arXiv subjects

Jia Zhao

Publications and source records attributed to Jia Zhao.

At least 37 records · Page 2Linked to original sources

arXiv2021

Improving the Accuracy and Consistency of the Scalar Auxiliary Variable (SAV) Method with Relaxation

The scalar auxiliary variable (SAV) method was introduced by Shen et al. and has been broadly used to solve thermodynamically consistent PDE problems. By utilizing scalar auxiliary variables, the original PDE problems are reformulated into equivalent PDE problems. The advantages of the SAV approach, such as linearity, unconditionally energy stability, and easy-to-implement, are prevalent. However, there is still an open issue unresolved, i.e., the numerical schemes resulted from the SAV method preserve a "modified" energy law according to the auxiliary variables instead of the original variables. Truncation errors are introduced during numerical calculations so that the numerical solutions of the auxiliary variables are no longer equivalent to their original continuous definitions. In other words, even though the SAV scheme satisfies a modified energy law, it does not necessarily satisfy the energy law of the original PDE models. This paper presents one essential relaxation technique to overcome this issue, which we named the relaxed-SAV (RSAV) method. Our RSAV method penalizes the numerical errors of the auxiliary variables by a relaxation technique. In general, the RSAV method keeps all the advantages of the baseline SAV method and improves its accuracy and consistency noticeably. Several examples have been presented to demonstrate the effectiveness of the RSAV approach.

math.NA↗

arXiv2021

A Revisit of The Energy Quadratization Method with A Relaxation Technique

This letter revisits the energy quadratization (EQ) method by introducing a novel and essential relaxation technique to improve its accuracy and stability. The EQ method has witnessed significant popularity in the past few years. Though acknowledging its effectiveness in designing energy-stable schemes for thermodynamically consistent models, the primary known drawback is apparent, i.e., its preserves a "modified" energy law represented by auxiliary variables instead of the original variables. Truncation errors are introduced during numerical calculations so that the numerical solutions of the auxiliary variables are no longer equivalent to their original continuous definitions. Even though the "modified" energy dissipation law is preserved, the original energy dissipation law is not guaranteed. In this paper, we overcome this issue by introducing a relaxation technique. The computational cost of this extra technique is negligible compared with the baseline EQ method. Meanwhile, the relaxed-EQ method holds all the baseline EQ method's good properties, such as linearity and unconditionally energy stability. Then we apply the relaxed-EQ method to several widely-used phase field models to highlight its effectiveness.

math.NA↗

arXiv2021

A General Framework to Derive Linear, Decoupled and Energy-stable Schemes for Reversible-Irreversible Thermodynamically Consistent Models: Part I Incompressible Hydrodynamic Models

In this paper, we present a general numerical platform for designing accurate, efficient, and stable numerical algorithms for incompressible hydrodynamic models that obeys the thermodynamical laws. The obtained numerical schemes are automatically linear in time. It decouples the hydrodynamic variable and other state variables such that only small-size linear problems need to be solved at each time marching step. Furthermore, if the classical velocity projection method is utilized, the velocity field and pressure field can be decoupled. In the end, only a few elliptic-type equations shall be solved in each time step. This strategy is made possible through a sequence of model reformulations by fully exploring the models' thermodynamic structures. The generalized Onsager principle directly guides these reformulation procedures. In the reformulated but equivalent models, the reversible and irreversible components can be identified, guiding the numerical platform to decouple the reversible and irreversible dynamics. This eventually leads to decoupled numerical algorithms, given that the coupling terms only involve irreversible dynamics. To further demonstrate the numerical platform's power, we apply it to several specific incompressible hydrodynamic models. The energy stability of the proposed numerical schemes is shown in detail. The second-order accuracy in time is verified numerically through time step refinement tests. Several benchmark numerical examples are presented to further illustrate the proposed numerical framework's accuracy, stability, and efficiency.

math.NA↗

arXiv2021

Second-order Decoupled Energy-stable Schemes for Cahn-Hilliard-Navier-Stokes equations

The Cahn-Hilliard-Navier-Stokes (CHNS) equations represent the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Due to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation, the CHNS system is non-trivial to solve numerically. Traditionally, a numerical extrapolation for the coupling terms is used. However, such brute-force extrapolation usually destroys the intrinsic thermodynamic structures of this CHNS system. This paper proposes a new strategy to reformulate the CHNS system into a constraint gradient flow formation. Under the new formulation, the reversible and irreversible structures are clearly revealed. This guides us to propose operator splitting schemes. The operator splitting schemes have several advantageous properties. First of all, the proposed schemes lead to several decoupled systems in smaller sizes to be solved at each time marching step. This significantly reduces computational costs. Secondly, the proposed schemes still guarantee the thermodynamic laws of the CHNS system at the discrete level. It ensures the thermodynamic laws, accuracy, and stability for the numerical solutions. In addition, unlike the recently populated IEQ or SAV approach using auxiliary variables, our resulting energy laws are formulated in the original variables. Our proposed framework lays out a foundation to design decoupled and energy stable numerical algorithms for hydrodynamic phase-field models. Furthermore, given different splitting steps, various numerical algorithms can be obtained, making this framework rather general. The proposed numerical algorithms are implemented. Their second-order accuracy in time is verified numerically. Some numerical examples and benchmark problems are calculated to verify the effectiveness of the proposed schemes.

math.NA↗

arXiv2021

Eigenvalues of Laplace operators on non-bipartite graphs

This paper considers the comparison between the eigenvalues of Laplace operators with the standard conditions and the anti-standard conditions on non-bipartite graphs which are equilateral or inequilateral. First of all, we show the calculation of the eigenvalues of Laplace operators on equilateral metric graphs with arbitrary edge length. Based on this method, we use the properties of the cosine function and the arccosine function to find the comparison between the eigenvalues of Laplace operators with the standard conditions and the anti-standard conditions on equilateral non-bipartite graphs. In addition, we give the inequalities between standard and anti-standard eigenvalues on a special inequilateral non-bipartite graph.

math.SP↗

arXiv2020

Metasurface tessellation for reconfigurable electromagnetic scattering

Metasurfaces have attracted significant research interest owing to their unprecedented control over the spatial distributions of electromagnetic fields. Herein we propose the concept of metasurface tessellation to achieve reconfigurable scattering functions. Square meta-tiles, composed of identical structures, are arranged to fill a surface. The electromagnetic scattering of the tiled surface is determined by the orientation distribution of the meta-tiles. We present three typical cases of meta-tiles consisting of binary elements to realize several distinct scattering patterns. This study provides an alternative method to build reconfigurable and multi-functional metasurface devices without external stimuli and complicated fabrication.

physics.class-ph↗

arXiv2020

The effects of simple density-dependent prey diffusion and refuge in a predator-prey system

We study a spatial (two-dimensional) Rosenzweig-MacArthur model under the following assumptions: $(1)$ prey movement follows a nonlinear diffusion, $(2)$ preys have a refuge zone (sometimes called "protection zone") where predators cannot enter, (3) predators move following linear diffusion. We present a bifurcation analysis for the system that shows the existence of positive solutions at the steady state. We complement the theoretical results with numerical computations and compare our results with those obtained in the case of having linear diffusion for the prey movement. Our results show that both models, with linear and nonlinear diffusion for the prey, have the same bifurcation point and the positive solution curves are virtually the same in a neighborhood of this point, but they get drastically different as the bifurcation parameter approaches to zero.

math.AP↗

arXiv2020

Discovery of Governing Equations with Recursive Deep Neural Networks

Model discovery based on existing data has been one of the major focuses of mathematical modelers for decades. Despite tremendous achievements of model identification from adequate data, how to unravel the models from limited data is less resolved. In this paper, we focus on the model discovery problem when the data is not efficiently sampled in time. This is common due to limited experimental accessibility and labor/resource constraints. Specifically, we introduce a recursive deep neural network (RDNN) for data-driven model discovery. This recursive approach can retrieve the governing equation in a simple and efficient manner, and it can significantly improve the approximation accuracy by increasing the recursive stages. In particular, our proposed approach shows superior power when the existing data are sampled with a large time lag, from which the traditional approach might not be able to recover the model well. Several widely used examples of dynamical systems are used to benchmark this newly proposed recursive approach. Numerical comparisons confirm the effectiveness of this recursive neural network for model discovery.

math.NA↗

arXiv2020

Discovering Phase Field Models from Image Data with the Pseudo-spectral Physics Informed Neural Networks

In this paper, we introduce a new deep learning framework for discovering the phase field models from existing image data. The new framework embraces the approximation power of physics informed neural networks (PINN), and the computational efficiency of the pseudo-spectral methods, which we named pseudo-spectral PINN or SPINN. Unlike the baseline PINN, the pseudo-spectral PINN has several advantages. First of all, it requires less training data. A minimum of two snapshots with uniform spatial resolution would be adequate. Secondly, it is computationally efficient, as the pseudo-spectral method is used for spatial discretization. Thirdly, it requires less trainable parameters compared with the baseline PINN. Thus, it significantly simplifies the training process and assures less local minima or saddle points. We illustrate the effectiveness of pseudo-spectral PINN through several numerical examples. The newly proposed pseudo-spectral PINN is rather general, and it can be readily applied to discover other PDE-based models from image data.

math.NA↗

arXiv2020

Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks

Phase field models, in particular, the Allen-Cahn type and Cahn-Hilliard type equations, have been widely used to investigate interfacial dynamic problems. Designing accurate, efficient, and stable numerical algorithms for solving the phase field models has been an active field for decades. In this paper, we focus on using the deep neural network to design an automatic numerical solver for the Allen-Cahn and Cahn-Hilliard equations by proposing an improved physics informed neural network (PINN). Though the PINN has been embraced to investigate many differential equation problems, we find a direct application of the PINN in solving phase-field equations won't provide accurate solutions in many cases. Thus, we propose various techniques that add to the approximation power of the PINN. As a major contribution of this paper, we propose to embrace the adaptive idea in both space and time and introduce various sampling strategies, such that we are able to improve the efficiency and accuracy of the PINN on solving phase field equations. In addition, the improved PINN has no restriction on the explicit form of the PDEs, making it applicable to a wider class of PDE problems, and shedding light on numerical approximations of other PDEs in general.

math.NA↗

arXiv2020

A Non-uniform Time-stepping Convex Splitting Scheme for the Time-fractional Cahn-Hilliard Equation

In this paper, a non-uniform time-stepping convex-splitting numerical algorithm for solving the widely used time-fractional Cahn-Hilliard equation is introduced. The proposed numerical scheme employs the $L1^+$ formula for discretizing the time-fractional derivative and a second-order convex-splitting technique to deal with the non-linear term semi-implicitly. Then the pseudospectral method is utilized for spatial discretization. As a result, the fully discrete scheme has several advantages: second-order accurate in time, spectrally accurate in space, uniquely solvable, mass preserving, and unconditionally energy stable. Rigorous proofs are given, along with several numerical results to verify the theoretical results, and to show the accuracy and effectiveness of the proposed scheme. Also, some interesting phase separation dynamics of the time-fractional Cahn-Hilliard equation has been investigated.

math.NA↗

arXiv2019

Arbitrarily High-order Linear Schemes for Gradient Flow Models

We present a paradigm for developing arbitrarily high order, linear, unconditionally energy stable numerical algorithms for gradient flow models. We apply the energy quadratization (EQ) technique to reformulate the general gradient flow model into an equivalent gradient flow model with a quadratic free energy and a modified mobility. Given solutions up to $t_n=n Δt$ with $Δt$ the time step size, we linearize the EQ-reformulated gradient flow model in $(t_n, t_{n+1}]$ by extrapolation. Then we employ an algebraically stable Runge-Kutta method to discretize the linearized model in $(t_n, t_{n+1}]$. Then we use the Fourier pseudo-spectral method for the spatial discretization to match the order of accuracy in time. The resulting fully discrete scheme is linear, unconditionally energy stable, uniquely solvable, and may reach arbitrarily high order. Furthermore, we present a family of linear schemes based on prediction-correction methods to complement the new linear schemes. Some benchmark numerical examples are given to demonstrate the accuracy and efficiency of the schemes.

math.NA↗

arXiv2019

Arbitrarily High-order Unconditionally Energy Stable Schemes for Thermodynamically Consistent Gradient Flow Models

We present a systematical approach to developing arbitrarily high order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization (EQ) method, We formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic energy, arriving at an arbitrarily high-order scheme for gradient flow models. Schemes derived using both approaches are proved rigorously to be unconditionally energy stable. The proposed schemes are then implemented in four gradient flow models numerically to demonstrate their accuracy and effectiveness. Detailed numerical comparisons among these schemes are carried out as well. These numerical strategies are rather general so that they can be readily generalized to solve any thermodynamically consistent PDE models.

math.NA↗

arXiv2019

Arbitrarily High-order Unconditionally Energy Stable Schemes for Gradient Flow Models Using the Scalar Auxiliary Variable Approach

In this paper, we propose a novel family of high-order numerical schemes for the gradient flow models based on the scalar auxiliary variable (SAV) approach, which is named the high-order scalar auxiliary variable (HSAV) method. The newly proposed schemes could be shown to reach arbitrarily high order in time while preserving the energy dissipation law without any restriction on the time step size (i.e., unconditionally energy stable). The HSAV strategy is rather general that it does not depend on the specific expression of the effective free energy, such that it applies to a class of thermodynamically consistent gradient flow models arriving at semi-discrete high-order energy-stable schemes. We then employ the Fourier pseudospectral method for spatial discretization. The fully discrete schemes are also shown to be unconditionally energy stable. Furthermore, we present several numerical experiments on several widely-used gradient flow models, to demonstrate the accuracy, efficiency and unconditionally energy stability of the HSAV schemes. The numerical results verify that the HSAV schemes can reach the expected order of accuracy, and it allows a much larger time step size to reach the same accuracy than the standard SAV schemes.

math.NA↗

arXiv2019

Analyticity of the spectrum and Dirichlet-to-Neumann operator technique for quantum graphs

In some previous works, the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph was established. However, a specific local coordinate chart on the Grassmanian of all possible vertex conditions was used, thus creating an erroneous impression that something ``wrong'' can happen at the boundaries of the chart. Here we show that the analyticity of the corresponding ``dispersion relation'' holds over the whole Grassmannian, as well as over other parameter spaces. We also address the Dirichlet-to-Neumann (DtN) technique of relating quantum and discrete graph operators, which allows one to transfer some results from the discrete to the quantum graph case, but which has issues at the Dirichlet spectrum. We conclude that this difficulty, as in the first part of the paper, stems from the use of specific coordinates in a Grassmannian and show how to avoid it to extend some of the consequent results to the general situation.

math-ph↗

arXiv2019

Bayesian Weighted Mendelian Randomization for Causal Inference based on Summary Statistics

The results from Genome-Wide Association Studies (GWAS) on thousands of phenotypes provide an unprecedented opportunity to infer the causal effect of one phenotype (exposure) on another (outcome). Mendelian randomization (MR), an instrumental variable (IV) method, has been introduced for causal inference using GWAS data. Due to the polygenic architecture of complex traits/diseases and the ubiquity of pleiotropy, however, MR has many unique challenges compared to conventional IV methods. We propose a Bayesian weighted Mendelian randomization (BWMR) for causal inference to address these challenges. In our BWMR model, the uncertainty of weak effects owing to polygenicity has been taken into account and the violation of IV assumption due to pleiotropy has been addressed through outlier detection by Bayesian weighting. To make the causal inference based on BWMR computationally stable and efficient, we developed a variational expectation-maximization (VEM) algorithm. Moreover, we have also derived an exact closed-form formula to correct the posterior covariance which is often underestimated in variational inference. Through comprehensive simulation studies, we evaluated the performance of BWMR, demonstrating the advantage of BWMR over its competitors. Then we applied BWMR to make causal inference between 130 metabolites and 93 complex human traits, uncovering novel causal relationship between exposure and outcome traits. The BWMR software is available at https://github.com/jiazhao97/BWMR.

stat.ME↗

arXiv2018

On Power Law Scaling Dynamics for Time-fractional Phase Field Models during Coarsening

In this paper, we study the phase field models with fractional-order in time. The phase field models have been widely used to study coarsening dynamics of material systems with microstructures. It is known that phase field models are usually derived from energy variation so that they obey some energy dissipation laws intrinsically. Recently, many works have been published on investigating fractional-order phase field models, but little is known of the corresponding energy dissipation laws. We focus on the time-fractional phase field models and report that the effective free energy and roughness obey a universal power-law scaling dynamics during coarsening. Mainly, the effective free energy and roughness in the time-fractional phase field models scale by following a similar power law as the integer phase field models, where the power is linearly proportional to the fractional order. This universal scaling law is verified numerically against several phase field models, including the Cahn-Hilliard equations with different variable mobilities and molecular beam epitaxy models. This new finding sheds light on potential applications of time fractional phase field models in studying coarsening dynamics and crystal growths.

math.NA↗

arXiv2018

An Accurate and Efficient Algorithm for The Time-fractional Molecular Beam Epitaxy Model with Slope Selection

In this paper, we propose a time-fractional molecular beam epitaxy (MBE) model with slope selection and its efficient, accurate, full discrete, linear numerical approximation. The numerical scheme utilizes the fast algorithm for the Caputo fractional derivative operator in time discretization and Fourier spectral method in spatial discretization. Refinement tests are conducted to verify the $2-α$ order of time convergence, with $α\in (0, 1]$ the fractional order of derivative. Several numerical simulations are presented to demonstrate the accuracy and efficiency of our newly proposed scheme. By exploring the fast algorithm calculating the Caputo fractional derivative, our numerical scheme makes it practice for long time simulation of MBE coarsening, which is essential for MBE model in practice. With the proposed fractional MBE model, we observe that the scaling law for the energy decays as $ O(t^{-\fracα{3}})$ and the roughness increases as $O(t^{\fracα{3}})$, during the coarsening dynamics with random initial condition. That is to say, the coarsening rate of MBE model could be manipulated by the fractional order $α$, and it is linearly proportional to $α$. This is the first time in literature to report/discover such scaling correlation. It provides a potential application field for fractional differential equations. Besides, the numerical approximation strategy proposed in this paper can be readily applied to study many classes of time-fractional and high dimensional phase field models.

math.NA↗