arXiv subjects

Ping Lin

Publications and source records attributed to Ping Lin.

At least 37 records · Page 2Linked to original sources

arXiv2022

A fast front-tracking approach and its analysis for a temporal multiscale flow problem with a fractional-order boundary growth

This paper is concerned with a blood flow problem coupled with a slow plaque growth at the artery wall. In the model, the micro (fast) system is the Navier-Stokes equation with a periodically applied force and the macro (slow) system is a fractional reaction equation, which is used to describe the plaque growth with memory effect. We construct an auxiliary temporal periodic problem and an effective time-average equation to approximate the original problem and analyze the approximation error of the corresponding linearized PDE (Stokes) system, where the simple front-tracking technique is used to update the slow moving boundary. An effective multiscale method is then designed based on the approximate problem and the front tracking framework. We also present a temporal finite difference scheme with a spatial continuous finite element method and analyze its temporal discrete error. Furthermore, a fast iterative procedure is designed to find the initial value of the temporal periodic problem and its convergence is analyzed as well. Our designed front-tracking framework and the iterative procedure for solving the temporal periodic problem make it easy to implement the multiscale method on existing PDE solving software. The numerical method is implemented by a combination of the finite element platform COMSOL Multiphysics and the mainstream software MATLAB, which significantly reduce the programming effort and easily handle the fluid-structure interaction, especially moving boundaries with more complex geometries. We present some numerical examples of ODEs and 2-D Navier-Stokes system to demonstrate the effectiveness of the multiscale method. Finally, we have a numerical experiment on the plaque growth problem and discuss the physical implication of the fractional order parameter.

math.NA↗

arXiv2022

Thermodynamically Consistent Diffuse Interface Model for Cell Adhesion and Aggregation

A thermodynamically consistent phase-field model is introduced for simulating multicellular deformation, and aggregation under flow conditions. In particular, a Lennard-Jones type potential is proposed under the phase-field framework for cell-cell, cell-wall interactions. A second-order accurate in both space and time $C^0$ finite element method is proposed to solve the model governing equations. Various numerical tests confirm the convergence, energy stability, and nonlinear mechanical properties of cells of the proposed scheme. Vesicles with different adhesion are also used to explain the pathological risk for patients with sickle cell disease.

math.NA↗

arXiv2022

A temporal multiscale method and its analysis for a system of fractional differential equations

In this paper, a nonlinear system of fractional ordinary differential equations with multiple scales in time is investigated. We are interested in the effective long-term computation of the solution. The main challenge is how to obtain the solution of the coupled problem at a lower computational cost. We analysize a multiscale method for the nonlinear system where the fast system has a periodic applied force and the slow equation contains fractional derivatives as a simplication of the atherosclerosis with a plaque growth. A local periodic equation is derived to approximate the original system and the error estimates are given. Then a finite difference method is designed to approximate the original and the approximate problems. We construct four examples, including three with exact solutions and one following the original problem setting, to test the accuracy and computational efficiency of the proposed method. It is observed that, the computational time is very much reduced and the multiscale method performs very well in comparison to fully resolved simulation for the case of small time scale separation. The larger the time scale separation is, the more effective the multiscale method is.

math.NA↗

arXiv2021

Causal State Feedback Representation for Linear Quadratic Optimal Control Problems of Singular Volterra Integral Equations

This paper is concerned with a linear quadratic optimal control for a class of singular Volterra integral equations. Under proper convexity conditions, optimal control uniquely exists, and it could be characterized via Frechet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions. However, these (equivalent) characterizations have a shortcoming that the current value of the optimal control depends on the future values of the optimal state. Practically, this is not feasible. The main purpose of this paper is to obtain a causal state feedback representation of the optimal control.

math.OC↗

arXiv2021

Feedback controllability for blowup points of heat equation

This paper concerns a controllability problem for blowup points on heat equation. It can be described as follows: In the absence of control, the solution to the linear heat system globally exists in a bounded domain $Ω$. While, for a given time $T>0$ and a point $a$ in this domain, we find a feedback control, which is acted on an internal subset $ω$ of this domain, such that the corresponding solution to this system blows up at time $T$ and holds unique point $a$. We show that $a\in ω$ can be the unique blowup point of the corresponding solution with a certain feedback control, and for any feedback control, $a\in Ω\setminus \overlineω$ could not be the unique blowup point.

math.OC↗

arXiv2021

An astronomical image content-based recommendation system using combined deep learning models in a fully unsupervised mode

We have developed a method that maps large astronomical images onto a two-dimensional map and clusters them. A combination of various state-of-the-art machine learning (ML) algorithms is used to develop a fully unsupervised image quality assessment and clustering system. Our pipeline consists of a data pre-processing step where individual image objects are identified in a large astronomical image and converted to smaller pixel images. This data is then fed to a deep convolutional autoencoder jointly trained with a self-organizing map (SOM). This part can be used as a recommendation system. The resulting output is eventually mapped onto a two-dimensional grid using a second, deep, SOM. We use data taken from ground-based telescopes and, as a case study, compare the system's ability and performance with the results obtained by supervised methods presented by Teimoorinia et al. (2020). The availability of target labels in this data allowed a comprehensive performance comparison between our unsupervised and supervised methods. In addition to image-quality assessments performed in this project, our method can have various other applications. For example, it can help experts label images in a considerably shorter time with minimum human intervention. It can also be used as a content-based recommendation system capable of filtering images based on the desired content.

astro-ph.IM↗

arXiv2020

Automatic Recognition of Landmarks on Digital Dental Models

Fundamental to improving Dental and Orthodontic treatments is the ability to quantitatively assess and cross-compare their outcomes. Such assessments require calculating distances and angles from 3D coordinates of dental landmarks. The costly and repetitive task of hand-labelling dental models impedes studies requiring large sample size to penetrate statistical noise. We have developed techniques and software implementing these techniques to map out automatically, 3D dental scans. This process is divided into consecutive steps - determining a model's orientation, separating and identifying the individual tooth and finding landmarks on each tooth - described in this paper. Examples to demonstrate techniques and the software and discussions on remaining issues are provided as well. The software is originally designed to automate Modified Huddard Bodemham (MHB) landmarking for assessing cleft lip/palate patients. Currently only MHB landmarks are supported, but is extendable to any predetermined landmarks. This software, coupled with intra-oral scanning innovation, should supersede the arduous and error prone plaster model and caliper approach to Dental research and provide a stepping-stone towards automation of routine clinical assessments such as "index of orthodontic treatment need" (IOTN).

math.NA↗

arXiv2020

An Energy Stable C0 Finite Element Scheme for A Phase-Field Model of Vesicle Motion and Deformation

A thermodynamically consistent phase-field model is introduced for simulating motion and shape transformation of vesicles under flow conditions. In particular, a general slip boundary condition is used to describe the interaction between vesicles and the wall of the fluid domain. A second-order accurate in both space and time C0 finite element method is proposed to solve the model governing equations. Various numerical tests confirm the convergence, energy stability, and conservation of mass and surface area of cells of the proposed scheme. Vesicles with different mechanical properties are also used to explain the pathological risk for patients with sickle cell disease.

math.NA↗

arXiv2019

An energy stable $C^0$ finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density

In this paper, we focus on modeling and simulation of two-phase flow with moving contact lines and variable density. A thermodynamically consistent phase-field model with General Navier Boundary Condition is developed based on the concept of quasi-incompressibility and the energy variational method. Then a mass conserving and energy stable C0 finite element scheme is developed to solve the PDE system. Various numerical simulation results show that the proposed schemes are mass conservative, energy stable and the 2nd order for P1 element and 3rd order for P2 element convergence rate in the sense of L2 norm.

math.NA↗

arXiv2019

Generative Adversarial Network with Multi-Branch Discriminator for Cross-Species Image-to-Image Translation

Current approaches have made great progress on image-to-image translation tasks benefiting from the success of image synthesis methods especially generative adversarial networks (GANs). However, existing methods are limited to handling translation tasks between two species while keeping the content matching on the semantic level. A more challenging task would be the translation among more than two species. To explore this new area, we propose a simple yet effective structure of a multi-branch discriminator for enhancing an arbitrary generative adversarial architecture (GAN), named GAN-MBD. It takes advantage of the boosting strategy to break a common discriminator into several smaller ones with fewer parameters, which can enhance the generation and synthesis abilities of GANs efficiently and effectively. Comprehensive experiments show that the proposed multi-branch discriminator can dramatically improve the performance of popular GANs on cross-species image-to-image translation tasks while reducing the number of parameters for computation. The code and some datasets are attached as supplementary materials for reference.

cs.CV↗

arXiv2018

A consistency study of coarse-grained dynamical chains through a nonlinear wave equation of mixed type

A dynamical atomistic chain to simulate mechanical properties of a one-dimensional material with zero temperature may be modelled by the molecular dynamics (MD) model. Because the number of particles (atoms) is huge for a MD model, in practice one often takes a much smaller number of particles to formulate a coarse-grained approximation. We shall mainly consider the consistency of the coarse-grained model with respect to the grain (mesh) size to provide a justification to the goodness of such an approximation. In order to reduce the characteristic oscillations with very different frequencies in such a model, we either add a viscous term to the coarse-grained MD model or apply a space average to the coarse-grained MD solutions for the consistency study. The coarse-grained solution is also compared with the solution of the (macroscopic) continuum model (a nonlinear wave equation of mixed type) to show how well the coarse-grained model can approximate the macroscopic behavior of the material. We also briefly study the instability of the dynamical atomistic chain and the solution of the Riemann problem of the continuum model which may be related to the defect of the atomistic chain under a large deformation in certain locations.

math.NA↗

arXiv2018

A Posteriori Error Estimate and Adaptive Mesh Refinement Algorithm for Atomistic/Continuum Coupling with Finite Range Interactions in Two Dimensions

In this paper, we develop the residual based a posteriori error estimates and the corresponding adaptive mesh refinement algorithm for atomistic/continuum (a/c) coupling with finite range interactions in two dimensions. We have systematically derived a new explicitly computable stress tensor formula for finite range interactions. In particular, we use the geometric reconstruction based consistent atomistic/continuum (GRAC) coupling scheme, which is optimal if the continuum model is discretized by $P^1$ finite elements. The numerical results of the adaptive mesh refinement algorithm is consistent with the optimal a priori error estimates.

math.NA↗

arXiv2018

Multiple solutions and their asymptotics for laminar flows through a porous channel with different permeabilities

The existence and multiplicity of similarity solutions for the steady, incompressible and fully developed laminar flows in a uniformly porous channel with two permeable walls are investigated. We shall focus on the so-called asymmetric case where the upper wall is with an amount of flow injection and the lower wall with a different amount of suction. We show that there exist three solutions designated as type $I$, type $II$ and type $III$ for the asymmetric case. The numerical results suggest that a unique solution exists for the Reynolds number $0\leq R<14.10$ and two additional solutions appear for $R>14.10$. The corresponding asymptotic solution for each of the multiple solutions is constructed by the method of boundary layer correction or matched asymptotic expansion for the most difficult high Reynolds number case. Asymptotic solutions are all verified by their corresponding numerical solutions.

math.CA↗

arXiv2017

Controlled Singular Volterra Integral Equations and Pontryagin Maximum Principle

This paper is concerned with a class of controlled singular Volterra integral equations, which could be used to describe problems involving memories. The well-known fractional order ordinary differential equations of the Riemann--Liouville or Caputo types are strictly special cases of the equations studied in this paper. Well-posedness and some regularity results in proper spaces are established for such kind of questions. For the associated optimal control problem, by using a Liapounoff's type theorem and the spike variation technique, we establish a Pontryagin's type maximum principle for optimal controls. Different from the existing literature, our method enables us to deal with the problem without assuming regularity conditions on the controls, the convexity condition on the control domain, and some additional unnecessary conditions on the nonlinear terms of the integral equation and the cost functional.

math.OC↗

arXiv2017

Size Scaling of Velocity Field in Granular Flows through Apertures

For vertical velocity field $v_{\rm z} (r,z;R)$ of granular flow through an aperture of radius $R$, we propose a size scaling form $v_{\rm z}(r,z;R)=v_{\rm z} (0,0;R)f (r/R_{\rm r}, z/R_{\rm z})$ in the region above the aperture. The length scales $R_{\rm r}=R- 0.5 d$ and $R_{\rm z}=R+k_2 d$, where $k_2$ is a parameter to be determined and $d$ is the diameter of granule. The effective acceleration, which is derived from $v_{\rm z}$, follows also a size scaling form $a_{\rm eff} = v_{\rm z}^2(0,0;R)R_{\rm z}^{-1} θ(r/R_{\rm r}, z/R_{\rm z})$. For granular flow under gravity $g$, there is a boundary condition $a_{\rm eff} (0,0;R)=-g$ which gives rise to $v_{\rm z} (0,0;R)= \sqrt{ λg R_{\rm z}}$ with $λ=-1/θ(0,0)$. Using the size scaling form of vertical velocity field and its boundary condition, we can obtain the flow rate $W =C_2 ρ\sqrt{g } R_{\rm r}^{D-1} R_{\rm z}^{1/2} $, which agrees with the Beverloo law when $R \gg d$. The vertical velocity fields $v_z (r,z;R)$ in three-dimensional (3D) and two-dimensional (2D) hoppers have been simulated using the discrete element method (DEM) and GPU program. Simulation data confirm the size scaling form of $v_{\rm z} (r,z;R)$ and the $R$-dependence of $v_{\rm z} (0,0;R)$.

cond-mat.soft↗

arXiv2017

Mass Conservative and Energy Stable Finite Difference Methods for the Quasi-incompressible Navier-Stokes-Cahn-Hilliard system: Primitive Variable and Projection-Type Schemes

In this paper we describe two fully mass conservative, energy stable, finite difference methods on a staggered grid for the quasi-incompressible Navier-Stokes-Cahn-Hilliard (q-NSCH) system governing a binary incompressible fluid flow with variable density and viscosity. Both methods, namely the primitive method (finite difference method in the primitive variable formulation) and the projection method (finite difference method in a projection-type formulation), are so designed that the mass of the binary fluid is preserved, and the energy of the system equations is always non-increasing in time at the fully discrete level. We also present an efficient, practical nonlinear multigrid method - comprised of a standard FAS method for the Cahn-Hilliard equation, and a method based on the Vanka-type smoothing strategy for the Navier-Stokes equation - for solving these equations. We test the scheme in the context of Capillary Waves, rising droplets and Rayleigh-Taylor instability. Quantitative comparisons are made with existing analytical solutions or previous numerical results that validate the accuracy of our numerical schemes. Moreover, in all cases, mass of the single component and the binary fluid was conserved up to 10 to -8 and energy decreases in time.

math.NA↗

arXiv2017

A Posteriori Error Estimation and Adaptive Algorithm for the Atomistic/Continuum Coupling in 2D

Atomistic/continuum coupling methods aim to achieve optimal balance between accuracy and efficiency. Adaptivity is the key for the efficient implementation of such methods. In this paper, we carry out a rigorous a posteriori analysis of the residual, the stability constant, and the error bound, for a consistent atomistic/continuum coupling method in 2D. We design and implement the corresponding adaptive mesh refinement algorithm, and the convergence rate with respect to degrees of freedom is optimal compare with a priori error estimates.

math.NA↗

arXiv2016

A multiscale mathematical model of tumour invasive growth

Known as one of the hallmarks of cancer [30], cancer cell invasion of human body tissue is a complicated spatio-temporal multiscale process which enables a localised solid tumour to transform into a systemic, metastatic and fatal disease. This process explores and takes advantage of the reciprocal relation that solid tumours establish with the extracellular matrix (ECM) components and other multiple distinct cell types from the surrounding microenvironment. Through the secretion of various proteolytic enzymes such as matrix metalloproteinases (MMP) or the urokinase plasminogen activator (uPA), the cancer cell population alters the configuration of the surrounding ECM composition and overcomes the physical barriers to ultimately achieve local cancer spread into the surrounding tissue. The active interplay between the tissue-scale tumour dynamics and the molecular mechanics of the involved proteolytic enzymes at the cell-scale underlines the biologically multiscale character of invasion, and raises the challenge of modelling this process with an appropriate multiscale approach. In this paper, we present a new two-scale moving boundary model of cancer invasion that explores the tissue scale tumour dynamics in conjunction with the molecular dynamics of the urokinase plasminogen activation system. Building on the multiscale moving boundary method proposed in [58], the modelling that we propose here allows us to study the changes in tissue scale tumour morphology caused by the cell-scale uPA micro-dynamics occurring along the invasive edge of the tumour. Our computational simulation results demonstrate a range of heterogeneous dynamics which are qualitatively similar to the invasive growth patterns observed in a number of different types of cancer, such as the tumour infiltrative growth patterns discussed in [33].

math.DS↗