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Research Interest

• Applied Mathematics, Numerical Analysis and Scientific Computing

High Order Numerical Methods for Solving Partial Differential Equations
Finite Element Methods; Finite Volume Methods; Adaptive Methods
A Priori and A Posteriori Error Estimations for Numerical Schemes
Multiscale Modeling Methods and Computational Methods
Domain Decomposition Methods and Faster Numerical Solver
Efficient Numerical Methods in High Dimensional Approximation
Numerical Methods for Uncertainty Quantification
Model Reduction in Stochastic Partial Differential Equations

• Math/Computational Biology

Geometric and Electrostatic Modeling
Topology Based Machine Learning of Biomolecular Systems

Optimal Control of Convection-Cooling

This project deals with enhancing convection-cooling via active control of the incompressible velocity. Following is one example of a stationary diffusion-convection model. The vector field plots the convection flow, which works on spreading out the temperature distribution. The color has been scale as the initial distribution of temperature.

Fig. Cost Values VS Iteration

Upscale Brinkman Flow Simulation

In this project, we validate empirical models of effective viscosity and permeability for a multi-scale porous media by analyzing flow field between fully resolved microporosity (Stokes flow) and upscaled model (Brinkman flow) and comparing parameters from our simulation against empirical models.

Following illustrates the resolved Stokes simulation and upscaled Brinkman simulation (Credit to Bin Wang, LSU).

Pressure Robust Schemes in CFD Modeling

Locking phenomenon to poor mass conservation may introduce computational errors in CFD modeling [1].

In the numerical scheme, discrete inf-sup stability ⇒ convergence & optimal convergence order
However, the vector L2 scalar product can excite arbitrarily large velocity errors

Pressure-robust scheme means the velocity error is independent of the continuous pressure. This project develops a locking-free weak Galerkin methods which achieves locking-free, pressure-robust by minimal efforts in modifying the previous non-pressure robust scheme without compromising the accuracy.

Fig. Brinkman simulation: wrong simulation by standard (left) and correct simulation by pressure robust (right) simulations.

The locking phenomenon: as small values in viscosity will produce large error in velocity: [1]

By employing divergence preserving velocity reconstruction, one can retrieve the pressure robustness.
For Stokes equations, one only needs to modify the right hand body force.
For Brinkman equations [2] and Navier-Stokes equations, only minimal efforts are needed in the development of numerical scheme.

Weak Galerkin WENO Limiter for Hyperbolic Equations

In this project [9], we consider the time dependent hyperbolic equations. There are two difficulties in the hyperbolic simulation: upwind scheme and sharp shock front. The upwind-type stabilizer is imposed to enforce the flux direction. Besides, for problems with strong discontinuities, 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.

Fig. Plot of numerical solution to linear convection equation on mesh h = 0.01 with for WG elements (a) k = 1; (b) k = 2; (c) k = 3.

Fig. Plot of numerical solution to linear convection equation on mesh h = 0.01 with WENO limiter for WG elements (a) k = 1; (b) k = 2; (c) k = 3.

Polygonal Meshes with Curved Edges

In this project [8], we consider the polygonal meshes with curved edges. By modifying the basis functions by the corresponding parametric equations on the curved edges, we can achieve the optimal rate in convergence for high order finite element methods.

Fig. Illustration of quadrature points on the curved hexagon (Left); Illustration of a curved polygonal mesh (Middle); Plot of the numerical solution on the curved mesh.

The mesh with flat edge (b) treats the boundary curve as piecewise line segment, which limits the approximation order to O(h2) even for high order numerical scheme. The curved mesh (a) can resolve this limitation and produce the optimal convergence rate.

Weak Galerkin Finite Element Methods

The weak Galerkin ﬁnite element method (WGFEM) is a novel numerical method that was first proposed and analyzed by Wang and Ye for general second order elliptic problems. Like discontinuous Galerkin finite element Method (DGFEM), WGFEM is making use of discontinuous basis functions for approximation. The Key of WGFEM is to define the weak derivative operators from integration by parts.

Advantages of WGFEM:

The weak Galerkin finite element methods represent advanced methodology for handling discontinuous approximation functions. The WG formulations are similar to the corresponding weak forms of the PDEs.
The weak Galerkin methodology provide a general framework for deriving new methods and simplifying the existing methods. For discontinuous approximation function, the gradient is well defined.
Simple formulations imply easy analysis and easy applications. The system is symmetric and positive definite.
The flexibility of choosing high-order scheme and arbitrary polygonal meshing enables that the h-p version of the adaptive refinement's capability in avoiding non-necessary mesh refinement.

Fig. Two Adaptivity Tests for Kellogg Problem and Convergence Results for Adaptive Polygonal Meshing [3].

Model Reduction for Stochastic PDEs

In the real world numerical simulations, one has to consider input data or coefficients in PDEs as random variables and represent them in probabilistic terms and thus solving stochastic PDEs accordingly. In such case, we denote our problem as a parametrized PDEs. However full-order model (FOM) approximation becomes prohibitive when one needs multi-query the solutions of PDEs. For example, parametrized PDEs involve a huge algebraic systems from discretization of full-order models. Model order reduction (ROM) approach is to replace the FOM problem by a much lower numerical simulation. The computation savings lie in that latter solution evaluation for any new parameter will be independent of the dimension of the original FOM problem. In our project, we explore the parametric dependence of the linear elliptic PDEs solution by combining a handful samplings (snapshots) for a set of parameter values to set up a reduced basis models. Then the huge algebraic system is replaced by a smaller system, whose dimension depends on the dominated modes of the derived snapshots [4].

Fig. Three snapshots of random convection dominate problems.

Table. Comparison for Computational Cost

Sparse Grids for High-dimensional PDEs

In the higher dimensional simulations, for example the 6D kinetic problems or etc, the standard methods are limited since the unaffordable computational cost as exponentially increasing at order O(Nd). Sparse grid (SG) methods have been developed to efficiently solving partial differential equations and integral equations. The fundamental of SG methods is the decomposition for the discretization of functional space into several low resolution grids and utilizing the sparse tensor product. SG can reduce the DOFs significantly while preserving the similar accuracy as that of full grids. We used sparse grids methods to reduce the DOFs to the order O(N|log2 N|d−1) in the calculation and developed the numerical solver to approximate the full 6D kinetic equations [5]. Comparing to the full grids simulation, SG is able to convert the large-sparse matrix into smaller sparse matrix with denser blocks.

Fig. An Adaptive Discontinuous Galerkin Finite Element Sparse Grids Approximation for Gaussian Distribution.

Computational Biomolecular

Geometric/electrostatic modeling is an essential component in computational biophysics and molecular biology. However, commonly used geometric representations admit geometric singularities such as cusps, tips and self-intersecting facets that lead to computational instabilities in the molecular modeling. In our project, we investigate the use of flexibility and rigidity index (FRI), which has a proved superiority in protein B-factor prediction, for biomolecular geometric representation and associated electrostatic analysis. Besides, FRI rigidity surfaces are free of geometric singularities. Polarized curvatures based surface modeling by using the product of minimum curvature and electrostatic potential can be utilized to accurately predict potential protein-ligand binding sites (shown as Figure) [6].

Fig. Protein binding site prediction using polarized curvatures.

Machine Learning of Biomolecular Systems

Machine learning is a major driving force behind the current data science, while its full potential in unveiling structure-function relationships of biomolecular systems is yet to be reached. Due to the geometric complexity and biological complexity in three-dimensional (3D) biomolecular data sets, there is no competitive deep learning algorithm for predicting protein-ligand binding affinities or docking prediction. Persistent homology, a relatively new branch of algebraic topology, has emerged as a new strategy for data analysis. In this project, we developed topology based machine learning approach to investigate the structures in biomolecular systems [7].

Fig. Flow Chart of Deep Learning for Biomolecular Systems

Deep Neural Network for Interface Problems

In this project, we investigate a mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the interface, we employ different neural networks in different sub-domains.

Fig. Illustration of neural network architecture diagram for interface problems.

Fig. (a) 20×20 grid; (b). Numerical solution UNN; (c). Error of u−UNN.

References:[1] Alexander Linke & Christian Merdon, Slides, http://www.pitt.edu/~trenchea/NAPFF/Presentations/Alexander_Linke-PressureRobustMixedNSE.pdf[2] L. Mu. Pressure Robust Weak Galerkin Finite Element Methods for Stokes Problems. SIAM Journal on Scientific Computing. 42 (2020), B608–B629. [3] F. Bao, L. Mu, J. Wang. A fully computable a posteriori error estimate for the Stokes equations on polytopal meshes. SIAM Journal on Numerical Analysis 57 (2019):458-477.[4] L. Mu, G. Zhang. A Domain-decomposition-based Sparse Approximation for Convection-diffusion Equations with Random Coefficients. SIAM Journal on Scientific Computing 41 (3), A1984-A2011[5] Technique report: DG-SG-Vlasov.[6] L. Mu, K. Xia, and G. Wei. Geometric and Electrostatic Modeling Using Molecular Rigidity Functions. Journal of Computational and Applied Mathematics, 313 (2017): 18-37. [7] Z. Cang, Lin Mu, and G. Wei. Representability of Algebraic Topology for Biomolecules in Machine Learning Based Scoring and Virtual Screening. PLOS Computational Biology, 14 (1): e1005929. [8] L. Mu. Weak Galerkin Finite Element with Curved Edges. Submitted.[9] L. Mu and Z. Chen. A New WENO Weak Galerkin Finite Element Method for Time Dependent Hyperbolic Equations. Submitted.[10] C. He, X. Hu, L. Mu. Mesh-free Method Using Piecewise Deep Neural Network for Elliptic Interface Problems. Submitted.

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