arXiv subjects

Ran Zhang

Publications and source records attributed to Ran Zhang.

125 records · Page 7Linked to original sources

arXiv2019

A High order Conservative Flux Optimization Finite Element Method for Diffusion Equations

This article presents a high order conservative flux optimization (CFO) finite element method for the elliptic diffusion equations. The numerical scheme is based on the classical Galerkin finite element method enhanced by a flux approximation on the boundary of a prescribed set of arbitrary control volumes (either the finite element partition itself or dual voronoi mesh, etc). The numerical approximations can be characterized as the solution of a constrained-minimization problem with constraints given by the flux conservation equations on each control volume. The discrete linear system is a typical saddle-point problem, but with less number of degrees of freedom than the standard mixed finite element method, particularly for elements of high order. Moreover, the numerical solution of the proposed scheme is of super-closeness with the finite element solution. Error estimates of optimal order are established for the numerical flux as well as the primary variable approximations. We present several numerical studies in order to verify convergence of the CFO schemes. A simplified two-phase flow in highly heterogeneous porous media model problem will also be presented. The numerical results show obvious advantages of applying high order CFO schemes.

math.NA↗

arXiv2018

On Virtual Network Embedding: Paths and Cycles

Network virtualization provides a promising solution to overcome the ossification of current networks, allowing multiple Virtual Network Requests (VNRs) embedded on a common infrastructure. The major challenge in network virtualization is the Virtual Network Embedding (VNE) problem, which is to embed VNRs onto a shared substrate network and known to be $\mathcal{NP}$-hard. The topological heterogeneity of VNRs is one important factor hampering the performance of the VNE. However, in many specialized applications and infrastructures, VNRs are of some common structural features $\textit{e.g.}$, paths and cycles. To achieve better outcomes, it is thus critical to design dedicated algorithms for these applications and infrastructures by taking into accounting topological characteristics. Besides, paths and cycles are two of the most fundamental topologies that all network structures consist of. Exploiting the characteristics of path and cycle embeddings is vital to tackle the general VNE problem. In this paper, we investigated the path and cycle embedding problems. For path embedding, we proved its $\mathcal{NP}$-hardness and inapproximability. Then, by utilizing Multiple Knapsack Problem (MKP) and Multi-Dimensional Knapsack Problem (MDKP), we proposed an efficient and effective MKP-MDKP-based algorithm. For cycle embedding, we proposed a Weighted Directed Auxiliary Graph (WDAG) to develop a polynomial-time algorithm to determine the least-resource-consuming embedding. Numerical results showed our customized algorithms can boost the acceptance ratio and revenue compared to generic embedding algorithms in the literature.

cs.NI↗

arXiv2018

Dropout during inference as a model for neurological degeneration in an image captioning network

We replicate a variation of the image captioning architecture by Vinyals et al. (2015), then introduce dropout during inference mode to simulate the effects of neurodegenerative diseases like Alzheimer's disease (AD) and Wernicke's aphasia (WA). We evaluate the effects of dropout on language production by measuring the KL-divergence of word frequency distributions and other linguistic metrics as dropout is added. We find that the generated sentences most closely approximate the word frequency distribution of the training corpus when using a moderate dropout of 0.4 during inference.

cs.CL↗

arXiv2018

The Shifted-inverse Power Weak Galerkin Method for Eigenvalue Problems

This paper proposes and analyzes a new weak Galerkin method for the eigenvalue problem by using the shifted-inverse power technique. A high order lower bound can be obtained at a relatively low cost via the proposed method. The error estimates for both eigenvalue and eigenfunction are provided and asymptotic lower bounds are shown as well under some conditions. Numerical examples are presented to validate the theoretical analysis.

math.NA↗

arXiv2017

Acceleration of weak Galerkin methods for the Laplacian eigenvalue problem

Recently, we proposed a weak Galerkin finite element method for the Laplace eigenvalue problem. In this paper, we present two-grid and two-space skills to accelerate the weak Galerkin method. By choosing parameters properly, the two-grid and two-space weak Galerkin method not only doubles the convergence rate, but also maintains the asymptotic lower bounds property of the weak Galerkin method. Some numerical examples are provided to validate our theoretical analysis.

math.NA↗

arXiv2017

$β$-expansion: A Theoretical Framework for Fast and Recursive Construction of Polar Codes

In this work, we introduce $β$-expansion, a notion borrowed from number theory, as a theoretical framework to study fast construction of polar codes based on a recursive structure of universal partial order (UPO) and polarization weight (PW) algorithm. We show that polar codes can be recursively constructed from UPO by continuously solving several polynomial equations at each recursive step. From these polynomial equations, we can extract an interval for $β$, such that ranking the synthetic channels through a closed-form $β$-expansion preserves the property of nested frozen sets, which is a desired feature for low-complex construction. In an example of AWGN channels, we show that this interval for $β$ converges to a constant close to $1.1892 \approx 2^{1/4}$ when the code block-length trends to infinity. Both asymptotic analysis and simulation results validate our theoretical claims.

cs.IT↗

arXiv2016

Supercloseness analysis and polynomial preserving recovery for a class of weak Galerkin method

In this paper, we analyze convergence and supercloseness properties of a class of weak Galerkin (WG) finite element methods for solving second-order elliptic problems. It is shown that the WG solution is superclose to the Lagrange type interpolation using Lobatto points. This supercloseness behavior is obtained through some newly designed stabilization terms. A post-processing technique using the polynomial preserving recovery (PPR) is introduced for WG approximation. Superconvergence analysis is carried out for the PPR approximation. Numerical examples are provided to verify our theoretical results.

math.NA↗

arXiv2016

Behavior of different numerical schemes for population genetic drift problems

In this paper, we focus on numerical methods for the genetic drift problems, which is governed by a degenerated convection-dominated parabolic equation. Due to the degeneration and convection, Dirac singularities will always be developed at boundary points as time evolves. In order to find a \emph{complete solution} which should keep the conservation of total probability and expectation, three different schemes based on finite volume methods are used to solve the equation numerically: one is a upwind scheme, the other two are different central schemes. We observed that all the methods are stable and can keep the total probability, but have totally different long-time behaviors concerning with the conservation of expectation. We prove that any extra infinitesimal diffusion leads to a same artificial steady state. So upwind scheme does not work due to its intrinsic numerical viscosity. We find one of the central schemes introduces a numerical viscosity term too, which is beyond the common understanding in the convection-diffusion community. Careful analysis is presented to prove that the other central scheme does work. Our study shows that the numerical methods should be carefully chosen and any method with intrinsic numerical viscosity must be avoided.

math.NA↗

arXiv2016

Angle-of-Attack Modulation in Trajectory Tracking for a Reusable Launch Vehicle

This paper deals with the problem of angle-of-attack modulation with the aim of enhancing transient performance of entry guidance during bank reversals, while compensating adverse effects of fast time-varying transient disturbances. An extended single-input/single-output system is developed in the velocity domain by means of a dynamic extension technique, and explicitly captures the trajectory dynamics of angle-of-attack modulation. A normal form for this extended system is derived for the sake of employing a feedback linearization controller. Further, the control characteristics of angle-of-attack modulation is found to be a non-minimum phase behavior under two common conditions in a near- equilibrium glide flight. Therefore, the issue of angle-of-attack modulation is formulated as robust output stabilization of the non-minimum phase system. A disturbance observer-based feedback linearization technique is used to design a robustly dynamical output-feedback controller for angle-of-attack modulation, and an internal-state feedback controller for bank-angle modulation is used to stabilize the unstable internal dynamics. Numerical simulations are conducted to demonstrate that the performance of the proposed method of angle-of-attack modulation is enhanced compared to the existing shuttle method.

eess.SY↗

arXiv2016

A Weak Galerkin Finite Element Scheme for the Biharmonic Equations by Using Polynomials of Reduced Order

A new weak Galerkin (WG) finite element method for solving the biharmonic equation in two or three dimensional spaces by using polynomials of reduced order is introduced and analyzed. The WG method is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^2$ norm and the standard $L^2$ norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.

math.NA↗

arXiv2016

A Weak Galerkin Finite Element Scheme for solving the stationary Stokes Equations

A weak Galerkin (WG) finite element method for solving the stationary Stokes equations in two- or three- dimensional spaces by using discontinuous piecewise polynomials is developed and analyzed. The variational form we considered is based on two gradient operators which is different from the usual gradient-divergence operators. The WG method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms. Numerical results are presented to illustrate the theoretical analysis of the new WG finite element scheme for Stokes problems.

math.NA↗

arXiv2015

A weak Galerkin finite element scheme with boundary continuity for second-order elliptic problems

A new weak Galerkin (WG) finite element method for solving the second-order elliptic problems on polygonal meshes by using polynomials of boundary continuity is introduced and analyzed. The WG method is utilizing weak functions and their weak derivatives which can be approximated by polynomials in different combination of polynomial spaces. Different combination gives rise to different weak Galerkin finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of certain combination of polynomial spaces that minimize the degree of freedom in the numerical scheme, yet without losing the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^1$ norm and the standard $L^2$ norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.

math.NA↗

arXiv2015

The weak Galerkin method for eigenvalue problems

This article is devoted to computing the eigenvalue of the Laplace eigenvalue problem by the weak Galerkin (WG) finite element method with emphasis on obtaining lower bounds. The WG method is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. We establish the optimal-order error estimates for the WG finite element approximation for the eigenvalue problem. Comparing with the classical nonconforming finite element method which can just provide lower bound approximation by linear elements with only the second order convergence, the WG methods can naturally provide lower bound approximation with a high order convergence (larger than $2$). Some numerical results are also presented to demonstrate the efficiency of our theoretical results.

math.NA↗

arXiv2015

A Locking-Free Weak Galerkin Finite Element Method for Elasticity Problems in the Primal Formulation

This paper presents an arbitrary order locking-free numerical scheme for linear elasticity on general polygonal/polyhedral partitions by using weak Galerkin (WG) finite element methods. Like other WG methods, the key idea for the linear elasticity is to introduce discrete weak strain and stress tensors which are defined and computed by solving inexpensive local problems on each element. Such local problems are derived from weak formulations of the corresponding differential operators through integration by parts. Locking-free error estimates of optimal order are derived in a discrete $H^1$-norm and the usual $L^2$-norm for the approximate displacement when the exact solution is smooth. Numerical results are presented to demonstrate the efficiency, accuracy, and the locking-free property of the weak Galerkin finite element method.

math.NA↗

arXiv2015

A Hybridized Weak Galerkin Finite Element Scheme for the Stokes Equations

In this paper a hybridized weak Galerkin (HWG) finite element method for solving the Stokes equations in the primary velocity-pressure formulation is introduced. The WG method uses weak functions and their weak derivatives which are defined as distributions. Weak functions and weak derivatives can be approximated by piecewise polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. A Lagrange multiplier is introduced to provide a numerical approximation for certain derivatives of the exact solution. With this new feature, HWG method can be used to deal with jumps of the functions and their flux easily. Optimal order error estimate are established for the corresponding HWG finite element approximations for both {\color{black}primal variables} and the Lagrange multiplier. A Schur complement formulation of the HWG method is derived for implementation purpose. The validity of the theoretical results is demonstrated in numerical tests.

math.NA↗

arXiv2013

A Bayesian linear model for the high-dimensional inverse problem of seismic tomography

We apply a linear Bayesian model to seismic tomography, a high-dimensional inverse problem in geophysics. The objective is to estimate the three-dimensional structure of the earth's interior from data measured at its surface. Since this typically involves estimating thousands of unknowns or more, it has always been treated as a linear(ized) optimization problem. Here we present a Bayesian hierarchical model to estimate the joint distribution of earth structural and earthquake source parameters. An ellipsoidal spatial prior allows to accommodate the layered nature of the earth's mantle. With our efficient algorithm we can sample the posterior distributions for large-scale linear inverse problems and provide precise uncertainty quantification in terms of parameter distributions and credible intervals given the data. We apply the method to a full-fledged tomography problem, an inversion for upper-mantle structure under western North America that involves more than 11,000 parameters. In studies on simulated and real data, we show that our approach retrieves the major structures of the earth's interior as well as classical least-squares minimization, while additionally providing uncertainty assessments.

stat.AP↗

arXiv2011

Maximum Principles for P1-Conforming Finite Element Approximations of Quasi-Linear Second Order Elliptic Equations

This paper derives some discrete maximum principles for $P1$-conforming finite element approximations for quasi-linear second order elliptic equations. The results are extensions of the classical maximum principles in the theory of partial differential equations to finite element methods. The mathematical tools are based on the variational approach that was commonly used in the classical PDE theory. The discrete maximum principles are established by assuming a property on the discrete variational form that is of global nature. In particular, the assumption on the variational form is verified when the finite element partition satisfies some angle conditions. For the general quasi-linear elliptic equation, these angle conditions indicate that each triangle or tetrahedron needs to be $\mathcal{O}(h^α)$-acute in the sense that each angle $α_{ij}$ (for triangle) or interior dihedral angle $α_{ij}$ (for tetrahedron) must satisfy $α_{ij}\le π/2-γh^α$ for some $α\ge 0$ and $γ>0$. For the Poisson problem where the differential operator is given by Laplacian, the angle requirement is the same as the existing ones: either all the triangles are non-obtuse or each interior edge is non-negative. It should be pointed out that the analytical tools used in this paper are based on the powerful De Giorgi's iterative method that has played important roles in the theory of partial differential equations. The mathematical analysis itself is of independent interest in the finite element analysis.

math.NA↗