Discontinuous Galerkin finite element heterogeneous multiscale method for advection–diffusion problems with multiple scales

A discontinuous Galerkin finite element heterogeneous multiscale method is proposed for advection–diffusion problems with highly oscillatory coefficients. The method is based on a coupling of a discontinuous Galerkin discretization for an effective advection–diffusion problem on a macroscopic mesh, whose a priori unknown data are recovered from micro finite element calculations on sampling domains within each macro element. The computational work involved is independent of the high oscillations in the problem at the smallest scale. The stability of our method (depending on both macro and micro mesh sizes) is established for both diffusion dominated and advection dominated regimes without any assumptions about the type of heterogeneities in the data. Fully discrete a priori error bounds are derived for locally periodic data. Numerical experiments confirm the theoretical error estimates.     

A Barzilai-Borwein-based heuristic algorithm for locating multiple facilities with regional demand

We are interested in locations of multiple facilities in the plane with the aim of minimizing the sum of weighted distance between these facilities and regional customers, where the distance between a facility and a regional customer is evaluated by the farthest distance from this facility to the demand region. By applying the well-known location-allocation heuristic, the main task for solving such a problem turns out to solve a number of constrained Weber problems (CWPs). This paper focuses on the computational contribution in this topic by developing a variant of the classical Barzilai-Borwein (BB) gradient method to solve the reduced CWPs. Consequently, a hybrid Cooper type method is developed to solve the problem under consideration. Preliminary numerical results are reported to verify the evident effectiveness of the new method.     

The existence of multiple positive solutions for a?class of?semipositone Dirichlet boundary value problems

In this paper, we consider the existence of multiple positive solutions for the following singular semipositone Dirichlet boundary value problem: $$\left\{\begin{array}{l}-x''(t)=p(t)f(t, x) +q(t),\quad t\in(0,1),\\[4pt]x(0) =0,\qquad x(1) = 0,\end{array}\right.$$ where p:(0,1)??[0,+??) and f:[0,1]×[0,+??)??[0,+??) are continuous, q:(0,1)??(???,+??) is Lebesgue integrable. Under certain local conditions and superlinear or sublinear conditions on f, by using the fixed point theorem, some sufficient conditions for the existence of multiple positive solutions are established for the case in which the nonlinearity is allowed to be sign-changing.     

A Littlewood–Richardson rule for Macdonald polynomials

Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric Macdonald polynomials are a common generalization of Schur functions, Macdonald spherical functions, and Jack polynomials. We use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine Hecke algebra. From this, we obtain a product formula for Macdonald polynomials of general Lie type.     

Speeding up Martins’ algorithm for multiple objective shortest path problems

The latest transportation systems require the best routes in a large network with respect to multiple objectives simultaneously to be calculated in a very short time. The label setting algorithm of Martins efficiently finds this set of Pareto optimal paths, but sometimes tends to be slow, especially for large networks such as transportation networks. In this article we investigate a number of speedup measures, resulting in new algorithms. It is shown that the calculation time to find the Pareto optimal set can be reduced considerably. Moreover, it is mathematically proven that these algorithms still produce the Pareto optimal set of paths.     

A method of adding–removing knots for solving smoothing problems with obstacles

We study a method of adding–removing knots that has been proposed in the literature for solving the smoothing problem with obstacles. The method uses the coefficients of natural splines in the expansion by radial basis functions. We present examples of cycling and counterexamples to possible use of some ideas. We also give some sufficient conditions for finiteness of the method.     

A Littlewood–Richardson rule for two-step flag varieties

This paper studies the geometry of one-parameter specializations of subvarieties of Grassmannians and two-step flag varieties. As a consequence, we obtain a positive, geometric rule for expressing the structure constants of the cohomology of two-step flag varieties in terms of their Schubert basis. A corollary is a positive, geometric rule for computing the structure constants of the small quantum cohomology of Grassmannians. We also obtain a positive, geometric rule for computing the classes of subvarieties of Grassmannians that arise as the projection of the intersection of two Schubert varieties in a partial flag variety. These rules have numerous applications to geometry, representation theory and the theory of symmetric functions. Mathematics Subject Classification (2000)  Primary 14M15, 14N35, 32M10     

Nonresonance conditions for generalised ?-Laplacian problems with jumping nonlinearities

We consider the boundary value problem

(0.1)     

A fourth–order orthogonal spline collocation method for two-dimensional Helmholtz problems with interfaces

Orthogonal spline collocation is implemented for the numerical solution of two-dimensional Helmholtz problems with discontinuous coefficients in the unit square. A matrix decomposition algorithm is used to solve the collocation matrix system at a cost of O(N² log N) on an N × N partition of the unit square. The results of numerical experiments demonstrate the efficacy of this approach, exhibiting optimal global estimates in various norms and superconvergence phenomena for a broad spectrum of wave numbers.     

A concrete example with multiple limit cycles for three dimensional Lotka–Volterra systems

The number of limit cycles for three dimensional Lotka–Volterra systems is an open problem. Recently, Yu et al. (2016) constructed some examples with the possibility of the existence of four limit cycles. Unfortunately, multiple limit cycles are not visible by numerical simulations, because all of them are very close to the interior equilibrium and extremely small. We present a concrete example with multiple limit cycles for three dimensional Lotka–Volterra systems which we can confirm them by numerical simulations. First we prepare the modified formula to compute coefficients of the normal form for the generalized Hopf bifurcation. Applying this formula to three dimensional Lotka–Volterra competitive systems with the aid of the computer algebra system, we derive the critical parameter values explicitly such that the interior equilibrium is exactly an unstable weak focus. Also we show that the heteroclinic cycle on the boundary of ${\mathbb{R}}_{+}^3$ is repelling. This implies that there exists a stable limit cycle by the Poincare–Bendixson theorem. Then, adding some suitable perturbations to parameters, we generate additional two limit cycles near the interior equilibrium by the generalized Hopf bifurcation. Finally we confirm that there exist three limit cycles by numerical simulations.     

Regularization method with two parameters for nonlinear ill-posed problems

This paper is devoted to the regularization of a class of nonlinear ill-posed problems in Banach spaces. The operators involved are multi-valued and the data are assumed to be known approximately. Under the assumption that the original problem is solvable, a strongly convergent approximation procedure is designed by means of the Tikhonov regularization method with two pa- rameters.     

Inverse problems for radial Schr?dinger operators with the missing part of eigenvalues

We study inverse spectral problems for radial Schr?dinger operators in L²(0, 1). It is well known that for a radial Schr?dinger operator, two spectra for the different boundary conditions can uniquely determine the potential. However, if the spectra corresponding to the radial Schr?dinger operators with the two potential functions miss a finite number of eigenvalues, what is the relationship between the two potential functions?Inspired by Hochstadt(1973)’s work, which handled the Stur...     

Heuristics and reduction methods for multiple constraints 0–1 linear programming problems

This work extends the efficient results relative to the 0–1 knapsack problem to the multiple inequality constraints 0–1 linear programming problems. The two crucial phases for the solving of this type of problems are presented: (i) Two linear expected time complexity greedy algorithms are proposed for the determination of a lower bound on the optimal value by using a cascade of surrogate relaxations of the original problem whose sizes are decreasing step by step. A comparative study with the best known heuristic methods is reported; it concerned the accuracy of the approximate solutions and the practical computational times. (ii) This greedy algorithm is inserted in an efficient reduction framework. Variables and constraints are eliminated by the conjunction of tests applied to Lagrangean and surrogate relaxations of the original problem. A lot of computational results are summarized by considering test problems of the literature.     

Multiple solutions for Neumann and periodic problems with singular ?-Laplacian

We use the critical point theory for convex, lower semicontinuous perturbations of C¹-functionals to establish existence of multiple radial solutions for some one parameter Neumann problems involving the operator . Similar results for periodic problems are also provided.     

A fast ‘Monte-Carlo cross-validation’ procedure for large least squares problems with noisy data

Summary We propose a fast Monte-Carlo algorithm for calculating reliable estimates of the trace of the influence matrixA involved in regularization of linear equations or data smoothing problems, where is the regularization or smoothing parameter. This general algorithm is simply as follows: i) generaten pseudo-random valuesw ₁, ...,w _n , from the standard normal distribution (wheren is the number of data points) and letw=(w ₁, ...,w _n ) ^T , ii) compute the residual vectorw–A w, iii) take the normalized inner-product (w ^T (w–A w))/(w ^T w) as an approximation to (1/n)tr(I–A ). We show, both by theoretical bounds and by numerical simulations on some typical problems, that the expected relative precision of these estimates is very good whenn is large enough, and that they can be used in practice for the minimization with respect to of the well known Generalized Cross-Validation (GCV) function. This permits the use of the GCV method for choosing in any particular large-scale application, with only a similar amount of work as the standard residual method. Numerical applications of this procedure to optimal spline smoothing in one or two dimensions show its efficiency.     

A smoothing sample average approximation method for stochastic optimization problems with CVaR risk measure

This paper is concerned with solving single CVaR and mixed CVaR minimization problems. A CHKS-type smoothing sample average approximation (SAA) method is proposed for solving these two problems, which retains the convexity and smoothness of the original problem and is easy to implement. For any fixed smoothing constant ε, this method produces a sequence whose cluster points are weak stationary points of the CVaR optimization problems with probability one. This framework of combining smoothing technique and SAA scheme can be extended to other smoothing functions as well. Practical numerical examples arising from logistics management are presented to show the usefulness of this method.     

On Descartes’ rule for polynomials with two variations of signs

Lithuanian Mathematical Journal - For sequences of d + 1 signs + and ? beginning with a + and having exactly two variations of sign, we give some sufficient conditions for the (non)existence...