arXiv subjects

Lin Mu

Publications and source records attributed to Lin Mu.

At least 37 records · Page 2Linked to original sources

arXiv2023

A uniform and pressure-robust enriched Galerkin method for the Brinkman equations

This paper presents a pressure-robust enriched Galerkin (EG) method for the Brinkman equations with minimal degrees of freedom based on EG velocity and pressure spaces. The velocity space consists of linear Lagrange polynomials enriched by a discontinuous, piecewise linear, and mean-zero vector function per element, while piecewise constant functions approximate the pressure. We derive, analyze, and compare two EG methods in this paper: standard and robust methods. The standard method requires a mesh size to be less than a viscous parameter to produce stable and accurate velocity solutions, which is impractical in the Darcy regime. Therefore, we propose the pressure-robust method by utilizing a velocity reconstruction operator and replacing EG velocity functions with a reconstructed velocity. The robust method yields error estimates independent of a pressure term and shows uniform performance from the Stokes to Darcy regimes, preserving minimal degrees of freedom. We prove well-posedness and error estimates for both the standard and robust EG methods. We finally confirm theoretical results through numerical experiments with two- and three-dimensional examples and compare the methods' performance to support the need for the robust method.

math.NA↗

arXiv2023

Pressure-robust enriched Galerkin methods for the Stokes equations

In this paper, we present a pressure-robust enriched Galerkin (EG) scheme for solving the Stokes equations, which is an enhanced version of the EG scheme for the Stokes problem proposed in [Son-Young Yi, Xiaozhe Hu, Sanghyun Lee, James H. Adler, An enriched Galerkin method for the Stokes equations, Computers and Mathematics with Applications, accepted, 2022]. The pressure-robustness is achieved by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. An a priori error analysis proves that the velocity error is independent of the pressure and viscosity. We also propose and analyze a perturbed version of our pressure-robust EG method that allows for the elimination of the degrees of freedom corresponding to the discontinuous component of the velocity vector via static condensation. The resulting method can be viewed as a stabilized $H^1$-conforming $\mathbb{P}_1$-$\mathbb{P}_0$ method. Further, we consider efficient block preconditioners whose performances are independent of the viscosity. The theoretical results are confirmed through various numerical experiments in two and three dimensions.

math.NA↗

arXiv2022

A low-cost, parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations

In this paper, we propose a low-cost, parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, the symmetric IPDG formulation, despite of its advantage of symmetry, requires a lot of computational effort to choose an optimal penalty parameter and to compute different trace terms. In order to reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a parameter-free numerical scheme which has reduced computational complexity as well as optimal rates of convergence. Moreover, we achieve pressure-robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples.

math.NA↗

arXiv2021

An Efficient High-order Numerical Solver for Diffusion Equations with Strong Anisotropy

In this paper, we present an interior penalty discontinuous Galerkin finite element scheme for solving diffusion problems with strong anisotropy arising in magnetized plasmas for fusion applications. We demonstrate the accuracy produced by the high-order scheme and develop an efficient preconditioning technique to solve the corresponding linear system, which is robust to the mesh size and anisotropy of the problem. Several numerical tests are provided to validate the accuracy and efficiency of the proposed algorithm.

math.NA↗

arXiv2021

Optimal Control of Convection-Cooling and Numerical Implementation

This paper is concerned with the problem of enhancing convection-cooling via active control of the incompressible velocity field, described by a stationary diffusion-convection model. This essentially leads to a bilinear optimal control problem. A rigorous proof of the existence of an optimal control is presented and the first order optimality conditions are derived for solving the control using a variational inequality. Moreover, the second order sufficient conditions are established to characterize the local minimizer. Finally, numerical experiments are conducted utilizing finite elements methods together with nonlinear iterative schemes, to demonstrate and validate the effectiveness of our control design.

math.OC↗

arXiv2021

Solving the linear transport equation by a deep neural network approach

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 of 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 on the convergence of the neural network and its approximated solution towards the analytic solution is shown. We demonstrate the accuracy and effectiveness of the proposed DNN method in the numerical experiments.

math.NA↗

arXiv2020

Accelerating linear solvers for Stokes problems with C++ metaprogramming

The efficient solution of large sparse saddle point systems is very important in computational fluid mechanics. The discontinuous Galerkin finite element methods have become increasingly popular for incompressible flow problems but their application is limited due to high computational cost. We describe the C++ programming techniques that may help to accelerate linear solvers for such problems. The approach is based on the policy-based design pattern and partial template specialization, and is implemented in the open source AMGCL library. The efficiency is demonstrated with the example of accelerating an iterative solver of a discontinuous Galerkin finite element method for the Stokes problem. The implementation allows selecting algorithmic components of the solver by adjusting template parameters without any changes to the codebase. It is possible to switch the system matrix to use small statically sized blocks to store the nonzero values, or use a mixed precision solution, which results in up to 4 times speedup, and reduces the memory footprint of the algorithm by about 40\%. We evaluate both monolithic and composite preconditioning strategies for the 3 benchmark problems. The performance of the proposed solution is compared with a multithreaded direct Pardiso solver and a parallel iterative PETSc solver.

cs.MS↗

arXiv2020

A Pressure-Robust Weak Galerkin Finite Element Method for Navier-Stokes Equations

In this paper, we develop and analyze a novel numerical scheme for the steady incompressible Navier-Stokes equations by the weak Galerkin methods. The divergence-preserving velocity reconstruction operator is employed in the discretization of momentum equation. By employing the velocity construction operator, our algorithm can achieve pressure-robust, which means, the velocity error is independent of the pressure and the irrotational body force. Error analysis is established to show the optimal rate of convergence. Numerical experiments are presented to validate the theoretical conclusions.

math.NA↗

arXiv2020

A stabilizer free, pressure robust, and superconvergence weak Galerkin finite element method for the Stokes Equations on polytopal mesh

In this paper, we propose a new stabilizer free and pressure robust WG method for the Stokes equations with super-convergence on polytopal mesh in the primary velocity-pressure formulation. Convergence rates with one order higher than the optimal-order for velocity in both energy-norm and the $L^2$-norm and for pressure in $L^2$-norm are proved in our proposed scheme. The $H$(div)-preserving operator has been constructed based on the polygonal mesh for arbitrary polynomial degrees and employed in the body source assembling to break the locking phenomenon induced by poor mass conservation in the classical discretization. Moreover, the velocity error in our proposed scheme is proved to be independent of pressure and thus confirm the pressure-robustness. For Stokes simulation, our proposed scheme only modifies the body source assembling but remains the same stiffness matrix. Four numerical experiments are conducted to validate the convergence results and robustness.

math.NA↗

arXiv2020

A Mesh-free Method Using Piecewise Deep Neural Network for Elliptic Interface Problems

In this paper, we propose a novel 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. By reformulating the interface problem as a least-squares problem, we discretize the objective function using mean squared error via sampling and solve the proposed deep least-squares method by standard training algorithms such as stochastic gradient descent. The discretized objective function utilizes only the point-wise information on the sampling points and thus no underlying mesh is required. Doing this circumvents the challenging meshing procedure as well as the numerical integration on the complex interface. To improve the computational efficiency for more challenging problems, we further design an adaptive sampling strategy based on the residual of the least-squares function and propose an adaptive algorithm. Finally, we present several numerical experiments in both 2D and 3D to show the flexibility, effectiveness, and accuracy of the proposed deep least-square method for solving interface problems.

math.NA↗

arXiv2018

Wave Climate from Spectra and its Connections with Local and Remote Wind Climate

Because wind-generated waves can propagate over large distances, wave spectra from a fixed point can record information about air-sea interactions in distant areas. In this study, the spectral wave climate is computed for a specific location in the tropical Eastern Pacific Ocean. Several well-defined partitions independent of each other, referred to as wave-climate systems, are observed in the annual mean wave spectrum. Significant seasonal cycling, long-term trends, and correlations with climate indices are observed in the local wave spectra, showing the abundant climatic information they contain. The projections of the wind vector on the direction pointing to the target location are used to link the spectral wave climate and basin-scale wind climate. The origins of all the identified wave climate systems are clearly shown in the wind projection maps and some are thousands of kilometers away from the target point, demonstrating the validity of this connection. Comparisons are made between wave spectra and the corresponding local and remote wind fields with respect to seasonal and interannual variability, as well as the long-term trends. The results show that each frequency and direction of ocean wave spectra at a certain location can be linked to the wind field for a geographical area from a climatological point of view, implying that it is feasible to reconstruct a spectral wave climate from global observational wind field data and wind climate monitoring using observations of wave spectrum geographically far away.

physics.ao-ph↗

arXiv2018

A Domain-Decomposition Model Reduction Method for Linear Convection-Diffusion Equations with Random Coefficients

We develop a domain-decomposition model reduction method for linear steady-state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion equations with random diffusivities, and the convection-dominated transport equations with random velocities. We investigate the equations with two types of random fields, i.e., colored noises and discrete white noises, both of which can lead to high-dimensional parametric dependence. The motivation is to use domain decomposition to exploit low-dimensional structures of local problems in the sub-domains, such that the total number of expensive PDE solves can be greatly reduced. Our objective is to develop an efficient model reduction method to simultaneously handle high-dimensionality and irregular behaviors of the stochastic PDEs under consideration. The advantages of our method lie in three aspects: (i) online-offline decomposition, i.e., the online cost is independent of the size of the triangle mesh; (ii) operator approximation for handling non-affine and high-dimensional random fields; (iii) effective strategy to capture irregular behaviors, e.g., sharp transitions of the PDE solution. Two numerical examples will be provided to demonstrate the advantageous performance of our method.

math.NA↗

arXiv2018

An Immersed Weak Galerkin Method For Elliptic Interface Problems

In this paper, we present an immersed weak Galerkin method for solving second-order elliptic interface problems. The proposed method does not require the meshes to be aligned with the interface. Consequently, uniform Cartesian meshes can be used for nontrivial interfacial geometry. We show the existence and uniqueness of the numerical algorithm, and prove the error estimates for the energy norm. Numerical results are reported to demonstrate the performance of the method.

math.NA↗

arXiv2017

Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening

This work introduces a number of algebraic topology approaches, such as multicomponent persistent homology, multi-level persistent homology and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes. Multicomponent persistent homology retains critical chemical and biological information during the topological simplification of biomolecular geometric complexity. Multi-level persistent homology enables a tailored topological description of inter- and/or intra-molecular interactions of interest. Electrostatic persistence incorporates partial charge information into topological invariants. These topological methods are paired with Wasserstein distance to characterize similarities between molecules and are further integrated with a variety of machine learning algorithms, including k-nearest neighbors, ensemble of trees, and deep convolutional neural networks, to manifest their descriptive and predictive powers for chemical and biological problems. Extensive numerical experiments involving more than 4,000 protein-ligand complexes from the PDBBind database and near 100,000 ligands and decoys in the DUD database are performed to test respectively the scoring power and the virtual screening power of the proposed topological approaches. It is demonstrated that the present approaches outperform the modern machine learning based methods in protein-ligand binding affinity predictions and ligand-decoy discrimination.

q-bio.QM↗

arXiv2016

Multiscale persistent functions for biomolecular structure characterization

In this paper, we introduce multiscale persistent functions for biomolecular structure characterization. The essential idea is to combine our multiscale rigidity functions with persistent homology analysis, so as to construct a series of multiscale persistent functions, particularly multiscale persistent entropies, for structure characterization. To clarify the fundamental idea of our method, the multiscale persistent entropy model is discussed in great detail. Mathematically, unlike the previous persistent entropy or topological entropy, a special resolution parameter is incorporated into our model. Various scales can be achieved by tuning its value. Physically, our multiscale persistent entropy can be used in conformation entropy evaluation. More specifically, it is found that our method incorporates in it a natural classification scheme. This is achieved through a density filtration of a multiscale rigidity function built from bond and/or dihedral angle distributions. To further validate our model, a systematical comparison with the traditional entropy evaluation model is done. It is found that our model is able to preserve the intrinsic topological features of biomolecular data much better than traditional approaches, particularly for resolutions in the mediate range. Moreover, our method can be successfully used in protein classification. For a test database with around nine hundred proteins, a clear separation between all-alpha and all-beta proteins can be achieved, using only the dihedral and pseudo-bond angle information. Finally, a special protein structure index (PSI) is proposed, for the first time, to describe the "regularity" of protein structures. Essentially, PSI can be used to describe the "regularity" information in any systems.

q-bio.BM↗

arXiv2016

A simple finite element method for the Stokes equations

The goal of this paper is to introduce a simple finite element method to solve the Stokes and the Navier-Stokes equations. This method is in primal velocity-pressure formulation and is so simple such that both velocity and pressure are approximated by piecewise constant functions. Implementation issues as well as error analysis are investigated. A basis for a divergence free subspace of the velocity field is constructed so that the original saddle point problem can be reduced to a symmetric and positive definite system with much fewer unknowns. The numerical experiments indicate that the method is accurate and robust.

math.NA↗

arXiv2016

A Discrete Divergence-Free Weak Galerkin Finite Element Method for the Stokes Equations

A discrete divergence-free weak Galerkin finite element method is developed for the Stokes equations based on a weak Galerkin (WG) method introduced in the reference [15]. Discrete divergence-free bases are constructed explicitly for the lowest order weak Galerkin elements in two and three dimensional spaces. These basis functions can be derived on general meshes of arbitrary shape of polygons and polyhedrons. With the divergence-free basis derived, the discrete divergence-free WG scheme can eliminate the pressure variable from the system and reduces a saddle point problem to a symmetric and positive definite system with many fewer unknowns. Numerical results are presented to demonstrate the robustness and accuracy of this discrete divergence-free WG method.

math.NA↗

arXiv2016

Effective Implementation of the Weak Galerkin Finite Element Methods for the Biharmonic Equation

The weak Galerkin (WG) methods have been introduced in the references [11, 16] for solving the biharmonic equation. The purpose of this paper is to develop an algorithm to implement the WG methods effectively. This can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size. In fact, this reduced global system is equivalent to the Schur complements of the WG methods. The unknowns of the Schur complement of the WG method are those defined on the element boundaries. The equivalence of the WG method and its Schur complement is established. The numerical results demonstrate the effectiveness of this new implementation technique.

math.NA↗