Bound-preserving discontinuous Galerkin method for compressible miscible displacement in porous media

In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacement problems. We consider the problem with two components and the (volumetric) concentration of the $i$th component of the fluid mixture, $c_i$, should be between $0$ and $1$. However, $c_i$ does not satisfy the maximum principle. Therefore, the numerical techniques introduced in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied directly. The main idea is to apply the positivity-preserving techniques to both $c_1$ and $c_2$, respectively and enforce $c_1+c_2=1$ simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure $dp/dt$ as a source in the concentration equation. Moreover, $c_i's$ are not the conservative variables, as a result, the classical bound-preserving limiter in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied. Therefore, another limiter will be introduced. Numerical experiments will be given to demonstrate the accuracy in $L^\infty$-norm and good performance of the numerical technique

Atomic Bright Soliton Interferometry

The properties of nonlinear interference pattern between atomic bright solitons are characterized analytically, with the aid of exact solutions of dynamical equation in mean-field approximation. It is shown that relative velocity, relative phase, and nonlinear interaction strength can be measured from the interference pattern. The nonlinear interference properties are proposed to design atomic soliton interferometry in Bose-Einstein condensate. As an example, we apply them to measure gravity acceleration in a ultra-cold atom systems with a high precision degree. The results are also meaningful for precise measurements in optical fiber, water wave tank, plasma, and other nonlinear systems.Comment: 7 pages, 3 figure

A lattice Boltzmann method for binary fluids based on mass-conserved quasi-incompressible phase-field theory

In this paper, a lattice Boltzmann equation (LBE) model is proposed for binary fluids based on a quasi-incompressible phase-field model [J. Shen et al, Comm. Comp. Phys. 13, 1045 (2013)]. Compared with the other incompressible LBE models based on the incompressible phase-field theory, the quasi-incompressible model conserves mass locally. A series of numerical simulations are performed to validate the proposed model, and comparisons with an incompressible LBE model [H. Liang et al, Phys. Rev. E 89, 053320 (2014)] are also carried out. It is shown that the proposed model can track the interface accurately, and the predictions by the quasi-incompressible and incompressible models agree qualitatively well as the distribution of chemical potential is uniform, otherwise differ significantly

Ekeland's Variational Principle for An Lˉ0−\bar{L}^{0}-Valued Function on A Complete Random Metric Space

Motivated by the recent work on conditional risk measures, this paper studies the Ekeland's variational principle for a proper, lower semicontinuous and lower bounded $\bar{L}^{0}-$valued function, where $\bar{L}^{0}$ is the set of equivalence classes of extended real-valued random variables on a probability space. First, we prove a general form of Ekeland's variational principle for such a function defined on a complete random metric space. Then, we give a more precise form of Ekeland's variational principle for such a local function on a complete random normed module. Finally, as applications, we establish the Bishop-Phelps theorem in a complete random normed module under the framework of random conjugate spaces.Comment: 26 page

The Control Complexity of rr-Approval: from the Single-Peaked Case to the General Case

We investigate the complexity of $r$-Approval control problems in $k$-peaked elections, where at most $k$ peaks are allowed in each vote with respect to an order of the candidates. We show that most NP-hardness results in general elections also hold in k-peaked elections even for $k=2,3$. On the other hand, we derive polynomial-time algorithms for some problems for $k=2$. All our NP-hardness results apply to Approval and sincere-strategy preference-based Approval as well. Our study leads to many dichotomy results for the problems considered in this paper, with respect to the values of $k$ and $r$. In addition, we study $r$-Approval control problems from the viewpoint of parameterized complexity and achieve both fixed-parameter tractability results and W-hardness results, with respect to the solution size. Along the way exploring the complexity of control problems, we obtain two byproducts which are of independent interest. First, we prove that every graph of maximum degree 3 admits a specific 2-interval representation where every 2-interval corresponding to a vertex contains a trivial interval and, moreover, 2-intervals may only intersect at the endpoints of the intervals. Second, we develop a fixed-parameter tractable algorithm for a generalized $r$-Set Packing problem with respect to the solution size, where each element in the given universal set is allowed to occur in more than one r-subset in the solution.Comment: 23 page

Measuring and Controlling Bias for Some Bayesian Inferences and the Relation to Frequentist Criteria

A common concern with Bayesian methodology in scientific contexts is that inferences can be heavily influenced by subjective biases. As presented here, there are two types of bias for some quantity of interest: bias against and bias in favor. Based upon the principle of evidence, it is shown how to measure and control these biases for both hypothesis assessment and estimation problems. Optimality results are established for the principle of evidence as the basis of the approach to these problems. A close relationship is established between measuring bias in Bayesian inferences and frequentist properties that hold for any proper prior. This leads to a possible resolution to an apparent conflict between these approaches to statistical reasoning. Frequentism is seen as establishing a figure of merit for a statistical study, while Bayesianism plays the key role in determining inferences based upon statistical evidence

Existence and BV-regularity for Neutron transport equation in non-convex domain

This paper considers the neutron transport equation in bounded domain with a combination of the diffusive boundary condition and the in-flow boundary condition. We firstly study the existence of solution in any fixed time by $L^2-L^{\infty}$ method, which was established to study Boltzmann equation in \cite{[Guo2]}. Based on the uniform estimates of the solution, we also consider the BV-regularity of the solution in non-convex domain. A cut-off function, which aims to exclude all the characteristics emanating from the grazing set $\mathfrak{S}_B$, has been constructed precisely.Comment: 50 pages. arXiv admin note: text overlap with arXiv:1409.016

Concentration on Surfaces for a Singularly Perturbed Neumann Problem in Three-Dimensional Domains

We consider the following singularly perturbed elliptic problem \varepsilon^2\triangle\tilde{u}-\tilde{u}+\tilde{u}^p=0, \ \tilde{u}>0\quad \mbox{in} \ \Omega,\ \ \ \frac{\partial\tilde{u}}{\partial \mathbf{n}}=0 \quad \mbox{on}\ \partial\Omega, where $\Omega$ is a bounded domain in $\mathbb{R}^3$ with smooth boundary, $\varepsilon$ is a small parameter, $\mathbf{n}$ denotes the inward normal of $\partial\Omega$ and the exponent $p>1$. Let $\Gamma$ be a hypersurface intersecting $\partial\Omega$ in the right angle along its boundary $\partial\Gamma$ and satisfying a {\em non-degenerate condition}. We establish the existence of a solution $u_\varepsilon$ concentrating along a surface $\tilde{\Gamma}$ close to $\Gamma$, exponentially small in $\varepsilon$ at any positive distance from the surface $\tilde{\Gamma}$, provided $\varepsilon$ is small and away from certain {\em critical numbers}. The concentrating surface $\tilde{\Gamma}$ will collapse to $\Gamma$ as $\varepsilon\rightarrow 0$

Gradient recovery for elliptic interface problem: III. Nitsche's method

This is the third paper on the study of gradient recovery for elliptic interface problem. In our previous works [H. Guo and X. Yang, 2016, arXiv:1607.05898 and {\it J. Comput. Phys.}, 338 (2017), 606--619], we developed {gradient recovery methods} for elliptic interface problem based on body-fitted meshes and immersed finite element methods. Despite the efficiency and accuracy that these methods bring to recover the gradient, there are still some cases in unfitted meshes where skinny triangles appear in the generated local body-fitted triangulation that destroy the accuracy of recovered gradient near the interface. In this paper, we propose a gradient recovery technique based on Nitsche's method for elliptic interface problem, which avoids the loss of accuracy of gradient near the interface caused by skinny triangles. We analyze the supercloseness between the gradient of the numerical solution by the Nitsche's method and the gradient of interpolation of the exact solution, which leads to the superconvergence of the proposed gradient recovery method. We also present several numerical examples to validate the theoretical results

Quantum Zeno and anti-Zeno effect in atom-atom entanglement induced by non-Markovian environment

The dynamic behavior of the entanglement for two two-level atoms coupled to a common lossy cavity is studied. We find that the speed of disentanglement is a decreasing (increasing) function of the damping rate of the cavity for on/near (far-off) resonant couplings. The quantitative explanations for these phenomena are given, and further, it is shown that they are related to the quantum Zeno and anti-Zeno effect induced by the non-Markovian environment.Comment: 4 pages, 2 figure