Shape and topology optimization for elliptic boundary value problems using a piecewise constant level set method

The aim of this paper is to propose a variational piecewise constant level set method for solving elliptic shape and topology optimization problems. The original model is approximated by a two-phase optimal shape design problem by the ersatz material approach. Under the piecewise constant level set framework, we first reformulate the two-phase design problem to be a new constrained optimization problem with respect to the piecewise constant level set function. Then we solve it by the projection Lagrangian method. A gradient-type iterative algorithm is presented. Comparisons between our numerical results and those obtained by level set approaches show the effectiveness, accuracy and efficiency of our algorithm.     

Dynamical level set method for parameter identification of nonlinear parabolic distributed parameter systems

This article considers a dynamical level set method for the identification problem of the nonlinear parabolic distributed parameter system, which is based on the solvability and stability of the direct PDE (partial differential equation) in Sobolev space. The dynamical level set algorithms have been developed for ill-posed problems in Hilbert space. This method can be regarded as a asymptotical regularization method as long as a certain stopping rule is satisfied. Hence, the convergence analysis of the method is established similar to the proof of convergence of asymptotical regularization. The level set converges to a solution as the artificial time evolves to infinity. Furthermore, the proposed level set method is proved to be stable by using Lyapunov stability theorem, which is constructed in my previous article.Numerical tests are discussed to demonstrate the efficacy of the dynamical level set method, which consequently confirm the level set method to be a powerful tool for the identification of the parameter.     

An improvement of level set equations via approximation of a distance function

In the classical level set method, the slope of solutions can be very small or large, and it can make it difficult to get the precise level set numerically. In this paper, we introduce an improved level set equation whose solutions are close to the signed distance function to evolving interfaces. The improved equation is derived via approximation of the evolution equation for the distance function. Applying the comparison principle, we give an upper- and lower bound near the zero level set for the viscosity solution to the initial value problem.     

Numerical simulation two phase flows of casting filling process using SOLA particle level set method

Mould filling process is a typical gas–liquid metal two phase flow phenomenon. Numerical simulation of the two phase flows of mould filling process can be used to properly predicate the back pressure effect, the gas entrapment defects, and better understand the complex motions of the gas phase and the liquid phase. In this paper, a novel sharp interface incompressible two phase numerical model for mould filling process is presented. A simple ghost fluid method like discretization method and a density evaluation method at face centers of finite difference staggered grid are proposed to overcome the difficulties when solving two phase Navier–Stokes equations with large-density ratio and large-viscosity ratio. A new mass conservation particle level set method is developed to capture the gas–liquid metal phase interface. The classical pressure-correction based SOLA algorithm is modified to solve the two phase Navier–Stokes equations. Two numerical tests including the Zalesak disk problem and the broken dam problem are used to demonstrate the accuracy of the present method. The numerical method is then adopted to simulate three mould filling examples including two high speed CCD camera imaging water filling experiments and an in situ X-ray imaging experiment of pure aluminum filling. The simulation results are in good agreement with the experiments.     

Using a level set approach for image segmentation under interpolation conditions

In this paper, we propose a new 2D segmentation model including geometric constraints, namely interpolation conditions, to detect objects in a given image. We propose to apply the deformable models to an explicit function using the level set approach (Osher and Sethian [24]); so, we avoid the classical problem of parameterization of both segmentation representation and interpolation conditions. Furthermore, we allow this representation to have topological changes. A problem of energy minimization on a closed subspace of a Hilbert space is defined and introducing Lagrange multipliers enables us to formulate the corresponding variational problem with interpolation conditions. Thus the explicit function evolves, while minimizing the energy and it stops evolving when the desired outlines of the object to detect are reached. The stopping term, as in the classical deformable models, is related to the gradient of the image. Numerical results are given. AMS subject classification 74G65, 46-xx, 92C55     

A new approach to the level function

This article devotes to design a hybrid Level Set Method which compromises the advantages of the level functions designed in [2] and [3].     

A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows

This paper reports a new meshless Integrated Radial Basis Function Network (IRBFN) approach to the numerical simulation of interfacial flows in which the two-way interaction between a moving interface and the ambient viscous flow is fully investigated. When an interface between two immiscible fluids moves, not only its position and shape but also the flow variables (i.e. velocity field and pressure) change due to the presence of surface tension along the moving interface. The velocity field of the ambient flow, on the other hand, causes the interface to move and deform as a result of momentum transport between the two immiscible fluids on both sides of the interface. Numerical investigations of such a two-way interaction is reported in this paper where the level set method is used in combination with high-order projection schemes in the meshless framework of the IRBFN method. Numerical investigations on the meshless projection schemes are performed with typical benchmark incompressible viscous flow problems for verification purposes. The approach is then demonstrated with the numerical simulation of two bubbles moving, stretching and merging in an incompressible ambient fluid under the action of buoyancy force.     

Segmentation under geometrical conditions using geodesic active contours and interpolation using level set methods

Let I: be a given bounded image function, where is an open and bounded domain which belongs to ⁿ. Let us consider n=2 for the purpose of illustration. Also, let S={x_i}_i be a finite set of given points. We would like to find a contour , such that is an object boundary interpolating the points from S. We combine the ideas of the geodesic active contour (cf. Caselles et al. [7,8]) and of interpolation of points (cf. Zhao et al. [40]) in a level set approach developed by Osher and Sethian [33]. We present modelling of the proposed method, both theoretical results (viscosity solution) and numerical results are given. AMS subject classification 49L25, 74G65, 68U10     

Application of the dual active set algorithm to quadratic network optimization

A new algorithm, the dual active set algorithm, is presented for solving a minimization problem with equality constraints and bounds on the variables. The algorithm identifies the active bound constraints by maximizing an unconstrained dual function in a finite number of iterations. Convergence of the method is established, and it is applied to convex quadratic programming. In its implementable form, the algorithm is combined with the proximal point method. A computational study of large-scale quadratic network problems compares the algorithm to a coordinate ascent method and to conjugate gradient methods for the dual problem. This study shows that combining the new algorithm with the nonlinear conjugate gradient method is particularly effective on difficult network problems from the literature.     

A method for constructing a recognition algorithm in algebra over an estimate calculation set

A method is proposed for constructing an algorithm in algebra over an estimate calculation set in an algebraic extension of the least degree.     

变测度的积分-水平集确定性算法

提出了一个求总极值的变测度确定性算法,对不同的箱子采用不同的测度,结合确定性数论方法选取一致分布佳点集来代替Monte-Carlo随机投点,使水平值充分地下降,更快地到达全局最小,从而提高算法的计算效率.在文中给出了算法的收敛性证明,并通过数值算例验证了它的有效性.     

A sparse proximal implementation of the LP dual active set algorithm

We present an implementation of the LP Dual Active Set Algorithm (LP DASA) based on a quadratic proximal approximation, a strategy for dropping inactive equations from the constraints, and recently developed algorithms for updating a sparse Cholesky factorization after a low-rank change. Although our main focus is linear programming, the first and second-order proximal techniques that we develop are applicable to general concave–convex Lagrangians and to linear equality and inequality constraints. We use Netlib LP test problems to compare our proximal implementation of LP DASA to Simplex and Barrier algorithms as implemented in CPLEX. This material is based upon work supported by the National Science Foundation under Grant No. 0203270.     

An approximation method for the efficiency set of multiobjective programming problems

In this paper so-called ε-approximations for the efficiency set of vector minimization problems are defined. A general generating algorithm for such E-approximations is given which will be modified for linear continuous problems by means of the Dual Simplex Method.     

油藏模型特征数值模拟的一个新算法

使用双Lelel set方法来重新表示油藏模型特征,渗透率可以通过解无限制的Lagrangian最优化问题得到.对Lagrangian范函加上Level Set函数的限制之后,鞍点就可以通过算子分裂格式得到.数值模拟表明新算法是稳定高效的.     

Stability and consistency of the semi-implicit co-volume scheme for regularized mean curvature flow equation in level set formulation

We show stability and consistency of the linear semi-implicit complementary volume numerical scheme for solving the regularized, in the sense of Evans and Spruck, mean curvature flow equation in the level set formulation. The numerical method is based on the finite volume methodology using the so-called complementary volumes to a finite element triangulation. The scheme gives the solution in an efficient and unconditionally stable way.     

Asymptotic properties of plug-in level set estimators for right censored data

We assume T1,...,Tn are i.i.d.data sampled from distribution function F with density function f and C1,...,Cn are i.i.d.data sampled from distribution function G.Observed data consists of pairs(Xi,δi),i=1,...,n,where Xi=min{Ti,Ci},δi=I(Ti Ci),I(A)denotes the indicator function of the set A.Based on the right censored data{Xi,δi},i=1,...,n,we consider the problem of estimating the level set{f c}of an unknown one-dimensional density function f and study the asymptotic behavior of the plug-in level set estimators.Under some regularity conditions,we establish the asymptotic normality and the exact convergence rate of theλg-measure of the symmetric difference between the level set{f c}and its plug-in estimator{fn c},where f is the density function of F,and fn is a kernel-type density estimator of f.Simulation studies demonstrate that the proposed method is feasible.Illustration with a real data example is also provided.     

New modifications of the generalized saddle version of the level method

New modifications are proposed for the iterative method that seeks a saddle point of a convex-concave function whose efficient set is contained in the Cartesian product of polyhedra. The convergence rate of these modifications is estimated.     

Adaptive level set evolution starting with a constant function

In this paper, we propose a novel level set evolution model in a partial differential equation (PDE) formulation. According to the governing PDE, the evolution of level set function is controlled by two forces, an adaptive driving force and a total variation (TV)-based regularizing force that smoothes the level set function. Due to the adaptive driving force, the evolving level set function can adaptively move up or down in accordance with image information as the evolution proceeds forward in time. As a result, the level set function can be simply initialized to a constant function rather than the widely-used signed distance function or piecewise constant function in existing level set evolution models. Our model completely eliminates the needs of initial contours as well as re-initialization, and so avoids the problems resulted from contours initialization and re-initialization. In addition, the evolution PDE can be solved numerically via a simple explicit finite difference scheme with a significantly larger time step. The proposed model is fast enough for near real-time segmentation applications while still retaining enough accuracy; in general, only a few iterations are needed to obtain segmentation results accurately.     

A counterexample to the dominating set conjecture

The metric polytope met _n is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities x _ij − x _ik − x _jk ≤ 0 and x _ij + x _ik + x _jk ≤ 2 for all triples i, j, k of {1,..., n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n ≤ 8 and, in particular, for the 1,550,825,600 vertices of met₈. While the overwhelming majority of the known vertices of met₉ satisfy the conjecture, we exhibit a fractional vertex not adjacent to any integral vertex.     

Stopping of functionals with discontinuity at the boundary of an open set

We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set O. The stopping horizon is either random, equal to the first exit from the set O, or fixed (finite or infinite). The payoff function is continuous with a possible jump at the boundary of O. Using a generalization of the penalty method, we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or ε-optimal stopping times.