Discrete Variable Problems Research Articles

Probability collectives for solving discrete and mixed variable problems

The approach of probability collectives (PC) in the collective intelligence (COIN) framework is one of the emerging artificial intelligence tools dealing with the complex problems in a distributed way. It decomposes the entire system into subsystems and treats them as a multi-agent system (MAS). These agents iteratively select their strategies to optimise their local goals which make the system to achieve the global optimum. This paper demonstrates the ability of PC solving discrete as well as mixed variable problems. The approach has produced competent and sufficiently robust results with comparatively higher computational cost. The associated strengths, weaknesses and possible real world extensions are also discussed.

BRANE Cut: biologically-related a priori network enhancement with graph cuts for gene regulatory network inference.

BackgroundInferring gene networks from high-throughput data constitutes an important step in the discovery of relevant regulatory relationships in organism cells. Despite the large number of available Gene Regulatory Network inference methods, the problem remains challenging: the underdetermination in the space of possible solutions requires additional constraints that incorporate a priori information on gene interactions.MethodsWeighting all possible pairwise gene relationships by a probability of edge presence, we formulate the regulatory network inference as a discrete variational problem on graphs. We enforce biologically plausible coupling between groups and types of genes by minimizing an edge labeling functional coding for a priori structures. The optimization is carried out with Graph cuts, an approach popular in image processing and computer vision. We compare the inferred regulatory networks to results achieved by the mutual-information-based Context Likelihood of Relatedness (CLR) method and by the state-of-the-art GENIE3, winner of the DREAM4 multifactorial challenge.ResultsOur BRANE Cut approach infers more accurately the five DREAM4 in silico networks (with improvements from 6 % to 11 %). On a real Escherichia coli compendium, an improvement of 11.8 % compared to CLR and 3 % compared to GENIE3 is obtained in terms of Area Under Precision-Recall curve. Up to 48 additional verified interactions are obtained over GENIE3 for a given precision. On this dataset involving 4345 genes, our method achieves a performance similar to that of GENIE3, while being more than seven times faster. The BRANE Cut code is available at: http://www-syscom.univ-mlv.fr/~pirayre/Codes-GRN-BRANE-cut.html.ConclusionsBRANE Cut is a weighted graph thresholding method. Using biologically sound penalties and data-driven parameters, it improves three state-of-the art GRN inference methods. It is applicable as a generic network inference post-processing, due to its computational efficiency.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-015-0754-2) contains supplementary material, which is available to authorized users.

Complementarity problem arising from static growth of multiple cracks and MLS-based numerical manifold method

Abstract The full response of a brittle structure containing multiple cracks to loading under the servo control is of vital importance in the evaluation of properties of the structure. During crack growth, the fracture toughness condition at all the crack tips as well as the equilibrium condition should be obeyed, leading to a nonlinear complementarity problem (NCP). The vector-valued function in the NCP depends implicitly on the cracking increments which in turn determine the stress field. The stress field can be obtained through solving a mixed variational problem. Since the degrees of freedom in the discrete variational problem vary with cracking not only in magnitude but also in number, the Jacobian matrix of the NCP is hard to compute and, it cannot be expected to be solved by the Newton methods that involve the calculation of Jacobian matrices. Therefore, a well-scaled projection-contraction algorithm is designed. The proposed procedure is able to simulate growth of multiple cracks in a natural way, allowing the crack tips to stop anywhere with no sensitivity to node configuration or cracking increments. Through the analysis of some examples that have been widely tested, many interesting and profound phenomena are found which have never been revealed.

Stabilized mixed approximation of axisymmetric Brinkman flows

This paper is devoted to the numerical analysis of an augmented finite element approximation of the axisymmetric Brinkman equations. Stabilization of the variational formulation is achieved by adding suitable Galerkin least-squares terms, allowing us to transform the original problem into a formulation better suited for performing its stability analysis. The sought quantities (here velocity, vorticity, and pressure) are approximated by Raviart−Thomas elements of arbitrary order k ≥ 0, piecewise continuous polynomials of degree k + 1, and piecewise polynomials of degree k , respectively. The well-posedness of the resulting continuous and discrete variational problems is rigorously derived by virtue of the classical Babuska–Brezzi theory. We further establish a priori error estimates in the natural norms, and we provide a few numerical tests illustrating the behavior of the proposed augmented scheme and confirming our theoretical findings regarding optimal convergence of the approximate solutions.

Two Algorithms for a Class of Elliptic Problems in Shape Optimization

We propose two algorithms for elliptic boundary value problems in shape optimization. With the finite element method, the optimization problem is replaced by a discrete variational problem. We give rules and use them to decide which elements are to be reserved. Those rules are determined by the optimization; as a result, we get the optimal design in shape. Numerical examples are provided to show the effectiveness of our algorithms.

Application of Improved Combined Shape Genetic Algorithm on Structural Optimization

A new approach to the generation of an initial point is proposed for discrete combined shape, which improves fully the local searching capability of discrete combined shape algorithm. Combined shape algorithm is embedded into genetic algorithm as a combined shape operator. Consequently a hybrid genetic algorithm for structural optimization with discrete variables is proposed. The constrained optimization problems were dealt with by adaptive annealing penalty factors and penalty function. The numerical results show that improved combined shape genetic algorithm for structural optimization with discrete variable problems has a faster convergence speed, which has advantages of local searching capability and globally searching capability of genetic algorithm. Improved combined shape genetic algorithm is an efficient optimal design method for engineering structure.

Classifier-guided sampling for discrete variable, discontinuous design space exploration: Convergence and computational performance

A classifier-guided sampling (CGS) method is introduced for solving engineering design optimization problems with discrete and/or continuous variables and continuous and/or discontinuous responses. The method merges concepts from metamodel-guided sampling and population-based optimization algorithms. The CGS method uses a Bayesian network classifier for predicting the performance of new designs based on a set of known observations or training points. Unlike most metamodelling techniques, however, the classifier assigns a categorical class label to a new design, rather than predicting the resulting response in continuous space, and thereby accommodates non-differentiable and discontinuous functions of discrete or categorical variables. The CGS method uses these classifiers to guide a population-based sampling process towards combinations of discrete and/or continuous variable values with a high probability of yielding preferred performance. Accordingly, the CGS method is appropriate for discrete/discontinuous design problems that are ill suited for conventional metamodelling techniques and too computationally expensive to be solved by population-based algorithms alone. The rates of convergence and computational properties of the CGS method are investigated when applied to a set of discrete variable optimization problems. Results show that the CGS method significantly improves the rate of convergence towards known global optima, on average, compared with genetic algorithms.

Shilnikov lemma for a nondegenerate critical manifold of a Hamiltonian system

We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic critical manifold $M\subset H^{-1}(0)$ of a Hamiltonian system. Using this result, trajectories with small energy $H=\mu>0$ shadowing chains of homoclinic orbits to $M$ are represented as extremals of a discrete variational problem, and their existence is proved. This paper is motivated by applications to the Poincar\'e second species solutions of the 3 body problem with 2 masses small of order $\mu$. As $\mu\to 0$, double collisions of small bodies correspond to a symplectic critical manifold of the regularized Hamiltonian system.

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

This paper deals with the numerical approximation of the stationary two-dimensional Stokes equations, formulated in terms of vorticity, velocity and pressure, with non-standard boundary conditions. Here, by introducing a Galerkin least-squares term, we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem. We propose two families of mixed finite elements to solve the discrete problem, in the first family, the unknowns are approximated by piecewise continuous and quadratic elements, Brezzi–Douglas–Marini, and piecewise constant finite elements, respectively, while in the second family, the unknowns are approximated by piecewise linear and continuous, Raviart–Thomas, and piecewise constant finite elements, respectively. The wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the Babuška–Brezzi theory. We establish a priori error estimates in the natural norms, and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes. Additional examples of cases not covered by our theory are also presented.

Higher-order discrete variational problems with constraints

An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this paper, we derive new variational integrators for higher-order Lagrangian mechanical system subjected to higher-order constraints. From the discretization of the variational principles, we show that our methods are automatically symplectic and, in consequence, with a very good energy behavior. Additionally, the symmetries of the discrete Lagrangian imply that momentum is conserved by the integrator. Moreover, we extend our construction to variational integrators where the Lagrangian is explicitly time-dependent. Finally, some motivating applications of higher-order problems are considered; in particular, optimal control problems for explicitly time-dependent underactuated systems and an interpolation problem on Riemannian manifolds.

Discrete Variational Calculus for B-Spline Curves

In this work, we study discrete variational problems, for B-spline curves, which are invariant under translation and rotation. We show this approach has advantages over studying smooth variational problems whose solutions are approximated by B-spline curves. The latter method has been well studied in the literature but leads to high order approximation problems. We are particularly interested in Lagrangians that are invariant under the special Euclidean group for which B-spline approximated curves are well suited. The main application we present here is the curve completion problem in 2D and 3D. Here, the aim is to find various aesthetically pleasing solutions as opposed to a solution of a physical problem. Smooth Lagrangians with special Euclidean symmetries involve curvature, torsion, and arc length. Expressions of these, in the original coordinates, are highly complex. We show that, by contrast, relatively simple discrete Lagrangians offer excellent results for the curve completion problem. The novel methods we develop for the discrete curve completion problem are general, and can be used to solve other discrete variational problems for B-spline curves. Our method completely avoids the difficulties of high order smooth differential invariants.

A Fully-Discrete Finite Element Approximation for the Eddy Currents Problem

The eddy current model is obtained from Maxwell’s equations by neglecting the displacement currents in the Amp`ere-Maxwell’s law and it is commonly used in many problems in sciences, engineering and industry (e.g, in induction heating, electromagnetic braking, and power transformers). The so-called “A, V −A potential formulation” (B´ır´o & Preis [1]) is nowadays one of the most accepted formulations to solve the eddy current equations numerically, and B´ır´o & Valli [2] have recently provided its well-posedness and convergence analysis for the time-harmonic eddy current problem. The aim of this paper is to extend the analysis performed by B´ır´o & Valli to the general transient eddy current model. We provide a backward-Euler fully-discrete approximation based on nodal finite elements and we show that the resulting discrete variational problem is well posed. Furthermore, error estimates that prove optimal convergence are settled.

Symmetry and asymmetry of solutions in discrete variable structural optimization

In this paper symmetry and asymmetry of optimal solutions in symmetric structural optimization problems is investigated, based on the choice of variables. A group theory approach is used to define the symmetry of the structural problems in a general way. This approach allows the set of symmetric structures to be described and related to the entire search space of the problem. A relationship between the design variables and the likelihood of finding symmetric or asymmetric solutions to problems is established. It is shown that an optimal symmetric solution (if any) does not necessarily exist in the case of discrete variable problems, regardless of the size of the discrete, countable set from which variables can be chosen. Finally a number of examples illustrate these principles on simple truss structures with discrete topology and sizing variables.

Sequential approximate optimization for discrete design variable problems using radial basis function network

This paper proposes a sequential approximate optimization (SAO) for discrete design variable problems using radial basis function (RBF) network. We assume that there are two important factors for successful SAO: one is parameter adjustment for good approximation, and the other is to find the unexplored regions for global approximation. The authors propose a simple estimate of the width in the Gaussian kernel for good approximation. In addition, in order to find the unexplored region, we develop a density function that, with the simple estimate of the width, works well in the case of continuous design variables. However, a simple application of the density function to discrete design variables often leads to the wrong result. In order to find the unexplored region of the discrete design variables with our density function, the permutation number is introduced. The density function with the permutation number can find out the unexplored region. As the result, the discrete optimum can find with a small number of function evaluations. The validity of proposed approach is examined by studying typical numerical examples.

Discrete variable structural optimization based on multidirectional fuzzy genetic algorithm

Round method is the common method for discrete variable optimization in optimal design of complex mechanical structures; however, it has some disadvantages such as poor precision, simple model and lacking of working conditions’ description, etc. To solve these problems, a new model is constructed by defining parameterized fuzzy entropy, and the rationality of parameterized fuzzy entropy is verified. And a new multidirectional searching algorithm is further put forward, which takes information of actual working conditions into consideration and has a powerful local searching capability. Then this new algorithm is combined with the GA by the fuzzy clustering algorithm (FCA). With the application of FCA, the optimal solution can be effectively filtered so as to retain the diversity and the elite of the optimal solution, and avoid the structural re-analysis phenomenon between the two algorithms. The structure design of a high pressure bypass-valve body is used as an example to make a structural optimization by the proposed HGA and finite element method (FEM), respectively. The comparison result shows that the improved HGA fully considers the characteristic of discrete variable and information of working conditions, and is more suitable to the optimal problems with complex working conditions. Meanwhile, the research provides a new approach for discrete variable structure optimization problems.

Sequential Approximate Optimization Using Radial Basis Function Network (Application to the Discrete Design Variable Problems)

This paper proposes a Sequential Approximate Optimization (SAO) for the discrete design variable problems using Radial Basis Function (RBF) Network. We consider that there are two important keys for SAO: One is the parameter adjustment for good approximation, and the other is to explore the sparse region for global approximation. The authors have proposed the simple estimate of the width in the Gaussian Kernel for good approximation. In addition, in order to explore the sparse region, we have developed the density function. The density function with the simple estimate of the width works well in the case of the continuous design variables. However, a simple application of the density function to the discrete design variable problems will cause some difficulties. In order to overcome these difficulties and find the sparse region of the discrete design variables with the density function, the new variables for the discrete design variables are introduced. By using the new variables, it is possible to find the sparse region of the discrete design variables. A simple sampling algorithm is shown, in which the Discrete Differential Evolution (DDE) for the discrete design variables is employed. Through typical numerical examples, the validity of proposed approach is examined.

Invariant Manifolds for Analytic Difference Equations

We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, and smooth dependence on parameters and study several singular limits, even if the difference equations do not define a dynamical system. This method also leads to efficient algorithms that we present with their implementations. The manifolds that we consider include not only the classical strong stable and unstable manifolds but also manifolds associated with nonresonant spaces. When the difference equations are the Euler--Lagrange equations of a discrete variational problem, we have sharper results. Note that, if the Legendre condition fails, the Euler--Lagrange equations cannot be treated as a dynamical system. If the Legendre condition becomes singular, the dynamical system may be singular while the difference equation remains regular. We present numerical applications to several examples in the physics literature: the Frenkel--Kontorova model with long-range ...

First variation formula and conservation laws in several independent discrete variables

This paper sets the scene for discrete variational problems on an abstract cellular complex that serves as discrete model of R p and for the discrete theory of partial differential operators that are common in the Calculus of Variations. A central result is the construction of a unique decomposition of certain partial difference operators into two components, one that is a vector bundle morphism and other one that leads to boundary terms. Application of this result to the differential of the discrete Lagrangian leads to unique discrete Euler and momentum forms not depending either on the choice of reference on the base lattice or on the choice of coordinates on the configuration manifold, and satisfying the corresponding discrete first variation formula. This formula leads to discrete Euler equations for critical points and to exact discrete conservation laws for infinitesimal symmetries of the Lagrangian density, with a clear physical interpretation.

First variation formula for discrete variational problems in two independent variables

We introduce a geometric setup for discrete variational problems in two independent variables based on the theory of bundles modelled on cell complexes, and characterize geometrically a first variation formula and a discrete Noether theorem for symmetries of the discrete Lagrangian. We explore the existence of discrete variational integrators, which will conserve the discrete Noether currents in the theory. The range of applications of the theory includes discrete versions of variational problems with or without PDE constraints and energy-conserving integrators in mechanics of continua. We illustrate the theory with a variational integrator for the movement of a string.

Development of Optimization Design Software for Bevel Gear Based on Integer Serial Number Encoding Genetic Algorithm

Bevel gear drive is widely used, quality of which not only affects its own transmission performance, size and weight, but also has some impact on the machine's performance. This paper introduces optimization design software for bevel gear, in which automatic optimization design is realized. In the paper mathematical model, programming of design data and realization of optimization design based on genetic algorithm are described in detail. The paper proposed integer serial number encoding genetic algorithm, which effectively deals with continuous and discrete variable optimization problem and reduces the code length of the string to improve the encoding and decoding efficiency, no invalid solution or duplicate solutions.