Research
Computational Neuroscience and High Performance Computing:
The following two movies are our 3D full cell simulations (on the HPC Cluster) for calcium dynamics in two different kinds of neurons both with endoplasmic reticulum (ER) inside. Geometric data of the two neurons are obtained from NeuroMorpho.org. The simulations are based on AnaMorph (for neuron construction), TetGen (for 3D mesh), FreeFem++ (for solving PDEs), and OpenMPI , PETSc (for parallel computing). Numerical results are visualized in ParaView and model equations can be found in PDF-file. (wait a while for loading the movies)
Publications:
Qingguang Guan and Gillian Queisser, Modeling calcium dynamics in neurons with endoplasmic reticulum: existence, uniqueness and an implicit-explicit finite element scheme, Communications in Nonlinear Science and Numerical Simulation, 109 (2022) 106354. PDF-file
Qingguang Guan and Gillian Queisser, Stable high order implicit-explicit multi-step finite element schemes for calcium dynamics in neurons with ER, in preparation.
Deep Learning with Neural Networks:
3dbinary-b2.pdfGlobal Optima in 3D
3dbinary-b2-6.pdfshrinking Compact Support
Publications:
Qingguang Guan, How Can Deep Neural Network Fail even with Global Optima? Submitted.
Abel Gurung (undergraduate) and Qingguang Guan, Hybrid PDE-Deep Neural Network Model for Calcium Dynamics in Neurons. Submitted.
Weak Galerkin FEM and VEM:
Domain with curved interface
Curvilinear element for WG-FEM
Publications:
Qingguang Guan, Max Gunzburger, and Wenju Zhao, Weak Galerkin finite element methods for a second-order elliptic variational inequality, Computer Methods in Applied Mechanics and Engineering, Volume 337, 2018, Pages 677-688.
Qingguang Guan, Weak Galerkin finite element method for Poisson's equation on polytopal meshes with small edges or faces, Journal of Computational and Applied Mathematics, Volume 368, 2020: 112584.
Susanne C. Brenner, Qingguang Guan, and Li-Yeng Sung, Some estimates for Virtual Element Methods, Computational Methods in Applied Mathematics, Volume 4, 2017, Pages:553 - 574.
Qingguang Guan, Some estimates of Virtual Element Methods for fourth order problem, Electronic Research Archive, 2021, published online.
Qingguang Guan, Weak Galerkin finite element method for second order problems on curvilinear polytopal meshes with Lipschitz continuous edges or faces, submitted, PDF-file
Wenju Zhao and Qingguang Guan, Numerical analysis of energy stable weak Galerkin schemes for the Cahn-Hilliard equation, submitted.
Numerical Analysis for Nonlocal PDEs:
Publications:
Qingguang Guan, Max Gunzburger, θ schemes for finite element methods for space-time fractional diffusion equation, Journal of Computational and Applied Mathematics, Volume 288, November 2015, Pages: 264-273.
Qingguang Guan, Max Gunzburger, Stability and accuracy of time-stepping schemes and dispersion relations for a nonlocal wave equation, Numerical Methods for Partial Differential Equations, Volume 31, Issue 2, March 2015, Pages: 500-516.
Qingguang Guan, Max Gunzburger, Stability and convergence of time-stepping methods for a nonlocal model for diffusion, Discrete and Continuous Dynamical Systems - Series B, Volume 20, Issue 5, July 2015, Pages: 1315-1335.
Qingguang Guan and Max Gunzburger, Analysis and approximation of a nonlocal obstacle problem, Journal of Computational and Applied Mathematics, Volume 313, 2017, Pages: 102-118.
Qingguang Guan, Max Gunzburger, Clayton Webster, and Guannan Zhang, Reduced basis methods for nonlocal diffusion problems with random input data, Computer Methods in Applied Mechanics and Engineering, Volume 317, 2017, Pages: 746-770.
Qingguang Guan, Max Gunzburger, and Xiaoping Zhang, Collocation method for one dimensional nonlocal diffusion equations, Numerical Methods for Partial Differential Equations, 2021, published online