On the discretization of singular systems

This note proposes two new discretization methods. The proposedsampled systems are described in terms of the Markov parametersof the system and therefore the proposed methods are easilyimplemented. The methodology we use is a zero-order hold discretizationfor the input and first-order approximation of its derivatives.     

Lipschitz Continuity of the Value Function in Optimal Control

For optimal control problems in with given target and free final time, we obtain a necessary and sufficient condition for local Lipschitz continuity of the optimal value as a function of the initial position. The target can be an arbitrary closed set, and the dynamics can depend in a measurable way on the time. As a limit case of this condition, we obtain a characterization of the viability property of the target, in terms of perpendiculars to the target instead of tangent cones. As an application, we analyze the convergence of certain discretization schemes for time-optimal problems.     

Core-generating approximate minimum entropy discretization for rough set feature selection in pattern classification

Rough set feature selection (RSFS) can be used to improve classifier performance. RSFS removes redundant attributes whilst retaining important ones that preserve the classification power of the original dataset. Reducts are feature subsets selected by RSFS. Core is the intersection of all the reducts of a dataset. RSFS can only handle discrete attributes, hence, continuous attributes need to be discretized before being input to RSFS. Discretization determines the core size of a discrete dataset. However, current discretization methods do not consider the core size during discretization. Earlier work has proposed core-generating approximate minimum entropy discretization (C-GAME) algorithm which selects the maximum number of minimum entropy cuts capable of generating a non-empty core within a discrete dataset. The contributions of this paper are as follows: (1) the C-GAME algorithm is improved by adding a new type of constraint to eliminate the possibility that only a single reduct is present in a C-GAME-discrete dataset; (2) performance evaluation of C-GAME in comparison to C4.5, multi-layer perceptrons, RBF networks and k-nearest neighbours classifiers on ten datasets chosen from the UCI Machine Learning Repository; (3) performance evaluation of C-GAME in comparison to Recursive Minimum Entropy Partition (RMEP), Chimerge, Boolean Reasoning and Equal Frequency discretization algorithms on the ten datasets; (4) evaluation of the effects of C-GAME and the other four discretization methods on the sizes of reducts; (5) an upper bound is defined on the total number of reducts within a dataset; (6) the effects of different discretization algorithms on the total number of reducts are analysed; (7) performance analysis of two RSFS algorithms (a genetic algorithm and Johnson’s algorithm).     

On the additive splitting procedures and their computer realization

Two additive splitting procedures are defined and studied in this paper. It is shown that these splitting procedures have good stability properties. Some other splitting procedures, which are traditionally used in mathematical models used in many scientific and engineering fields, are sketched. All splitting procedures are tested by using six different numerical methods for solving differential equations. Many conclusions, which are related both to the comparison of the additive splitting procedures with the other splitting procedures and to the influence of the numerical methods for solving differential equations on the accuracy of the splitting procedures, are drawn.     

A non-uniform discretization of stochastic heat equations with multiplicative noise on the unit sphere

We investigate a discretization of a class of stochastic heat equations on the unit sphere with multiplicative noise. A spectral method is used for the spatial discretization and the truncation of the Wiener process, while an implicit Euler scheme with non-uniform steps is used for the temporal discretization. Some numerical experiments inspired by Earth’s surface temperature data analysis GISTEMP provided by NASA are given.     

High order numerical simulation of the thermal load on a lobed strut injector for scramjet applications

In this paper, the thermal load on an actively cooled lobed strut injector for scramjet (supersonic combustion ramjet) applications is investigated numerically. This requires coupled simulations of the strut internal and external flow fields together with the heat conduction in the solid injector body. In order to achieve a fast mixing, the lobed strut is positioned at the channel axis to inject hydrogen into the core of a Mach 3 air stream. There it is exposed to the extremely high temperatures of the high speed flow. While the external air and hydrogen flows are supersonic, the strut internal hydrogen flow is mainly subsonic, in some regions at very low Mach numbers. To enable a simulation of the internal flow field which ranges from very low to very high Mach numbers (approximately Mach 2.25 at the nozzle exit), a preconditioning technique is employed. The compressible finite‐volume scheme uses a spatially fourth order multi‐dimensional limiting process discretization, which is used here for a first time to simulate a geometrically and fluid mechanically highly complex problem. It will be demonstrated that besides its high accuracy the multi‐dimensional limiting process scheme is numerically stable even in case of demanding practical applications. The coupled simulation of the lobed strut injector delivers unique insight into the flow phenomena inside and outside the strut, the heat fluxes, the temperature distribution in the solid material, the required hydrogen mass flux with respect to cooling requirements and details concerning the conditions at the exit of the injector. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonlinear parametric identification using periodic equilibrium states

The subject of this paper is the development and assessment of a new nonlinear parametric identification method for dynamic systems using periodic equilibrium states or outputs. The method consists of a modified Bayesian point estimation technique which can be used in combination with a time discretization method or a shooting method to obtain the periodic equilibrium states. It is assumed that the specific excited and measured periodic solution can be computed directly from a static initial guess. An important feature of this estimator is the possibility to estimate the best parameters based on all experiments of the complete experimental set-up. The choice of using periodic states appears to be computationally efficient compared to using transient states. The new method is applied to multiple sets of nonlinear shaker-test measurements of a uniaxially loaded F-16 nose landing gear damper, for which a standard 1 dof mechanical model and a more comprehensive 2 dof thermo-mechanical model are postulated. Finally, the predictive power of the method is assessed by comparison of predictions for a transient drop-test load case of the best 2 dof model with corresponding parameters and real drop-test measurements.Notation a,b partial derivative of a with respect to vector b - % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaca% WGHbaaleqabaGaaiOlaaaaaaa!37CD!\[\mathop a\limits^. \] total derivative of a with respect to time t - a total derivative of a with respect to the dimensionless time - â predicted value of a - mean value of a - ã approximate equations a - e residuals - f output equations - f _e external excitation frequency - F force - g ordinary differential equations - l number of parameters - n number of samples in each experiment - N number of experiments - % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuGacqWFneVtdaWgaaWcbaGa% amyyaaqabaGccaGGOaGaamOyaiaacYcacqWFce-qcaGGPaaaaa!4710!\[{\cal N}_a (b,{\cal C})\] a-dimensional normal distribution (mean b and covariance C) - p pressure - q degrees of freedom - s number of outputs - t time - T absolute temperature     

Dynamic contact of a beam against rigid obstacles: Convergence of a velocity-based approximation and numerical results

Motivated by the study of vibrations due to looseness of joints, we consider the motion of a beam between rigid obstacles. Due to the non-penetrability condition, the dynamics is described by a hyperbolic fourth order variational inequality. We build a family of fully discretized approximations of this problem by combining some classical space discretizations with velocity based time-stepping algorithms for discrete mechanical systems subjected to unilateral constraints. We prove the stability and the convergence of these numerical methods. Finally we propose some examples of implementation using either Hermite or B-spline finite element approximations.     

Post processing of solution and flux for the nodal mimetic finite difference method

We develop and analyze a post processing technique for the family of low‐order mimetic discretizations based on vertex unknowns for the numerical treatment of diffusion problems on unstructured polygonal and polyhedral meshes. The post processing works in two steps. First, from the nodal degrees of freedom, we reconstruct an elemental‐based vector field that approximates the gradient of the exact solution. Second, we solve a local problem for each mesh vertex associated with a scheme degree of freedom to determine a post processed normal flux that is conservative and divergence preserving. Theoretical results and numerical experiments for two‐dimensional (2D) and 3D benchmark problems show optimal convergence rates. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 336–363, 2015     

Front-fixing FEMs for the pricing of American options based on a PML technique

In this paper, efficient numerical methods are developed for the pricing of American options governed by the Black–Scholes equation. The front-fixing technique is first employed to transform the free boundary of optimal exercise prices to some a priori known temporal line for a one-dimensional parabolic problem via the change of variables. The perfectly matched layer (PML) technique is then applied to the pricing problem for the effective truncation of the semi-infinite domain. Finite element methods using the respective continuous and discontinuous Galerkin discretization are proposed for the resulting truncated PML problems related to the options and Greeks. The free boundary is determined by Newton’s method coupled with the discrete truncated PML problem. Stability and nonnegativeness are established for the approximate solution to the truncated PML problem. Under some weak assumptions on the PML medium parameters, it is also proved that the solution of the truncated PML problem converges to that of the unbounded Black–Scholes equation in the computational domain and decays exponentially in the perfectly matched layer. Numerical experiments are conducted to test the performance of the proposed methods and to compare them with some existing methods.