Discrete versions of continuous isoperimetric problems

For a general family of graphs on $\mathbb{Z}^n$, we translate the edge-isoperimetric problem into a continuous isoperimetric problem in $\mathbb{R}^n$. We then solve the continuous isoperimetric problem using the Brunn-Minkowski inequality and Minkowski's theorem on mixed volumes. This translation allows us to conclude, under a reasonable assumption about the discrete problem, that the shapes of the optimal sets in the discrete problem approach the shape of the optimal set in the continuous problem as the size of the set grows. The solution is the zonotope defined as the Minkowski sum of the edges of the original graph. We demonstrate the efficacy of this method by revisiting some previously solved classical edge-isoperimetric problems. We then apply our method to some discrete isoperimetric problems which had not previously been solved. The complexity of those solutions suggest that it would be quite difficult to find them using discrete methods only.

The investigation of the problem of integrability of polynomial-nonlinear evolution equations, in particular, verifying the existence of the higher symmetries and conservation laws can often be reduced to the problem of finding the exact solution of a complicated system of nonlinear algebraic equations. It is remarkable that these algebraic equations can be not only obtained completely automatically by computer [1] but also often not only completely solved by computer, in spite of their complicated structure and often infinitely many solutions.We demonstrate this fact using the Groebner basis method [2] and obtain all (infinitely many) solutions of the systems of algebraic equations which are equivalent to integrability of three different multiparametric families of NLEEs [1]: the seventh order scalar KdV-like equations, the seventh order MKdV-like equations, and the third order coupled KdV-like systems.All our computations have been carried out by using the computer algebra system REDUCE (version 3.2) on an IBM PC AT-like computer. Because of the fact that the computer algebra system REDUCE 3.2 (in particular on IBM PC, and unlike REDUCE 3.3), has no built-in package for computation of Groebner basis, we have written our own program in Rlisp in order to solve systems of algebraic equations using Buchberger's algorithm [2]. To make the program effecient we have used the distributive form for the internal representation of polynomials together with multivariate factorization. In order to obtain the (infinitely many) solutions, we construct a lexicographic Groebner basis, then we compute, according to [3], the dimension and independent sets of variables for the ideal which is generated by the input system.Thereafter, we consider each set of variables as free parameters and compute a Groebner basis leaving the order of the others unchanged. As a result we obtain a set of Groebner bases with a simple structure, and the solution can be found in an easy way.Our analysis shows that the Groebner basis method allows us to obtain the complete set of exact solutions for systems of nonlinear algebraic equations which are the necessary integrability conditions for NLEEs and therefore to select all integrable evolution equations. It is clear that the solvability of the above systems and of even more complicated ones is closely connected with the property of integrability. In addition to their importance in the theory and application of NLEEs, such systems are very useful for testing different computer algebra algorithms. One of our system is in a list of examples for Groebner basis computations [4].

The reader has been told that the great twentieth-century quests in the calculus of variations have involved existence and multiplier rules. Progress in functional analysis, together with the direct method, has largely resolved the former issue; we turn now to that of multipliers. We consider the classical problem of Lagrange for this purpose. It consists of the basic problem (P) to which has been grafted an additional pointwise equality constraint φ(t,x,x ′) = 0. The additional constraint makes this problem much more complex than (P), or even the isoperimetric problem. In part, this is because we now have infinitely many constraints, one for each t. Given our experience in optimization, we expect the multiplier rule to assert, in a now familiar pattern, that if x ∗ solves this problem, then there exist multipliers η,λ, not both zero, with η= 0 or 1, such that x ∗ satisfies the necessary conditions for the Lagrangian ηΛ+λ(t)φ. Note that λ is a function of t here, which is to be expected, since there is a constraint φ(t,x,x ′) = 0 for each t.KeywordsRank ConditionConstraint QualificationIsoperimetric ProblemMultiplier RuleFamiliar PatternThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We introduce the discrete counterpart of the vakonomic method in Lagrangian mechanics with non-holonomic constraints. After defining the concepts of “admissible section” and “admissible infinitesimal variation” of a discrete vakonomic system, we aim to determinate those admissible sections that are critical for the Lagrangian of the system with respect to admissible infinitesimal variations. For sections that satisfy a certain regularity condition, we prove that critical sections are extremals of a variational problem without constraints canonically associated to the initial system (Lagrange multiplier rule). We introduce a notion of “constrained variational integrator”, which is characterized by a Cartan equation that ensures its simplecticity. Moreover, under certain regularity conditions we prove that these integrators can be locally constructed from a generating function of the second kind in the sense of symplectic geometry. Finally, the theory is illustrated with two elementary examples: an isoperimetric problem and an optimal control problem.

7 - Numerical Solution of Finite Element Equations

Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases

In this paper, we applied a direct method for a solution of isoperimetric fractional variational problems. We use shifted Legendre orthonormal polynomials as basis function of operational matrices of fractional differentiation and fractional integration in combination with the Lagrange multipliers technique for converting such isoperimetric fractional variational problems into solving a system of algebraic equations. Also, we show the convergence analysis of the presented technique and introduce some test problems with comparisons between our numerical results with those introduced using different methods.

In geodesy and geoinformatics, most problems are nonlinear in nature and often require the solution of systems of polynomial equations. Before 2002, solutions of such systems of polynomial equations, especially of higher degree remained a bottleneck, with iterative solutions being the preferred approach. With the entry of Groebner basis as algebraic solution to nonlinear systems of equations in geodesy and geoinformatics in the pioneering work “Grobner bases, multipolynomial resultants and the Gauss Jacobi combinatorial algorithms : adjustment of nonlinear GPS/LPS observations”, the playing field changed. Most of the hitherto unsolved nonlinear problems, e.g., coordinate transformation, global navigation satellite systems (GNSS)’s pseudoranges, resection-intersection in photogrammetry, and most recently, plane fitting in point clouds in laser scanning have been solved. A comprehensive overview of such applications are captured in the first and second editions of our book Algebraic Geodesy and Geoinformatics published by Springer. In the coming third edition, an updated summary of the newest techniques and methods of combination of Groenbner basis with symbolic as well as numeric methods will be treated. To quench the appetite of the reader, this presentation considers an illustrative example of a two-dimension coordinate transformation problem solved through the combination of symbolic regression and Groebner basis.

PREFACE. INTRODUCTION. Generalities. Ordinary differential equations. Supplementary conditions associated to ODEs. The Cauchy (initial) problem.The two-point problem. 1: LINEAR ODEs OF FIRST AND SECOND ORDER. 1.1 Linear first order ODEs. 1.1.1 Equations of the form . 1.1.2 The linear homogeneous equation. 1.1.3 The general case. 1.1.4 The method of variation of parameters (Lagrange's method). 1.1.5 Differential polynomials. 1.2 Linear second order ODEs. 1.2.1 Homogeneous equations. 1.2.2 Non-homogeneous equations. Lagrange's method. 1.2.3 ODEs with constant coefficients. 1.2.4 Order reduction. 1.2.5 The Cauchy problem. Analytical methods to obtain the solution. 1.2.6 Two-point problems (Picard). 1.2.7 Sturm-Liouville problems. 1.2.8 Linear ODEs of special form. 1.3. Applications 2: LINEAR ODEs OF HIGHER ORDER (n >2). 2.1 The general study of linear ODEs of order . 2.1.1 Generalities. 2.1.2 Linear homogeneous ODEs. 2.1.3 The general solution of the non-homogeneous ODE. 2.1.4 Order reduction. 2.2 Linear ODEs with constant coefficients. 2.2.1 The general solution of the homogeneous equation. 2.2.2 The non-homogeneous ODE. 2.2.3 Euler type ODEs. 2.3 Fundamental solution. Green function. 2.3.1 The fundamental solution. 2.3.2 The Green function. 2.3.3 The non-homogeneous problem. 2.3.4 The homogeneous two-point problem. Eigenvalues. 2.4 Applications. 3: LINEAR ODSs OF FIRST ORDER. 3.1 The general study of linear first order ODSs. 3.1.1 Generalities. 3.1.2 The general solution of the homogeneous ODS. 3.1.3 The general solution of the non-homogeneous ODS. 3.1.4 Order reduction of homogeneous ODSs. 3.1.5 Boundary value problems for ODSs. 3.2 ODSs with constant coefficients. 3.2.1 The general solution of the homogeneous ODS. 3.2.2 Solutions in matrix form for linear ODSs with constant coefficients. 3.3 Applications. 4: NON-LINEAR ODEs OF FIRST AND SECOND ORDER. 4.1 First order non-linear ODEs. 4.1.1 Forms of first order ODEs and oftheir solutions. 4.1.2 Geometric interpretation. The theorem of existence and uniqueness. 4.1.3 Analytic methods for solving first order non-linear ODEs. 4.1.4. First order ODEs integrable by quadratures. 4.2 Non-linear second order ODEs. 4.2.1 Cauchy problems. 4.2.2 Two-point problems. 4.2.3 Order reduction of second order ODEs. 4.2.4 The Bernoulli-Euler equation. 4.2.5 Elliptic integrals. 4.3 Applications. 5: NON-LINEAR ODSs OF FIRST ORDER. 5.1 Generalities. 5.1.1 The general form of a first order ODS. 5.1.2 The existence and uniqueness theorem for the solution of the Cauchy problem. 5.1.3 The particle dynamics. 5.2 First integrals of an ODS. 5.2.1 Generalities. 5.2.2 The theorem of conservation of the kinetic energy. 5.2.3 The symmetric form of an ODS. Integral combinations. 5.2.4 Jacobi's multiplier. The method of the last multiplier. 5.3 Analytical methods of solving the Cauchy problem for non-linear ODSs. 5.3.1 The method of successive approximations (Picard-Lindeloff). 5.3.2 The method of the Taylor series expansion. 5.3.3 The linear equivalence method (LEM). 5.4 Applications. 6: VARIATIONAL CALCULUS. 6.1 Necessary condition of extremum for functionals of integral type. 6.1.1 Generalities. 6.1.2 Functionals of the form..... 6.1.3 Functionals of the form..... 6.1.4 Functionals of integral type, depending on n functions. 6.2 Conditional extrema. 6.2.1 Isoperimetric problems. 6.2.2 Lagrange's problem. 6.3 Applications. 7: STABILITY. 7.1 Lyapunov Stability. 7.1.1 Generalities. 7.1.2 Lyapunov's theorem of stability. 7.2 The stability of the solutions of dynamical systems. 7.2.1 Autonomous dynamical systems. 7.2.2 Long term behaviour of the solutions. 7.3 Applications. INDEX. REFERENCES.

This paper considers the formulation of a discrete inverse problem for a hyperbolic equation. First, the continuous inverse problem is reduced to a convenient form for research. In the inverse problem, the required function is considered even. Since the Dirac delta function is present in the problem data, the structure of a generalized solution to the Cauchy problem for a hyperbolic equation is determined. The solution to the Cauchy problem for a hyperbolic equation is determined only for positive values in time, therefore the solution to the Cauchy problem for negative values in time is determined using odd continuation. After some transformations, the formulation of the continuous inverse problem is reduced to a form convenient for research. A grid domain is introduced, and for all functions in the problem statement the corresponding grid functions and a discrete analogue of the Dirac delta function are determined. Differential operators, initial conditions and additional data of the inverse problem are approximated by finite differences. Assuming that a solution to the discrete inverse problem exists, we prove the data lemma of the discrete inverse problem. In order to study the discrete inverse problem for a hyperbolic equation, a theorem on the existence and uniqueness of the discrete direct problem is proved, as well as on the properties of the solution to this discrete problem. In the course of proving the theorem, a discrete analogue of d'Alembert's formula for solving the Cauchy problem for a hyperbolic equation was obtained. The theorem on the existence of a unique solution to the auxiliary discrete problem and its properties is proved.

Optimizing production planning has tremendous economic impact for many industrial production systems. In this paper, the planning problem of a class of production systems with hybrid dynamics and constraints is considered with practical background of power generation planning and other applications. The problem is solved within the Lagrangian relaxation framework, with the system wide demand and resource limit constraints relaxed by Lagrange multipliers. A new method is developed in this paper to obtain the exact optimal solutions to the subproblems with hybrid dynamics and constraints efficiently without discretizing the continuous production levels or introducing intermediate levels of relaxation. A novel definition of the discrete state associated with a consecutive time span is introduced so that solving each subproblem is converted into solving a number of continuous optimization problems and a discrete optimization problem separately. An efficient double dynamic programming (DP) method is developed to solve these subproblems and the principle of optimality is guaranteed for both the continuous and discrete problem. The production levels in a consecutive running span with non-convex piecewise linear cost functions are determined in a DP forward sweep without discretization. The DP method is then applied to determine the optimal discrete operating states across time efficiently. Numerical testing results demonstrate that the new method is efficient and effective for optimization based production planning with the complex hybrid dynamics and constraints.

Due to the complexity of the multi constraint and multi closed-loop of parallel mechanisms (PMs), their dynamic equations require additional Lagrangian multipliers and constraint equations, which is unfavorable for solving the dynamic equations. In this paper, a multibody elastodynamic modeling of PMs based on the Lagrangian equations is proposed to fix this issue. First, we decompose the closed-loop PMs into open-loop ones by cutting open at the joints, and we propose the subassembly element that considers the flexible links and joints to avoid the singularity of the traditional constraint equations. Second, we extract the nonsingular independent coordinates for these nodes that are constrained by joints and rigid element through multi-point constraint theory and singularity analysis, and then summarized to establish the global independent generalized displacement coordinates (IGDCs). Third, we formulate the Lagrangian equations through the global IGDCs by closing the open-loop of the mechanism. The Guyuan, improved reduction system, subspace iteration method, and precision integration method are respectively presented to analyze the natural frequencies and dynamic response of the mechanism. The proposed method is computationally efficient because the computation concerning the Lagrangian multipliers are not required. Finally, the 3PRRR PM is presented to implement the proposed method.

Some human-machine systems are designed so that machines (robots) gather and deliver data to remotely located operators (humans) through an interface to aid them in classification. The performance of a human as a (binary) classifier-in-the-loop is characterized by probabilities of correctly classifying objects (or points of interest) as a true target or a false target. These two probabilities depend on the time spent collecting information at a point of interest (POI), known as dwell time. The information gain associated with collecting information at a POI is then a function of dwell time and discounted by the revisit time, i.e., the duration between consecutive revisits to the same POI, to ensure that the vehicle covers all POIs in a timely manner. The objective of the routing problem for classification is to route the vehicles optimally, which is a discrete problem, and determine the optimal dwell time at each POI, which is a continuous optimization problem, to maximize the total discounted information gain while visiting every POI at least once. Due to the coupled discrete and continuous problem, which makes the problem hard to solve, we make a simplifying assumption that the information gain is discounted exponentially by the revisit time; this assumption enables one to decouple the problem of routing with the problem of determining optimal dwell time at each POI for a single vehicle problem. For the multi-vehicle problem, since the problem involves task partitioning between vehicles in addition to routing and dwell time computation, we provide a fast heuristic to obtain high-quality feasible solutions.

Swarm intelligence (SI) is widely applied for optimizing both continuous and discrete problems. Many papers have investigated them for continuous optimizations since most swarm-based algorithms are designed based on continuous movements, which are simply calculated using vector-based mathematical operations. It is quite easy to select the best SI algorithm for a given continuous problem. However, it is quite hard to pick an optimum SI algorithm for a discrete problem since the individual movement is difficult to develop. Therefore, in this paper, three SI algorithms: particle swarm optimization (PSO), firefly algorithm (FA), and bat algorithm (BA), are compared to solve some cases of traveling salesman problem (TSP). Evaluation on four TSP cases show that FA is the most effective and efficient since it dynamically evolves some individuals' groups and balances the exploitative-explorative movements.

In this paper we consider singular perturbation problems for ordinary differential operators of ordern and their discrete counterparts on arbitrary nonuniform grids. We prove that the singularly perturbed initial value problem is stable uniformly in the perturbation parameter ? in both the continuous and the discrete case. We use this result to characterize the stability of the corresponding continuous and discrete boundary value problems. If the continuous problem is stable and if the consistency error is smaller than a certain constant, the discrete problem is also stable.