arXiv subjects

Zhonghua Qiao

Publications and source records attributed to Zhonghua Qiao.

At least 55 records · Page 3Linked to original sources

arXiv2022

Multi-phase image segmentation by the Allen--Cahn Chan--Vese model

This paper proposes an Allen-Cahn Chan-Vese model to settle the multi-phase image segmentation. We first integrate the Allen--Cahn term and the Chan--Vese fitting energy term to establish an energy functional, whose minimum locates the segmentation contour. The subsequent minimization process can be attributed to variational calculation on fitting intensities and the solution approximation of several Allen-Cahn equations, wherein $n$ Allen-Cahn equations are enough to partition $m = 2^n$ segments. The derived Allen-Cahn equations are solved by efficient numerical solvers with exponential time integrations and finite difference space discretization. The discrete maximum bound principle and energy stability of the proposed numerical schemes are proved. Finally, the capability of our segmentation method is verified in various experiments for different types of images.

math.NA↗

arXiv2022

An Active Contour Model with Local Variance Force Term and Its Efficient Minimization Solver for Multi-phase Image Segmentation

In this paper, we propose an active contour model with a local variance force (LVF) term that can be applied to multi-phase image segmentation problems. With the LVF, the proposed model is very effective in the segmentation of images with noise. To solve this model efficiently, we represent the regularization term by characteristic functions and then design a minimization algorithm based on a modification of the iterative convolution-thresholding method (ICTM), namely ICTM-LVF. This minimization algorithm enjoys the energy-decaying property under some conditions and has highly efficient performance in the segmentation. To overcome the initialization issue of active contour models, we generalize the inhomogeneous graph Laplacian initialization method (IGLIM) to the multi-phase case and then apply it to give the initial contour of the ICTM-LVF solver. Numerical experiments are conducted on synthetic images and real images to demonstrate the capability of our initialization method, and the effectiveness of the local variance force for noise robustness in the multi-phase image segmentation.

cs.CV↗

arXiv2022

Stabilized exponential-SAV schemes preserving energy dissipation law and maximum bound principle for the Allen-Cahn type equations

It is well-known that the Allen-Cahn equation not only satisfies the energy dissipation law but also possesses the maximum bound principle (MBP) in the sense that the absolute value of its solution is pointwise bounded for all time by some specific constant under appropriate initial/boundary conditions. In recent years, the scalar auxiliary variable (SAV) method and many of its variants have attracted much attention in numerical solution for gradient flow problems due to their inherent advantage of preserving certain discrete analogues of the energy dissipation law. However, existing SAV schemes usually fail to preserve the MBP when applied to the Allen-Cahn equation. In this paper, we develop and analyze new first- and second-order stabilized exponential-SAV schemes for a class of Allen-Cahn type equations, which are shown to simultaneously preserve the energy dissipation law and MBP in discrete settings. In addition, optimal error estimates for the numerical solutions are rigorously obtained for both schemes. Extensive numerical tests and comparisons are also conducted to demonstrate the performance of the proposed schemes.

math.NA↗

arXiv2022

Generalized SAV-exponential integrator schemes for Allen-Cahn type gradient flows

The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen-Cahn equation. While some commonly-used first-order time stepping schemes have turned out to preserve unconditionally both energy dissipation law and MBP for the equation, restrictions on the time step size are still needed for existing second-order or even higher-order schemes in order to have such simultaneous preservation. In this paper, we develop and analyze novel first- and second-order linear numerical schemes for a class of Allen-Cahn type gradient flows. Our schemes combine the generalized scalar auxiliary variable (SAV) approach and the exponential time integrator with a stabilization term, while the standard central difference stencil is used for discretization of the spatial differential operator. We not only prove their unconditional preservation of the energy dissipation law and the MBP in the discrete setting, but also derive their optimal temporal error estimates under fixed spatial mesh. Numerical experiments are also carried out to demonstrate the properties and performance of the proposed schemes.

math.NA↗

arXiv2022

Strong Approximation of Monotone Stochastic Partial Differential Equations Driven by Multiplicative Noise

We establish a general theory of optimal strong error estimation for numerical approximations of a second-order parabolic stochastic partial differential equation with monotone drift driven by a multiplicative infinite-dimensional Wiener process. The equation is spatially discretized by Galerkin methods and temporally discretized by drift-implicit Euler and Milstein schemes. By the monotone and Lyapunov assumptions, we use both the variational and semigroup approaches to derive a spatial Sobolev regularity under the $L_ω^p L_t^\infty \dot H^{1+γ}$-norm and a temporal Hölder regularity under the $L_ω^p L_x^2$-norm for the solution of the proposed equation with an $\dot H^{1+γ}$-valued initial datum for $γ\in [0,1]$. Then we make full use of the monotonicity of the equation and tools from stochastic calculus to derive the sharp strong convergence rates $O(h^{1+γ}+τ^{1/2})$ and $O(h^{1+γ}+τ^{(1+γ)/2})$ for the Galerkin-based Euler and Milstein schemes, respectively.

math.NA↗

arXiv2021

Two-phase segmentation for intensity inhomogeneous images by the Allen-Cahn Local Binary Fitting Model

This paper proposes a new variational model by integrating the Allen-Cahn term with a local binary fitting energy term for segmenting images with intensity inhomogeneity and noise. An inhomogeneous graph Laplacian initialization method (IGLIM) is developed to give the initial contour for two-phase image segmentation problems. To solve the Allen-Cahn equation derived from the variational model, we adopt the exponential time differencing (ETD) method for temporal discretization, and the central finite difference method for spatial discretization. The energy stability of proposed numerical schemes can be proved. Experiments on various images demonstrate the necessity and superiority of proper initialization and verify the capability of our model for two-phase segmentation of images with intensity inhomogeneity and noise.

math.NA↗

arXiv2021

Two-phase image segmentation by the Allen-Cahn equation and a nonlocal edge detection operator

Based on a nonlocal Laplacian operator, a novel edge detection method of the grayscale image is proposed in this paper. This operator utilizes the information of neighbor pixels for a given pixel to obtain effective and delicate edge detection. The nonlocal edge detection method is used as an initialization for solving the Allen-Cahn equation to achieve two-phase segmentation of the grayscale image. Efficient exponential time differencing (ETD) solvers are employed in the time integration, and finite difference method is adopted in space discretization. The maximum bound principle and energy stability of the proposed numerical schemes are proved. The capability of our segmentation method has been verified in numerical experiments for different types of grayscale images.

math.NA↗

arXiv2020

Maximum bound principle preserving integrating factor Runge-Kutta methods for semilinear parabolic equations

A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. Investigation on numerical schemes of these equations with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge-Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP-preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen-Cahn type of equations. In addition, error estimates for these numerical schemes are proved theoretically and verified numerically, as well as their efficiency by simulations of long-time evolutional behavior.

math.NA↗

arXiv2020

Maximum bound principles for a class of semilinear parabolic equations and exponential time differencing schemes

The ubiquity of semilinear parabolic equations has been illustrated in their numerous applications ranging from physics, biology, to materials and social sciences. In this paper, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form $u_t=\mathcal{L}u+f[u]$ with $\mathcal{L}$ being a linear dissipative operator and $f$ being a nonlinear operator in space, namely a time-invariant maximum bound principle, in the sense that the time-dependent solution $u$ preserves for all time a uniform pointwise bound in absolute value imposed by its initial and boundary conditions. We first study an analytical framework for some sufficient conditions on $\mathcal{L}$ and $f$ that lead to such a maximum bound principle for the time-continuous dynamic system of infinite or finite dimensions. Then, we utilize a suitable exponential time differencing approach with a properly chosen generator of contraction semigroup to develop first- and second-order accurate temporal discretization schemes, that satisfy the maximum bound principle unconditionally in the time-discrete setting. Error estimates of the proposed schemes are derived along with their energy stability. Extensions to vector- and matrix-valued systems are also discussed. We demonstrate that the abstract framework and analysis techniques developed here offer an effective and unified approach to study the maximum bound principle of the abstract evolution equation that cover a wide variety of well-known models and their numerical discretization schemes. Some numerical experiments are also carried out to verify the theoretical results.

math.NA↗

arXiv2020

Strong Approximation of Stochastic Allen-Cahn Equation with White Noise

We establish an optimal strong convergence rate of a fully discrete numerical scheme for second order parabolic stochastic partial differential equations with monotone drifts, including the stochastic Allen-Cahn equation, driven by an additive space-time white noise. Our first step is to transform the original stochastic equation into an equivalent random equation whose solution possesses more regularity than the original one. Then we use the backward Euler in time and spectral Galerkin in space to fully discretize this random equation. By the monotone assumption, in combination with the factorization method and stochastic calculus in martingale-type 2 Banach spaces, we derive a uniform maximum norm estimation and a Hölder-type regularity for both stochastic and random equations. Finally, the strong convergence rate of the proposed fully discrete scheme under the $l_t^\infty L^2_ωL^2_x \cap l_t^q L^q_ωL^q_x$-norm is obtained. Several numerical experiments are carried out to verify the theoretical result.

math.NA↗

arXiv2020

Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn-Hilliard equation

In this paper, we provide a detailed convergence analysis for a first order stabilized linear semi-implicit numerical scheme for the nonlocal Cahn-Hilliard equation, which follows from consistency and stability estimates for the numerical error function. Due to the complicated form of the nonlinear term, we adopt the discrete $H^{-1}$ norm for the error function to establish the convergence result. In addition, the energy stability obtained in [Du et al., J. Comput. Phys., 363:39--54, 2018] requires an assumption on the uniform $\ell^\infty$ bound of the numerical solution and such a bound is figured out in this paper by conducting the higher order consistency analysis. Taking the view that the numerical solution is indeed the exact solution with a perturbation, the error function is $\ell^\infty$ bounded uniformly under a loose constraint of the time step size, which then leads to the uniform maximum-norm bound of the numerical solution.

math.NA↗

arXiv2019

A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability

In this paper we propose and analyze a (temporally) third order accurate exponential time differencing (ETD) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. A linear splitting is applied to the physical model, and an ETD-based multistep approximation is used for time integration of the corresponding equation. In addition, a third order accurate Douglas-Dupont regularization term, in the form of $-A \dt^2 ϕ_0 (L_N) Δ_N^2 ( u^{n+1} - u^n)$, is added in the numerical scheme. A careful Fourier eigenvalue analysis results in the energy stability in a modified version, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the optimal rate convergence analysis and error estimate are derived in details, in the $\ell^\infty (0,T; H_h^1) \cap \ell^2 (0,T; H_h^3)$ norm, with the help of a careful eigenvalue bound estimate, combined with the nonlinear analysis for the NSS model. This convergence estimate is the first such result for a third order accurate scheme for a gradient flow. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for $\varepsilon=0.02$ (up to $T=3 \times 10^5$) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

math.NA↗

arXiv2019

Maximum principle preserving exponential time differencing schemes for the nonlocal Allen-Cahn equation

The nonlocal Allen-Cahn (NAC) equation is a generalization of the classic Allen-Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, and satisfies the maximum principle as its local counterpart. In this paper, we develop and analyze first and second order exponential time differencing (ETD) schemes for solving the NAC equation, which unconditionally preserve the discrete maximum principle. The fully discrete numerical schemes are obtained by applying the stabilized ETD approximations for time integration with the quadrature-based finite difference discretization in space. We derive their respective optimal maximum-norm error estimates and further show that the proposed schemes are asymptotically compatible, i.e., the approximate solutions always converge to the classic Allen-Cahn solution when the horizon, the spatial mesh size and the time step size go to zero. We also prove that the schemes are energy stable in the discrete sense. Various experiments are performed to verify these theoretical results and to investigate numerically the relation between the discontinuities and the nonlocal parameters.

math.NA↗

arXiv2018

Wong--Zakai Approximations of Stochastic Allen-Cahn Equation

We establish a unconditional and optimal strong convergence rate of Wong--Zakai type approximations in Banach space norm for a parabolic stochastic partial differential equation with monotone drift, including the stochastic Allen--Cahn equation, driven by an additive Brownian sheet. The key ingredient in the analysis is the fully use of additive nature of the noise and monotonicity of the drift to derive a priori estimation for the solution of this equation, in combination with the factorization method and stochastic calculus in martingale type 2 Banach spaces applied to deduce sharp error estimation between the exact and approximate Ornstein--Uhlenbeck processes, in Banach space norm.

math.PR↗

arXiv2017

Resolving Knudsen Layer by High Order Moment Expansion

We model the Knudsen layer in Kramers' problem by linearized high order hyperbolic moment system. Due to the hyperbolicity, the boundary conditions of the moment system is properly reduced from the kinetic boundary condition. For Kramers' problem, we give the analytical solutions of moment systems. With the order increasing of the moment model, the solutions are approaching to the solution of the linearized BGK kinetic equation. The velocity profile in the Knudsen layer is captured with improved accuracy for a wide range of accommodation coefficients.

math.NA↗

arXiv2016

Acceleration for Microflow Simulations of High-Order Moment Models by Using Lower-Order Model Correction

We study the acceleration of steady-state computation for microflow, which is modeled by the high-order moment models derived recently from the steady-state Boltzmann equation with BGK-type collision term. By using the lower-order model correction, a novel nonlinear multi-level moment solver is developed. Numerical examples verify that the resulting solver improves the convergence significantly thus is able to accelerate the steady-state computation greatly. The behavior of the solver is also numerically investigated. It is shown that the convergence rate increases, indicating the solver would be more efficient, as the total levels increases. Three order reduction strategies of the solver are considered. Numerical results show that the most efficient order reduction strategy would be $m_{l-1} = \lceil m_{l} / 2 \rceil$.

math.NA↗

arXiv2016

Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations

Recent results in the literature provide computational evidence that stabilized semi-implicit time-stepping method can efficiently simulate phase field problems involving fourth-order nonlinear dif- fusion, with typical examples like the Cahn-Hilliard equation and the thin film type equation. The up-to-date theoretical explanation of the numerical stability relies on the assumption that the deriva- tive of the nonlinear potential function satisfies a Lipschitz type condition, which in a rigorous sense, implies the boundedness of the numerical solution. In this work we remove the Lipschitz assumption on the nonlinearity and prove unconditional energy stability for the stabilized semi-implicit time-stepping methods. It is shown that the size of stabilization term depends on the initial energy and the perturba- tion parameter but is independent of the time step. The corresponding error analysis is also established under minimal nonlinearity and regularity assumptions.

math.NA↗

arXiv2016

Gradient bounds for a thin film epitaxy equation

We consider a gradient flow modeling the epitaxial growth of thin films with slope selection. The surface height profile satisfies a nonlinear diffusion equation with biharmonic dissipation. We establish optimal local and global wellposedness for initial data with critical regularity. To understand the mechanism of slope selection and the dependence on the dissipation coefficient, we exhibit several lower and upper bounds for the gradient of the solution in physical dimensions $d\le 3$.

math.AP↗