore no convergence in the maximum norm, and convergence in smooth regions away from the discontinuity is also slow. In Gottlieb and Shu (1995), a methodology is proposed to completely overcome this difficulty in the context of spectral collocation methods, resulting in the recovery of exponential accuracy from collocation point values of a piecewise analytic function. In this paper, we extend this methodology to handle spectral collocation methods for functions which are analytic in the open interval but have singularities at end-points. With this extension, we are able to obtain exponential accuracy from collocation point values of such functions. Similar to Gottlieb and Shu (1995), the proof is constructive and uses the Gegenbauer polynomials $C_n^\lambda(x)$. The result implies that the Gibbs phenomenon can be overcome for smooth functions with endpoint singularities.