Auxiliary Variable Technique Research Articles

An efficient unconditional energy stable scheme for the simulation of droplet formation

We have developed an efficient and unconditionally energy-stable method for simulating droplet formation dynamics. Our approach involves a novel time-marching scheme based on the scalar auxiliary variable technique, specifically designed for solving the Cahn-Hilliard-Navier-Stokes phase field model with variable density and viscosity. We have successfully applied this method to simulate droplet formation in scenarios where a Newtonian fluid is injected through a vertical tube into another immiscible Newtonian fluid. To tackle the challenges posed by nonhomogeneous Dirichlet boundary conditions at the tube entrance, we have introduced additional nonlocal auxiliary variables and associated ordinary differential equations. These additions effectively eliminate the influence of boundary terms. Moreover, we have incorporated stabilization terms into the scheme to enhance its numerical effectiveness. Notably, our resulting scheme is fully decoupled, requiring the solution of only linear systems at each time step. We have also demonstrated the energy decaying property of the scheme, with suitable modifications. To assess the accuracy and stability of our algorithm, we have conducted extensive numerical simulations. Additionally, we have examined the dynamics of droplet formation and explored the impact of dimensionless parameters on the process. Overall, our work presents a refined method for simulating droplet formation dynamics, offering improved efficiency, energy stability, and accuracy.

Asymptotically compatible energy of two variable-step fractional BDF2 schemes for the time fractional Allen–Cahn model

In this paper, the variable-step fractional BDF2 (FBDF2) formula is applied to solve the time fractional Allen–Cahn (TFAC) model. By the discrete gradient structure of the FBDF2 operator and the local–nonlocal splitting technique, the asymptotically compatible energy of the nonlinear fully implicit FBDF2 scheme and the linearly FBDF2-SAV scheme by the scalar auxiliary variable technique are established for the TFAC model. Finally, several numerical experiments are provided to verify the theoretical results and compare the effectiveness of two numerical schemes.

Energy conserving discontinuous Galerkin method with scalar auxiliary variable technique for the nonlinear Dirac equation

In this paper, we propose a fully-discrete energy-conserving scheme for the nonlinear Dirac equation, by combining the scalar auxiliary variable (SAV) technique with discontinuous Galerkin (DG) discretization. We start by discussing the semi-discrete DG discretization, and show that, with suitable choices of numerical fluxes, the resulting method conserves the charge, energy exactly and preserves the multi-symplectic structure. The optimal error estimate of semi-discrete DG scheme is carried out. We combine it with the energy conserving SAV technique, and demonstrate that the fully-discrete scheme conserves the discrete global energy exactly. Both second order SAV method based on the midpoint rule and its high order extension have been studied. The proposed methods have been tested on some numerical experiments, which confirm the optimal rates of convergence and the energy conserving property. Numerical comparison with energy dissipative DG method is also provided to demonstrate that the numerical error of energy conserving method does not grow significantly in long time simulations.

Blind Image Deblurring via a Novel Sparse Channel Prior

Blind image deblurring (BID) is a long-standing challenging problem in low-level image processing. To achieve visually pleasing results, it is of utmost importance to select good image priors. In this work, we develop the ratio of the dark channel prior (DCP) to the bright channel prior (BCP) as an image prior for solving the BID problem. Specifically, the above two channel priors obtained from RGB images are used to construct an innovative sparse channel prior at first, and then the learned prior is incorporated into the BID tasks. The proposed sparse channel prior enhances the sparsity of the DCP. At the same time, it also shows the inverse relationship between the DCP and BCP. We employ the auxiliary variable technique to integrate the proposed sparse prior information into the iterative restoration procedure. Extensive experiments on real and synthetic blurry sets show that the proposed algorithm is efficient and competitive compared with the state-of-the-art methods and that the proposed sparse channel prior for blind deblurring is effective.

Transformed denoising autoencoder prior for image restoration

Image restoration problem is generally ill-posed, which can be alleviated by learning image prior. Inspired by the considerable performance of utilizing priors in pixel domain and wavelet domain jointly, we propose a novel transformed denoising autoencoder as prior (TDAEP). The core idea behind TDAEP is to enhance the classical denoising autoencoder (DAE) via transform domain, which captures complementary information from multiple views. Specifically, 1-level nonorthogonal wavelet coefficients are used to form 4-channel feature images. Moreover, a 5-channel tensor is obtained by stacking the original image under the pixel domain and 4-channel feature images under the wavelet domain. Then we train the transformed DAE (TDAE) with the 5-channel tensor as the network input. The optimized image prior is obtained based on the trained autoencoder, and it is incorporated into an iterative restoration procedure with the aid of the auxiliary variable technique. The resulting model is affiliationed by proximal gradient descent technique. Numerous experiments demonstrated that the TDAEP outperforms a set of image restoration benchmark algorithms.

Linearly Implicit and High-Order Energy-Conserving Schemes for Nonlinear Wave Equations

A key issue in developing efficient numerical schemes for nonlinear wave equations is the energy-conserving. Most existing schemes of the energy-conserving are fully implicit and the schemes require an extra iteration at each time step and considerable computational cost for a long time simulation, while the widely-used q-stage (implicit) Gauss scheme (method) only preserves polynomial Hamiltonians up to degree 2q. In this paper, we present a family of linearly implicit and high-order energy-conserving schemes for solving nonlinear wave equations. The construction of schemes is based on recently-developed scalar auxiliary variable technique with a combination of classical high-order Gauss methods and extrapolation approximation. We prove that the proposed schemes are unconditionally energy-conserved for a general nonlinear wave equation. Numerical results are given to show the energy-conserving and the effectiveness of schemes.

Uncertainty decomposition in climate-change impact assessments: a Bayesian perspective

A climate-impact projection usually consists of several stages, and the uncertainty of the projection is known to be quite large. It is necessary to assess how much each stage contributed to the uncertainty. We call an uncertainty quantification method in which relative contribution of each stage can be evaluated as uncertainty decomposition. We propose a new Bayesian model for uncertainty decomposition in climate change impact assessments. The proposed Bayesian model can incorporate uncertainty of natural variability and utilize data in control period. We provide a simple and efficient Gibbs sampling algorithm using the auxiliary variable technique. We compare the proposed method with other existing uncertainty decomposition methods by analyzing streamflow data for Yongdam Dam basin located at Geum River in South Korea.

A stable high-order absorbing boundary based on continued fraction for scalar wave propagation in 2D and 3D unbounded layers

PurposeThe purpose of this paper is to propose a stable high-order absorbing boundary condition (ABC) based on new continued fraction for scalar wave propagation in 2D and 3D unbounded layers.Design/methodology/approachThe ABC is obtained based on continued fraction (CF) expansion of the frequency-domain dynamic stiffness coefficient (DtN kernel) on the artificial boundary of a truncated infinite domain. The CF which has been used to the thin layer method in [69] will be applied to the DtN method to develop a time-domain high-order ABC for the transient scalar wave propagation in 2D. Furthermore, a new stable composite-CF is proposed in this study for 3D unbounded layers by nesting the above CF for 2D layer and another CF.FindingsThe ABS has been transformed from frequency to time domain by using the auxiliary variable technique. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the ABC-FEM coupled system is discussed and cured.Originality/valueThis manuscript establishes a stable high-order time-domain ABC for the scalar wave equation in 2D and 3D unbounded layers, which is based on the new continued fraction. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the coupled system is discussed and cured.

Multi-Channel and Multi-Model-Based Autoencoding Prior for Grayscale Image Restoration.

Image restoration (IR) is a long-standing challenging problem in low-level image processing. It is of utmost importance to learn good image priors for pursuing visually pleasing results. In this paper, we develop a multi-channel and multi-model-based denoising autoencoder network as image prior for solving IR problem. Specifically, the network that trained on RGB-channel images is used to construct a prior at first, and then the learned prior is incorporated into single-channel grayscale IR tasks. To achieve the goal, we employ the auxiliary variable technique to integrate the higher-dimensional network-driven prior information into the iterative restoration procedure. In addition, according to the weighted aggregation idea, a multi-model strategy is put forward to enhance the network stability that favors to avoid getting trapped in local optima. Extensive experiments on image deblurring and deblocking tasks show that the proposed algorithm is efficient, robust, and yields state-of-the-art restoration quality on grayscale images.

Time-Domain Stability of Artificial Boundary Condition Coupled with Finite Element for Dynamic and Wave Problems in Unbounded Media

The finite element modeling of the dynamic and wave problems in unbounded media requires an artificial boundary condition to simulate the truncated infinite domain. The Dirichlet-to-Neumann boundary condition has been transformed from frequency to time domain by using the rational function approximation and auxiliary variable technique. It is extended to three-dimensional layer problem here. The resulting artificial boundary condition is stable itself in time domain, whereas the time-domain instability of the artificial boundary condition coupled with the finite element method is found for the foundation vibration recently and for the wave propagation here. A simple and effective method that introduces the damping proportional to the stiffness matrix in the finite element method is given to cure such coupling instability completely. The stabilized damping is so small that it does not affect the solution accuracy nearly. The numerical examples show the instability phenomenon and indicate the effectiveness of the damping method. The time-domain stability studies here can be a reference for the other artificial boundary conditions.

Delay‐dependent H∞ control of parameter‐varying delayed systems via parameter‐dependent Lyapunov function approach

AbstractThe H∞ control problem of parameter‐varying delayed systems is investigated in this paper. The state‐space matrices of the systems are assumed to be dependent on a vector of time‐varying real parameters which are assumed to be real‐time measurable. The delays related to the parameter‐varying systems are assumed to be unknown but with known upper bounds and to be in the states and control inputs. A delay‐dependent H∞ performance condition of the system under consideration is derived by using a new parameter‐dependent Lyapunov function. Based on the H∞ performance condition, a linear matrix inequality (LMI) based H∞ control strategy is proposed by using auxiliary variable technique. The combined parameter‐dependent and delay‐dependent results are less conservative due to the generality of the parameter‐dependent and delay‐dependent Lyapunov function used, which includes the parameter‐independent one as a special case. It is shown that the underling H∞ control problem can be solved as LMI optimization problems that can be numerically computed very efficiently. A numerical example is also given to demonstrate the applicability of the proposed approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

NON‐PARAMETRIC BAYESIAN INFERENCE FOR INHOMOGENEOUS MARKOV POINT PROCESSES

SummaryWith reference to a specific dataset, we consider how to perform a flexible non‐parametric Bayesian analysis of an inhomogeneous point pattern modelled by a Markov point process, with a location‐dependent first‐order term and pairwise interaction only. A priori we assume that the first‐order term is a shot noise process, and that the interaction function for a pair of points depends only on the distance between the two points and is a piecewise linear function modelled by a marked Poisson process. Simulation of the resulting posterior distribution using a Metropolis–Hastings algorithm in the ‘conventional’ way involves evaluating ratios of unknown normalizing constants. We avoid this problem by applying a recently introduced auxiliary variable technique. In the present setting, the auxiliary variable used is an example of a partially ordered Markov point process model.

Adaptive rejection Metropolis sampling using Lagrange interpolation polynomials of degree 2

A crucial problem in Bayesian posterior computation is efficient sampling from a univariate distribution, e.g. a full conditional distribution in applications of the Gibbs sampler. This full conditional distribution is usually non-conjugate, algebraically complex and computationally expensive to evaluate. We propose an alternative algorithm, called ARMS2, to the widely used adaptive rejection sampling technique ARS [Gilks, W.R., Wild, P., 1992. Adaptive rejection sampling for Gibbs sampling. Applied Statistics 41 (2), 337–348; Gilks, W.R., 1992. Derivative-free adaptive rejection sampling for Gibbs sampling. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (Eds.), Bayesian Statistics, Vol. 4. Clarendon, Oxford, pp. 641–649] for generating a sample from univariate log-concave densities. Whereas ARS is based on sampling from piecewise exponentials, the new algorithm uses truncated normal distributions and makes use of a clever auxiliary variable technique [Damien, P., Walker, S.G., 2001. Sampling truncated normal, beta, and gamma densities. Journal of Computational and Graphical Statistics 10 (2) 206–215]. Furthermore, we extend this algorithm to deal with non-log-concave densities to provide an enhanced alternative to adaptive rejection Metropolis sampling, ARMS [Gilks, W.R., Best, N.G., Tan, K.K.C., 1995. Adaptive rejection Metropolis sampling within Gibbs sampling. Applied Statistics 44, 455–472]. The performance of ARMS and ARMS2 is compared in simulations of standard univariate distributions as well as in Gibbs sampling of a Bayesian hierarchical state-space model used for fisheries stock assessment.

A General Framework for the Parametrization of Hierarchical Models

In this paper, we describe centering and noncentering methodology as complementary techniques for use in parametrization of broad classes of hierarchical models, with a view to the construction of effective MCMC algorithms for exploring posterior distributions from these models. We give a clear qualitative understanding as to when centering and noncentering work well, and introduce theory concerning the convergence time complexity of Gibbs samplers using centered and noncentered parametrizations. We give general recipes for the construction of noncentered parametrizations, including an auxiliary variable technique called the state-space expansion technique. We also describe partially noncentered methods, and demonstrate their use in constructing robust Gibbs sampler algorithms whose convergence properties are not overly sensitive to the data.