We propose a graph cut based global minimization method for image segmentation by representing the segmentation label function with a series of nested binary super-level set functions. This representation enables us to use K-1 binary functions to partition any images into K phases. Both continuous and discretized formulations will be treated. For the discrete model, we propose a new graph cut algorithm which is faster than the existing graph cut methods to obtain the exact global solution. In the continuous case, we further improve the segmentation accuracy using a number of techniques that are unique to the continuous segmentation models. With the convex relaxation and the dual method, the related continuous dual model is convex and we can mathematically show that the global minimization can be achieved. The corresponding continuous max-flow algorithms are easy and stable. Experimental results

We propose a simple Eulerian approach to compute the moderate to long time flow map for approximating the Lyapunov exponent of a (periodic or aperiodic) dynamical system. The idea is to generalize a recently proposed backward phase flow method which is specially designed for long time level set propagation. Unlike the original phase flow method or the backward phase flow method, which is applicable only to autonomous systems, the current approach can also be applied to any time-dependent (periodic or aperiodic) flow. We will discuss the stability of the proposed method. Numerical examples will be given to demonstrate the effectiveness of the algorithm.

We propose efficient Eulerian methods for approximating the finite-time Lyapunov exponent (FTLE). The idea is to compute the related flow map using the Level Set Method and the Liouville equation. There are several advantages of the proposed approach. Unlike the usual Lagrangian-type computations, the resulting method requires the velocity field defined only at discrete locations. No interpolation of the velocity field is needed. Also, the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain. The computational complexity of the algorithm is <i>O</i>(<i>x</i>^(<i>d</i>+1)) with <i>d</i> the dimension of the physical space. Since there are the same number of mesh points in the <i>x</i><i>t</i> space, the computational complexity of the proposed Eulerian approach is optimal in the sense that each grid point is visited for only <i>O</i>(1) time. We also extend the algorithm to compute the FTLE

We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an mimes n rank mimes n matrix from mimes n number of linear measurements. The algorithms are first interpreted as iterative hard thresholding algorithms with subspace projections. Based on this connection, we show that provided the restricted isometry constant mimes n of the sensing operator is less than mimes n, the Riemannian gradient descent algorithm and a restarted variant of the Riemannian conjugate gradient algorithm are guaranteed to converge linearly to the underlying rank mimes n matrix if they are initialized by one step hard thresholding. Empirical evaluation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements necessary.

We develop numerical methods for solving partial differential equations (PDE) defined on an evolving interface represented by the grid based particle method (GBPM) recently proposed in [S. Leung, H.K. Zhao, A grid based particle method for moving interface problems, J. Comput. Phys. 228 (2009) 77067728]. In particular, we develop implicit time discretization methods for the advectiondiffusion equation where the time step is restricted solely by the advection part of the equation. We also generalize the GBPM to solve high order geometrical flows including surface diffusion and Willmore-type flows. The resulting algorithm can be easily implemented since the method is based on meshless particles quasi-uniformly sampled on the interface. Furthermore, without any computational mesh or triangulation defined on the interface, we do not require remeshing or reparametrization in the case of highly distorted motion

Different from conventional spaceborne or airborne synthetic aperture radar (SAR) with optimal aperture length, an imaging radar with highly suboptimal aperture length acquires the data in short bursts by a geometry spreading over a large range. A polarlike or pseudopolar format grid is introduced to sample data close to the resolution, which presents the design of a separable kernel for efficient FFT implementation. The proposed imaging algorithm formulates the reflectivity image of the target scene as an interpolation-free double image series expansion with two usual approximation-induced phase error terms being taken into account, whereby more generalized application scenarios with high frequency, large bandwidth or larger aperture length for imaging a target scene located within either the far-field or the near-field of the radar aperture are processable with high accuracy. In addition, convergence

Laser cladding has been widely used in direct laser fabrication and laser refabrication for metal parts. It is a high precision and complicated process due to multiple process parameters involved. These parameters are strongly tied to each other and their effects are still far from being completely understood. In order to make the optimal choice, a feasible mathematical model is established to describe the interaction between laser and powder particles and to predict and control the cross sectional area. Additionally, analysis concentrating on the effect of the various parameters on the cross sectional area is made. Results show that laser power, powder feed rate, scan speed and radius of focused powder stream are the main factors. Correlations between main process parameters and cross sectional area of an individual clad track have been found. The model also provides predictions of range of process parameters for melting powder to form specific cross sectional area, which will guide the experiment and greatly reduce the cost of the experimental investigations.

The accurate control of process parameters is particularly important for the growth of high quality structures during laser cladding. Melt pool size and cross sectional area are key process characteristics that can be controlled to allow the precise deposition of complex features. A feasible mathematical model is established to describe the interaction between laser and powder particles and to predict and control the cross sectional area. Also, a vision-based sensing system is proposed to detect the transient shape and size of the melt pool. The system focuses on the measuring of the geometric parameters of the melt pool by using a digital image processing method and thus giving researchers the parameters on-line, which gives us a necessary tool to gain insight into the cladding process and to optimize the material quality

In this paper, we consider a class of optimization problems of minimizing a (at least) twice continuously dierentiable function (probably nonconvex) f(x) : Rn ! R over a product of multiple balls/spheres constraints. Upon rescaling the balls/spheres, we cast without loss of generality such class of minimization problems in the following form: (BCOP) 8>< >: min x2Rn f(x) s:t: ci(x) := kx[i]k2 􀀀 1 = 0; i 2 E; ci(x) := kx[i]k2 􀀀 1  0; i 2 I; where E = f1; 2; : : : ;m1g, I = fm1 + 1;m1 + 2; : : : ;mg, x[i] 2 Rpi , x = (xT [1]; xT [2]; : : : ; xT [m])T , n = Pm i=1 pi, and k
k stands for the `2 vector norm. Here, we introduce the notation x[i] 2 Rpi to represent the ith subvector of x 2 Rn and formulate the product of multiple ball/sphere constraints as a set of equality and inequality constraints. To simplify subsequent presentation, we call the above programming the ball/sphere constrained optimization problem (BCOP). We emphasize that this problem does not allow overlap among the variables x[i] and therefore the constraints are separable. However, these variables x[i] may be linked together through the objective function f(x).

In this paper, we illustrate a new concept regarding unitary elements defined on Lorentz cone, and establish some basic properties under the so-called unitary transformation associated with Lorentz cone. As an application of unitary transformation, we achieve a weaker version of the triangle inequality and several (weak) majorizations defined on Lorentz cone.

In the paper, we consider a continuation approach for the binary quadratic program (BQP) based on a class of NCP-functions. More specifically, we recast the BQP as an equivalent minimization and then seeks its global minimizer via a global continuation method. Such approach had been considered in [11] which is based on the FischerBurmeister function. We investigate this continuation approach again by using a more general function, called the generalized FischerBurmeister function. However, the theoretical background for such extension can not be easily carried over. Indeed, it needs some subtle analysis.

In this paper, we explore a unified way to construct smoothing functions for solving the absolute value equation associated with second-order cone (SOCAVE). Numerical comparisons are presented, which illustrate what kinds of smoothing functions work well along with the smoothing Newton algorithm. In particular, the numerical experiments show that the well known loss function widely used in engineering community is the worst one among the constructed smoothing functions, which indicates that the other proposed smoothing functions can be employed for solving engineering problems.

Consider the single-facility Euclidean -centrum location problem in R n. This problem is a generalization of the classical Euclidean 1-median problem and 1-center problem. In this paper, we develop two efficient algorithms that are particularly suitable for problems where n is large by using unconstrained optimization techniques. The first algorithm is based on the neural networks smooth approximation for the plus function and reduces the problem to an unconstrained smooth convex minimization problem. The second algorithm is based on the FischerBurmeister merit function for the second-order cone complementarity problem and transforms the KKT system of the second-order cone programming reformulation for the problem into an unconstrained smooth minimization problem. Our computational experiments indicate that both methods are extremely efficient for large problems and the first algorithm is able to

This paper proposes a neural network approach to efficiently solve nonlinear convex programs with the second-order cone constraints. The neural network model is designed by the generalized FischerBurmeister function associated with second-order cone. We study the existence and convergence of the trajectory for the considered neural network. Moreover, we also show stability properties for the considered neural network, including the Lyapunov stability, the asymptotic stability and the exponential stability. Illustrative examples give a further demonstration for the effectiveness of the proposed neural network. Numerical performance based on the parameter being perturbed and numerical comparison with other neural network models are also provided. In overall, our model performs better than two comparative methods.

In this paper, we present a measure of distance in a second-order cone based on a class of continuously differentiable strictly convex functions on ₊₊. Since the distance function has some favorable properties similar to those of the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451464 [1992]), we refer to it as a quasi D-function. Then, a proximal-like algorithm using the quasi D-function is proposed and applied to the second-cone programming problem, which is to minimize a closed proper convex function with general second-order cone constraints. Like the proximal point algorithm using the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451464 [1992]; Chen and Teboulle in SIAM J. Optim. 3:538543 [1993]), under some mild assumptions we establish the global convergence of the algorithm expressed in terms of function values; we show that the sequence generated by the

Recently there have two different effective methods proposed by Kanzow et al. in (Kanzow, 2001 [8]) and (Kanzow and Petra, 2004 [9]), respectively, which commonly use the FischerBurmeister (FB) function to recast the mixed complementarity problem (MCP) as a constrained minimization problem and a nonlinear system of equations, respectively. They all remark that their algorithms may be improved if the FB function is replaced by other NCP functions. Accordingly, in this paper, we employ the generalized FischerBurmeister (GFB) where the 2-norm in the FB function is relaxed to a general p-norm (p&gt; 1) for the two methods and investigate how much the improvement is by changing the parameter p as well as which method is influenced more when we do so, by the performance profiles of iterations and function evaluations for the two methods with different p on MCPLIB collection.

In the paper [4], the authors proposed a derivative-free descent algorithm for the nonlinear complementarity problems (NCPs) by the generalized Fischer-Burmeister merit function: p (a, b)= 1

We show that the gradient mapping of the squared norm of FischerBurmeister function is globally Lipschitz continuous and semismooth, which provides a theoretical basis for solving nonlinear second-order cone complementarity problems via the conjugate gradient method and the semismooth Newtons method.

Shape optimization aims to optimize an objective function by changing the shape of the computational domain. In recent years, shape optimization has received considerable attentions. On the theoretical side there are several publications dealing with the existence of solution and the sensitivity analysis of the problem; see e.g., [6] and references therein.

Wind power is an increasingly popular renewable energy. In the design process of the wind turbine blade, the accurate aerodynamic simulation is important. In the past, most of the wind turbine simulations were carried out with some low fidelity methods, such as the blade element momentum method [9]. Recently, with the rapid development of the supercomputers, high fidelity simulations based on 3D unsteady Navier-Stokes (N-S) equations become more popular. For example, Sorensen et al. studied the 3D wind turbine rotor using the Reynolds-Averaged Navier-Stokes (RANS) framework where a finite volume method and a semi-implicit method are used for the spatial and temporal discretization, respectively [17]. Bazilevs et al. investigated the aerodynamic of the NREL 5MW offshore baseline wind turbine rotor using large eddy simulation built with a deforming-spatial-domain/stabilized space-time

We propose and study a new parallel one-shot Lagrange--Newton--Krylov--Schwarz (LNKSz) algorithm for shape optimization problems constrained by steady incompressible Navier--Stokes equations discretized by finite element methods on unstructured moving meshes. Most existing algorithms for shape optimization problems iteratively solve the three components of the optimality system: the state equations for the constraints, the adjoint equations for the Lagrange multipliers, and the design equations for the shape parameters. Such approaches are relatively easy to implement, but generally are not easy to converge as they are basically nonlinear Gauss--Seidel algorithms with three large blocks. In this paper, we introduce a fully coupled, or the so-called one-shot, approach which solves the three components simultaneously. First, we introduce a moving mesh finite element method for the shape optimization

In this paper, we consider an online basis enrichment mixed generalized multiscale method with oversampling, for solving flow problems in highly heterogeneous porous media. This is an exten-sion of the online mixed generalized multiscale method [6]. The multiscale online basis functions are computed by solving a Neumann problem in an over-sampled domain, instead of a standard neighborhood of a coarse face. We are motivated by the restricted domain decomposition method. Extensive numerical experiments are presented to demonstrate the performance of our methods for both steady-state flow, and two-phase flow and transport problems.

We present a new stabilization technique for multiscale convection diffusion problems. Stabilization for these problems has been a challenging task, especially for the case with high Peclet numbers. Our method is based on a constraint energy minimization idea and the discontinuous Petrov-Galerkin formulation. In particular, the test functions are constructed by minimizing an appropriate energy subject to certain orthogonality conditions, and are related to the trial space. The resulting test functions have a localization property, and can therefore be computed locally. We will prove the stability, and present several numerical results. Our numerical results confirm that our test space gives a good stability, in the sense that the solution error is close to the best approximation error.

In this paper we investigate staggered discontinuous Galerkin method for the Helmholtz equation with large wave number on general quadrilateral and polygonal meshes. The method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids, and at the same time, is numerical flux free. Furthermore, it allows hanging nodes, which can be simply treated as additional vertices. By exploiting a modified duality argument, the stability and convergence can be proved under the condition that \kappa h is sufficiently small, where \kappa h is the wave number and \kappa h is the mesh size. Error estimates for both the scalar and vector variables in \kappa h norm are established. Several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions.

In this paper, we present a constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for parabolic equations with multiscale coefficients, arising from applications in porous media. We will present the construction of CEM-GMsFEM and rigorously analyze its convergence for the parabolic equations. The convergence rate is characterized by the coarse grid size and the eigenvalue decay of local spectral problems, but is independent of the scale length and contrast of the media. The analysis shows that the method has a first order convergence rate with respect to coarse grid size in the energy norm and second order convergence rate with respect to coarse grid size in L^2 norm under some appropriate assumptions. For the temporal discretization, finite difference techniques are used and the convergence analysis of full discrete scheme is given. Moreover, a posteriori error estimator is