Relative error long-time behavior in matrix exponential approximations for numerical integration: the stiff situation

In the stiff situation, we consider the long-time behavior of the relative error $\gamma_n$ in the numerical integration of a linear ordinary differential equation $y^\prime(t)=Ay(t),\ t\ge 0$, where $A$ is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g. a polynomial or a rational approximation. We study the long-time behavior of $\gamma_n$ by comparing it to the relative error $\gamma_n^{\rm long}$ in the numerical integration of the long-time solution, i.e. the projection of the solution on the eigenspace of the rightmost eigenvalues. The error $\gamma_n^{\rm long}$ grows linearly in time, it is small and it remains small in the long-time. We give a condition under which $\gamma_n\approx \gamma_n^{\rm long}$, i.e. $\frac{\gamma_n}{\gamma_n^{\rm long}}\approx 1$, in the long-time. When this condition does not hold, the ratio $\frac{\gamma_n}{\gamma_n^{\rm long}}$ is large for all time. These results describe the long-time behavior of the relative error $\gamma_n$ in the stiff situation

Time-transformations for the event location in discontinuous ODEs

In this paper, we consider numerical methods for the location of events of ordinary differential equations. These methods are based on particular changes of the independent variable, called time-transformations. Such a time-transformation reduces the integration of an equation up to the unknown point, where an event occurs, to the integration of another equation up to a known point. This known point corresponds to the unknown point by means of the time-transformation. This approach extends the one proposed by Dieci and Lopez [BIT 55 (2015), no. 4, 987-1003], but our generalization permits, amongst other things, to deal with situations where the solution approaches the event in a tangential way. Moreover, we also propose to use this approach in a different manner with respect to that of Dieci and Lopez

On discretizing the semigroup of solution operators for linear time invariant - time delay systems

in this paper we give an account of the basic facts to be considered when one attempts to discretize the semigroup of solution operators for Linear Time Invariant - Time Delay Systems (LTI-TDS). Two main approaches are presented, namely pseudospectral and spectral, based respectively on classic interpolation when the state space is C = C(-\u3c4,0;C) and generalized Fourier projection when the state space is \u3c7 = C 7 L2(-\u3c4,0;C). Full discretization details for constructing the approximation matrices are given. Moreover, concise, yet fundamental, convergence results are discussed, with particular attention to their similarities and differences as well as pros and cons with regards to solution approximation and asymptotic stability detection

Numerical computation of characteristic multipliers for linear time periodic coefficients delay differential equations

In this work we address the question of asymptotic stability of linear delay differential equations (DDEs) with time periodic coefficients, a class which is recognized to be fundamental in machining tool. Since the dynamics of such a class of delay systems is governed by the dominant eigenvalues (multipliers) of the monodromy operator associated to the system of DDEs, i.e. the solution operator over the period of the coefficients, we discretize it by using pseudospectral differencing techniques based on collocation and approximate the dominant multipliers by the eigenvalues of the resulting matrix. The use of pseudospectral methods has already been proposed in the context of simpler DDEs. Here we fully generalize the method to the class of linear time periodic coefficients DDEs with arbitrary period and multiple discrete and distributed delays. The scheme is shown to have spectral accuracy by means of several numerical examples

Особенности попередельного способа калькуляции себестоимости продукции в перерабатывающем производстве

We are interested in the asymptotic stability of equilibria of structured populations modelled in terms of systems of Volterra functional equations coupled with delay differential equations. The standard approach based on studying the characteristic equation of the linearized system is often involved or even unattainable. Therefore, we propose and investigate a numerical method to compute the eigenvalues of the associated infinitesimal generator. The latter is discretized by using a pseudospectral approach, and the eigenvalues of the resulting matrix are the sought approximations. An algorithm is presented to explicitly construct the matrix from the model coefficients and parameters. The method is tested first on academic examples, showing its suitability also for a class of mathematical models much larger than that mentioned above, including neutral- and mixed-type equations. Applications to cannibalism and consumer\u2013resource models are then provided in order to illustrate the efficacy of the proposed technique, especially for studying bifurcations

4D-SFM photogrammetry for monitoring sediment dynamics in a debris-flow catchment: Software testing and results comparison

In recent years, the combination of Structure-from-Motion (SfM) algorithms and UAV-based aerial images has revolutionised 3D topographic surveys for natural environment monitoring, offering low-cost, fast and high quality data acquisition and processing. A continuous monitoring of the morphological changes through multi-temporal (4D) SfM surveys allows, e.g., to analyse the torrent dynamic also in complex topography environment like debris-flow catchments, provided that appropriate tools and procedures are employed in the data processing steps. In this work we test two different software packages (3DF Zephyr Aerial and Agisoft Photoscan) on a dataset composed of both UAV and terrestrial images acquired on a debris-flow reach (Moscardo torrent - North-eastern Italian Alps). Unlike other papers in the literature, we evaluate the results not only on the raw point clouds generated by the Structure-from- Motion and Multi-View Stereo algorithms, but also on the Digital Terrain Models (DTMs) created after post-processing. Outcomes show differences between the DTMs that can be considered irrelevant for the geomorphological phenomena under analysis. This study confirms that SfM photogrammetry can be a valuable tool for monitoring sediment dynamics, but accurate point cloud post-processing is required to reliably localize geomorphological changes

INVESTIGATING THE PERFORMANCE OF A HANDHELD MOBILE MAPPING SYSTEM IN DIFFERENT OUTDOOR SCENARIOS

In recent years, portable Mobile Mapping Systems (MMSs) are emerging as valuable survey instruments for fast and efficient mapping of both internal and external environments. The aim of this work is to assess the performance of a commercial handheld MMS, Gexcel HERON Lite, in two different outdoor applications. The first is the mapping of a large building, which represents a standard use-case scenario of this technology. Through the second case study, that consists in the survey of a torrent reach, we investigate instead the applicability of the handheld MMS for natural environment monitoring, a field in which portable systems are not yet widely employed. Quantitative and qualitative assessment is presented, comparing the point clouds obtained from the HERON Lite system against reference models provided by traditional techniques (i.e., Terrestrial Laser Scanning and Photogrammetry)

CEREBROSPINAL FLUID DRAINAGE DEVICES: EXPERIMENTAL CARACTERIZATION

Hydrocephalus is a pathophysiology due to the excess of cerebrospinal fluid in the brain ventricles and it can be caused by congenital defects, brain abnormalities, tumors, inflammations, infections, intracranial hemorrhage and others. Hydrocephalus can be followed by significant rise of intraventricular pressure due to the excess of production of cerebrospinalfluid over the absorption, resulting in a weakening of intellectual functions, serious neurological damage (decreased movement, sensation and functions), critical physical disabilities and even death. A procedure for treatment involves the placement of a ventricular catheter into the cerebral ventricles to divert/drain the cerebrospinal fluid flow to a bag outside of the patient body – provisory treatment known as external ventricular drainage (EVD). Another option is the permanent treatment, internal ventricular drainage (IVD), promoting the cerebrospinal fluid drainage to other body cavity, being more commonly the abdominal cavity. In both cases, EVD and IVD, it is necessary to use of some type of neurological valve in order to control the flow of cerebrospinal fluid. In the present work is proposed an experimental procedure to test the hydrodynamic behavior of a complete drainage system, or parts of them, in order to verify its performance when subjected to pressure gradients found in the human body. Results show that the method is well adapted to quantify the pressure drop in neurological systems

Role of Multipoles in Counterion-Mediated Interactions between Charged Surfaces: Strong and Weak Coupling

We present general arguments for the importance, or lack thereof, of the structure in the charge distribution of counterions for counterion-mediated interactions between bounding symmetrically charged surfaces. We show that on the mean field or weak coupling level, the charge quadrupole contributes the lowest order modification to the contact value theorem and thus to the intersurface electrostatic interactions. The image effects are non-existent on the mean-field level even with multipoles. On the strong coupling level the quadrupoles and higher order multipoles contribute additional terms to the interaction free energy only in the presence of dielectric inhomogeneities. Without them, the monopole is the only multipole that contributes to the strong coupling electrostatics. We explore the consequences of these statements in all their generality.Comment: 12 pages, 3 figure