“Splashes” in Fredholm integro-differential equations with rapidly varying kernels

We consider a singularly perturbed Fredholm integro-differential equation with a rapidly varying kernel. We derive an algorithmfor constructing regularized asymptotic solutions. It is shown that, given a rapidly decreasing multiplier of the kernel, the original problem does no involve the spectrum (i.e., it is solvable for any right-hand side).     

Regularized asymptotics of solutions of systems of Fredholm integro-differential equations with rapidly varying kernels

We consider a system of singularly perturbed Fredholm integro-differential equations with rapidly varying kernel and develop an algorithm for constructing regularized asymptotic solutions. It is shown that, in the presence of a rapidly decaying factor multiplying the kernel, the original problem is not on the spectrum (i.e., is solvable for any right-hand side). To prove this, we obtain and use a representation of the resolvent (for sufficiently small ? > 0) in the form of a function series that is uniformly convergent in the ordinary sense.     

Substantiation of a Theorem on Limit Transition in Singularly Perturbed Integral System With Diagonal Degeneration of a Kernel

We study a problem of limit transition (as the small parameter tends to zero) in integral singularly perturbed system with diagonal degeneration of a kernel. In the proof of the corresponding theorem on the limit transition we essentially use the structure of the main term of asymptotic behavior, the construction of which is performed by use of algorithm of regularization method developed by S. A. Lomov for integro-differential equations. The spectrum of the operator responsible for the regularization is composed of purely imaginary points, therefore the passage to the limit in the classical sense (i.e., in a continuous metric) in general case is impossible. In work we allocate the class of right parts in which a uniform transition in the classical sense will take place.     

Limit cycles for a perturbation of a quadratic center with symmetry

To estimate the number of limit cycles appearing under a perturbation of a quadratic system that has a center with symmetry, we use the method of generalized Dulac functions. To this end, we reduce the perturbed system to a Liénard system with a small parameter, for which we construct a Dulac function depending on the parameter. This permits one to estimate the number of limit cycles in the perturbed system for all sufficiently small parameter values. We find the Dulac function by solving a linear programming problem. The suggested method is used to analyze four specific perturbed systems that globally have exactly three limit cycles [i.e., the limit cycle distribution 3 or (3, 0)] and two systems that have the limit cycle distribution (3, 1) (i.e., one nest around each of the two foci).     

Construction of an asymptotic limit mode in a system of integral equations

We study the passage to the limit in a singularly perturbed integral system with a small parameter and a rapidly decaying kernel. In contrast to classical systems with a small parameter, the exact solution of such a system tends to infinity as the small parameter tends to zero, and hence the limit mode must be constructed in a special way. The construction of the limit mode based on the analysis of the asymptotics of the solution of the equivalent regularized (in the sense of S.A. Lomov) integro-differential system requires laborious computations. We suggest an approach to the construction of the limit mode in such systems based on the original data of the system without the construction of the corresponding asymptotic solution and not requiring heavy computations.     

On One class of Hermite projectors

We conjecture that every ideal projector on \({\mathbb {C}}\left[ x_1,\ldots ,x_d\right] \) whose kernel is generated by precisely d polynomials is Hermite (i.e., the limit of Lagrange interpolation projectors). We validate this conjecture in case the d generators of the kernel have no roots at infinity.     

On criticality problems in energy dependent neutron transport theory

In this paper, we consider the criticality problem for energy dependent neutron transport in an isotropically scattering, homogeneous slab. Under a positivity assumption on the scattering kernel, we can find an expression relating the thickness of the slab to a parameter characterizing production by fission. This is accomplished by exploiting the Perron-Frobenius-Jentsch characterization of positive operators (i.e. those leaving invariant a normal, reproducing cone in a Banach space). We point out that those techniques work for classes of multigroup problems where the Case singular eigenfunction approach is not as feasible as in the one-group theory, which is also analyzed.     

Aircraft Take-Off in Windshear: A Viability Approach

This paper is devoted to the analysis of aircraft dynamics during tage-off in the presence of windshear. We formulate the take-off problem as a differential game against Nature. Here, the first player is the relative angle of attack of the aircraft (considered as the control variable) and the second player is the disturbance caused by a windshear. We impose state constraints on the state variables of the game, which represents aircraft safety constraints (minimum altitude, given altitude rate). By using viability theory, we address the question of existence of an open loop control assuring a viable trajectory (i.e. satisfying the state constraints) no matter the disturbance is, i.e. for all admissible disturbances causeed by the windshear. Through numerical simulations of the viability kernel algorithm, we demonstrate the capabilities of this approach for determining safe flight domains of an aircraft during take-off within windshear.     

About the non-convex optimization problem induced by non-positive semidefinite kernel learning

During the last years, kernel based methods proved to be very successful for many real-world learning problems. One of the main reasons for this success is the efficiency on large data sets which is a result of the fact that kernel methods like support vector machines (SVM) are based on a convex optimization problem. Solving a new learning problem can now often be reduced to the choice of an appropriate kernel function and kernel parameters. However, it can be shown that even the most powerful kernel methods can still fail on quite simple data sets in cases where the inherent feature space induced by the used kernel function is not sufficient. In these cases, an explicit feature space transformation or detection of latent variables proved to be more successful. Since such an explicit feature construction is often not feasible for large data sets, the ultimate goal for efficient kernel learning would be the adaptive creation of new and appropriate kernel functions. It can, however, not be guaranteed that such a kernel function still leads to a convex optimization problem for Support Vector Machines. Therefore, we have to enhance the optimization core of the learning method itself before we can use it with arbitrary, i.e., non-positive semidefinite, kernel functions. This article motivates the usage of appropriate feature spaces and discusses the possible consequences leading to non-convex optimization problems. We will show that these new non-convex optimization SVM are at least as accurate as their quadratic programming counterparts on eight real-world benchmark data sets in terms of the generalization performance. They always outperform traditional approaches in terms of the original optimization problem. Additionally, the proposed algorithm is more generic than existing traditional solutions since it will also work for non-positive semidefinite or indefinite kernel functions.     

Limit cycles under perturbations of a quadratic Hamiltonian system

The perturbed quadratic Hamiltonian system is reduced to a Lienard system with a small parameter for which a Dulac function depending on it is constructed. This permits one to estimate the number of limit cycles of the perturbed system for all sufficiently small parameter values. To find the Dulac function, we use the solution of a linear programming problem. The suggested method is used for studying three specific perturbed systems that have exactly two limit cycles, i.e., the distribution 2 or (0, 2), and one system with distribution (1, 1).     

Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems

This paper is concerned with an inhomogeneous nonlocal dispersal equation. We study the limit of the re-scaled problem of this nonlocal operator and prove that the solutions of the re-scaled equation converge to a solution of the Fokker-Planck equation uniformly. We then analyze the nonlocal dispersal equation of an inhomogeneous diffusion kernel and find that the heterogeneity in the classical diffusion term coincides with the inhomogeneous kernel when the scaling parameter goes to zero.     

Probabilistic analysis of a differential equation for linear programming

In this paper we address the complexity of solving linear programming problems with a set of differential equations that converge to a fixed point that represents the optimal solution. Assuming a probabilistic model, where the inputs are i.i.d. Gaussian variables, we compute the distribution of the convergence rate to the attracting fixed point. Using the framework of Random Matrix Theory, we derive a simple expression for this distribution in the asymptotic limit of large problem size. In this limit, we find the surprising result that the distribution of the convergence rate is a scaling function of a single variable. This scaling variable combines the convergence rate with the problem size (i.e., the number of variables and the number of constraints). We also estimate numerically the distribution of the computation time to an approximate solution, which is the time required to reach a vicinity of the attracting fixed point. We find that it is also a scaling function. Using the problem size dependence of the distribution functions, we derive high probability bounds on the convergence rates and on the computation times to the approximate solution.     

Linear and convex aggregation of density estimators

We study the problem of finding the best linear and convex combination of M estimators of a density with respect to the mean squared risk. We suggest aggregation procedures and we prove sharp oracle inequalities for their risks, i.e., oracle inequalities with leading constant 1. We also obtain lower bounds showing that these procedures attain optimal rates of aggregation. As an example, we consider aggregation of multivariate kernel density estimators with different bandwidths. We show that linear and convex aggregates mimic the kernel oracles in asymptotically exact sense. We prove that, for Pinsker’s kernel, the proposed aggregates are sharp asymptotically minimax simultaneously over a large scale of Sobolev classes of densities. Finally, we provide simulations demonstrating performance of the convex aggregation procedure.      

Intrinsic ultracontractivity and small perturbation for one-dimensional generalized diffusion operators

We consider a one-dimensional generalized diffusion operator G on an open interval which is not necessarily bounded. We characterize [SP] (i.e., 1 is a small perturbation of the operator G) in terms of the classification of the boundary points from the view point of diffusion processes and give a sufficient condition for [IU] (i.e., the associated heat kernel is intrinsically ultracontractive).     

Exact linearization of the sliding problem for a dilute gas in the Bhatnagar-Gross-Krook model

We consider the nonlinear Boltzmann equation in the Bhathnagar-Gross-Krook model for the gas flow in a half-space (the Kramers problem). The problem can be exactly linearized, and its solution can be reduced to a linear integral equation with an addition-difference kernel and a simple nonlinear relation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp. 339–342, November, 2000.     

Limit theorems for random symmetric functions

In this paper, based on theorems for limit distributions of empirical power processes for the i.i.d. case and for the case with independent triangular arrays of random variables, we prove limit theorems for U- and V-statistics determined by generalized polynomial kernel functions. We also show that under some natural conditions the limit distributions can be represented as functionals on the limit process of the normed empirical power process. We consider the one-sample case, as well as multi-sample cases. Dedicated to Professor V. M. Zolotarev on his sixty-fifth birthday. Supported by the Hungarian National Foundation for Scientific Research (grant No. T1666). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

The average optimality of a repair-limit replacement policy

We consider a minimal-repair and replacement problem of a reliability system whose state at a failure is described by a pair of two attributes, i.e., the total number of its past failures and the current failure level. It is assumed that the system is bothered by more frequent and more costly failures as time passes. Our problem is to find and/or characterize a minimal-repair and replacement policy of minimizing the long-run average expected maintenance cost per unit time over the infinite time horizon. Formulating the problem as a semi-Markov decision process, we show that a repairlimit replacement policy is average optimal. That is, for each total number of past system failures, there exists a threshold, called a repair limit, such that it is optimal to repair minimally if the current failure level is lower than the repair limit, and to replace otherwise. Furthermore, the repair limit is decreasing in the total number of past system failures.     

A new model for nonlinear elastic plates with rapidly varying thickness,ii: the effect of the behavior of the forces when the thickness approaches zero

We consider a family of nonlinear elastic plates with rapidly varying thickness under the assumption that the three-dimensional constitutive equation is linear with respect to the "full" strain tensor (St. Venant-Kirchhoff material). The main goal of this paper is to shown that the limit problem, when the mean plate thickness converges to zero, may be a ill posed problem if the forces do not behave in an appropriate manner     

Accelerating viability kernel computation with CUDA architecture: application to bycatch fishery management

Computing a viability kernel consumes time and memory resources which increase exponentially with the dimension of the problem. This curse of dimensionality strongly limits the applicability of this approach, otherwise promising. We report here an attempt to tackle this problem with Graphics Processing Units (GPU). We design and implement a version of the viability kernel algorithm suitable for General Purpose GPU (GPGPU) computing using Nvidia’s architecture, CUDA (Computing Unified Device Architecture). Different parts of the algorithm are parallelized on the GPU device and we test the algorithm on a dynamical system of theoretical population growth. We study computing time gains as a function of the number of dimensions and the accuracy of the grid covering the state space. The speed factor reaches up to 20 with the GPU version compared to the Central Processing Unit (CPU) version, making the approach more applicable to problems in 4 to 7 dimensions. We use the GPU version of the algorithm to compute viability kernel of bycatch fishery management problems up to 6 dimensions.     

Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton–Jacobi–Bellman equation in our non-Markovian framework.