Ellipsoidal Approximation Research Articles

Treatment of Point Defects in Nanowire MOSFETs Using the Nonequilibrium Green’s Function Formalism

This paper describes the point defect treatment in nanowire MOSFETs. The 3D Poisson equation is self-consistently coupled to the Schrodinger equation. We adopt the mode-space representation in which the 3D Schrodinger equation is divided into a 2D equation, including the confinement of the cross-section, and an ID equation describing the ballistic transport based on the nonequilibrium Green's function formalism (NEGF). We use the simplest form of the tight-binding theory, with one orbital per atom, and considering only the first-neighbor-interactions. In this framework, equivalent to the ellipsoidal energy band approximation, influence of point defects on electronic transport is discussed.

Localization of periodic orbits of polynomial systems by ellipsoidal estimates

Abstract In this paper we study the localization problem of periodic orbits of multidimensional continuous-time systems in the global setting. Our results are based on the solution of the conditional extremum problem and using sign-definite quadratic and quartic forms. As examples, the Rikitake system and the Lamb’s equations for a three-mode operating cavity in a laser are considered.

On the accuracy of the ellipsoid norm approximation of the joint spectral radius

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper an approximation ρ ˆ that is based on ellipsoid norms, that can be computed by convex optimization, and that is such that the joint spectral radius belongs to the interval [ ρ ˆ / n , ρ ˆ ] , where n is the dimension of the matrices. We also provide a simple approximation for the special case where the entries of the matrices are non-negative; in this case the approximation is proved to be within a factor at most m ( m is the number of matrices) of the exact value.

On Convergence of External Ellipsoidal Approximations of the Reachability Domains of Discrete Dynamic Linear Systems

The ellipsoid technique is widely used in the guaranteed estimation for approximation of the reachability domains of dynamic systems. The present paper considered the issues of external ellipsoidal estimation of the current and limiting reachability sets of a stable discrete dynamic linear system. Recurrent estimation algorithms using the criterion of minimum trace of the “weighted” ellipsoid matrix were developed for these systems, and their limiting properties were considered.

Magnetic dipole anomalies as indicators of mantle wedge serpentinization

Fine‐scale magnetic anomaly data based on high‐density airborne surveys have revealed several dipole anomalies along the Chichibu Zone (CZ), which lies parallel to the Nankai Trough and forms a part of the accretionary complex of southwest Japan. A plausible explanation of magnetic sources for the anomalies is a series of fossil serpentine diapirs involved during the accretion process. Serpentine diapirs are known to be developed in several forearc regions as mantle wedge serpentinized material associated with dehydration of the subducting oceanic slab. Surface geological and paleomagnetic evidence also suggests the existence of serpentine bodies in the CZ. We applied a magnetic inversion to each of the dipole anomalies and determined the magnetic bodies in a triaxial ellipsoid approximation. Magnetic bodies are interpreted to lie nearly parallel to the CZ and are inclined southward. Intensities of magnetization are compatible with those expected by measured susceptibilities of samples in the subareal CZ, and their directions are roughly the same as the current geomagnetic field, implying that the induced magnetization is dominant rather than the remanent component.

The Active Vertice method: a performant filtering approach to high-dimensional indexing

The problem of finding nearest neighbors has emerged as an important foundation of feature-based similarity search in multimedia databases. Most spatial index structures based on the R-tree have failed to efficiently support nearest neighbor search in arbitrarily distributed high-dimensional data sets. In contrast, the so-called filtering principle as represented by the popular VA-file has turned out to be a more promising approach. Query processing is based on a flat file of compact vector approximations. In a first stage, those approximations are sequentially scanned and filtered so that in a second stage the nearest neighbors can be determined from a relatively small fraction of the data set. In this paper, we propose the Active Vertice method as a novel filtering approach. As opposed to the VA-file, approximation regions are arranged in a quad-tree like structure. High-dimensional feature vectors are assigned to ellipsoidal approximation regions on different levels of the tree. A compact approximation of a vector corresponds to the path within the index from the root to the respective tree node. When compared to the VA-file, our method enhances the discriminatory power of the approximations while maintaining their compactness in terms of storage consumption. To demonstrate its effectiveness, we conduct extensive experiments with synthetic as well as real-life data and show the superiority of our method over existing filtering approaches.

External Ellipsoidal Estimation of the Union of Two Concentric Ellipsoids and Its Applications

For an arbitrary set representable as the convex hull formed by the union of two concentric ellipsoids we propose a method to construct a family of external undominated ellipsoidal approximations and represent the estimated set as the intersection of all estimates from a given family. A sufficient condition of undominated guaranteed ellipsoidal approximation of a convex compactum is derived. A method is described that for certain classes of sets (such as the intersection of an ellipsoid or a cone with two halfspaces) constructs a family of internal undominated ellipsoidal approximations using the previous formulas for the external estimates of the union of concentric ellipsoids.

Contact resolution algorithm for an ellipsoid approximation for discrete element modeling

The efficiency of a discrete element implementation relies on several factors, including the particle representation, neighbor‐sorting algorithm, contact resolution, and force generation. The focus of this paper is on the four‐arc approximation for an ellipsoid – a geometrical representation useful in simulations of large numbers of smoothly shaped particles. A new contact resolution algorithm based on the four‐arc approximation is presented, which takes advantage of the properties of the geometry to provide favorable empirical convergence properties compared with the method proposed earlier. Special attention is given to the software implementation of the algorithm, and a discussion of the computational efficiency of the algorithm is provided.

Properties of the Optimal Ellipsoids Approximating the Reachable Sets of Uncertain Systems

The ellipsoidal estimation of reachable sets is an efficient technique for the set-membership modelling of uncertain dynamical systems. In the paper, the optimal outer-ellipsoidal approximation of reachable sets is considered, and attention is paid to the criterion associated with the projection of the approximating ellipsoid onto a given direction. The nonlinear differential equations governing the evolution of ellipsoids are analyzed and simplified. The asymptotic behavior of the ellipsoids near the initial point and at infinity is studied. It is shown that the optimal ellipsoids under consideration touch the corresponding reachable sets at all time instants. A control problem for a system subjected to uncertain perturbations is investigated in the framework of the optimal ellipsoidal estimation of reachable sets.

Ellipsoidal approximation of the stability domain of a polynomial

The stability domain in the space of coefficients of a polynomial is a nonconvex set in general. In this note, we propose a new convex ellipsoidal inner approximation of this set derived via optimization over linear matrix inequalities. As a by-product, we obtain new simple sufficient conditions for stability that may prove useful in robust control design.

Using the radon transform to solve inclusion problems in elasticity

From the Fourier transform method, the modified Green operator integral over a bounded domain in an infinite elastic medium takes, on each domain point, the form of a weighted average over an angular distribution of a single elementary operator. The Radon transform provides a geometric definition of the weight function characteristic of the domain shape, in terms of the domain intersections with all planes passing through the point. It allows a geometrically more meaningful analytical resolution of the general inclusion problem in an infinite medium of general elasticity symmetry, the “inclusion” being any bounded domain possibly made of groups (or distributions) of inclusions. The method is also likely to provide insights in the related problem of effective moduli estimates for heterogeneous microstructures. The determination of the weight functions characteristics of the involved inclusional domain shapes is therefore a key step of the resolution, the mean values of these weight functions being of first-order interest. Here, it is exemplified, on the case of cuboidal domain shapes, that for material morphologies involving shapes of hardly accessible exact mean weight functions, one can make use of approximate (conveniently analytical) expressions, to remain more accurate than using ellipsoidal approximations of the shapes.

Robust Estimation in Multivariate Control Charts for Individual Observations

Multivariate control charts, such as Hotelling's T2 control chart, are used for monitoring several quality characteristics measured simultaneously on a product or process. In the retrospective stage (Phase I) analysis of a historical data set, the presence of multiple outliers may go undetected by the conventional T2 chart, due to masking. In this paper we propose T2 control charts based on robust estimators of location and dispersion. Simulation studies show that the T2 control chart using the minimum volume ellipsoid (MVE) estimators is effective in detecting any reasonable number of outliers.

Fault Tolerant Control and Classification for Longitudinal Vehicle Control

In this paper, a new method of analyzing for the performance loss caused by faults in the systems is presented, and applied to the design of a fault tolerant longitudinal controller for a transit bus. Based on the amount of performance loss measured by a quadratic function, fault impact assessment is developed for both single and multiple faults. More specifically, ellipsoidal approximation of the tracking error bounds via dynamic surface control (DSC) is obtained via convex optimization technique for the nonlinear closed-loop system. Relying on the fault impact to the closed loop system and its isolatability on a fault detection and diagnosis system, the fault classification is proposed to provide a switching logic in the framework of a switched hierarchical structure. Finally, simulation results of the fault tolerant controller and corresponding fault classification are shown for multiple multiplicative faults.

WetUp: a software tool to display approximate wetting patterns from drippers

Knowledge of the wetted perimeter of soil arising from infiltration of water from trickle irrigation drippers is important in the design and management of efficient systems. A user-friendly software tool, WetUp, has been developed to help highlight the impact of soils on water distribution in trickle-irrigated systems. WetUp determines the approximate radial and vertical wetting distances from an emitter in homogeneous soils calculated using analytical methods, and then uses an elliptical plotting function to approximate the expected wetted perimeter. In this paper we describe WetUp and use examples to demonstrate how it can be applied. We also compare the wetted perimeter predicted using WetUp with that predicted by other methods. Results show that the wetting pattern is well described by the ellipsoidal approximation for slowly permeable soils, but that it tends to underestimate the radial wetting in highly permeable soils, particularly as the volume of applied water increases. The error is, however, small in most cases, and of minimal concern when applying WetUp to illustrate the important role that soil hydraulic properties play in determining wetting patterns.

Sp(3,R)mean field theory for heavy deformed nuclei

Algebraic mean field theory (AMFT) for the symplectic $\mathrm{sp}(3,R)$ algebra is used to derive collective rotational bands in the Riemann ellipsoidal approximation. AMFT is formulated in terms of symplectic density matrices that are defined by the quantum mechanical expectations of the $\mathrm{sp}(3,R)$ operators. The mean field approximation restricts the densities to a coadjoint orbit of the canonical transformation group $\mathrm{Sp}(3,R).$ For principal axis rotation, a system of three algebraic equations is derived from energy minimization on an orbit surface. The system is solved self-consistently for the axis lengths and the potential tensor.

Recursive algorithms for inner ellipsoidal approximation of convex polytopes

In this paper, fast recursive algorithms for the approximation of an n-dimensional convex polytope by means of an inscribed ellipsoid are presented. These algorithms consider at each step a single inequality describing the polytope and, under mild assumptions, they are guaranteed to converge in a finite number of steps. For their recursive nature, the proposed algorithms are better suited to treat a quite large number of constraints than standard off-line solutions, and have their natural application to problems where the set of constraints is iteratively updated, as on-line estimation problems, nonlinear convex optimization procedures and set membership identification.

On the ellipsoidal correction to the spherical Stokes solution of the gravimetric geoid

The solutions of four ellipsoidal approximations for the gravimetric geoid are reviewed: those of Molodenskii et al., Moritz, Martinec and Grafarend, and Fei and Sideris. The numerical results from synthetic tests indicate that Martinec and Grafarend’s solution is the most accurate, while the other three solutions contain an approximation error which is characterized by the first-degree surface spherical harmonic. Furthermore, the first 20 degrees of the geopotential harmonic series contribute approximately 90% of the ellipsoidal correction. The determination of a geoid model from the generalized Stokes scheme can accurately account for the ellipsoidal effect to overcome the first-degree surface spherical harmonic error regardless of the solution used.

Nearest-Neighbor Variance Estimation (NNVE)

Robust covariance estimation commonly proceeds by downweighting outliers. In this article we measure the “outlyingness” of a data point by the standardized distance between the point and its Kth nearest neighbor. The appropriate weights for robust estimation are found by a model-based mixture modeling approach that follows from considering the data cloud as a realization of a high-dimensional point process. To correct a potential bias when there are no outliers, we introduce a boundary correction procedure that artificially adds in extra outlying points; the resulting methodology is called nearest-neighbor variance estimation (NNVE). The strength of NNVE is its robustness against a large proportion of noise points and against deviation from normality of the signal. A consistency result for the method is established. Under some reasonable assumptions, it is shown that the covariance estimate is bounded and that each point has only bounded influence on the final estimates. NNVE outperformed the popular minimum volume ellipsoid (MVE) estimator in simulation studies, with a big improvement when the proportion of outliers was very large (> 50%). In our simulation study, when the proportion of outliers was ≥ 50%, the mean squared error of the NNVE estimator of variance was at least 100 times smaller than that of the MVE estimator. The proposed estimator also outperformed MVE in cases where the underlying data distribution was not normal. Good performance of NNVE in several real examples is demonstrated. A potential drawback of NNVE is that data points condensed in moderate-sized clusters would be classified as signal. Even though we do not support approaches discarding moderate-sized clusters as outliers without checking, this feature of NNVE could be problematic, particularly when only the main data cloud is of interest. A simple diagnostic tool built on existing model-based clustering procedure is proposed. This procedure allows us to check whether there is more than one separate data cloud in the data after cleaning. It also supplies the central locations of the separated moderate-sized clusters, which allows further investigation. Finally, because NNVE reduces the problem of finding the robustness weights to a one-dimensional problem, it may be useful in high-dimensional problems, such as those encountered in data mining.

Measurement of right ventricular volume in healthy term and preterm neonates

Background: Pulmonary hypertension is associated with worse perinatal outcomes in infants with respiratory disorders. In such infants right ventricular dysfunction may result in poor pulmonary blood flow. Objective: To evaluate...

The height datum/geodetic datum problem

SUMMARY The height datum problem is present in the geodetic literature since the times of Pizzetti, when it was realized that as only differences of the gravity potential can be derived from measurements, there was still one global parameter to be settled in order to determine a global model. Several more realistic formulations of the problem have been introduced into the geodetic literature. After reviewing them, an ultimate formulation is attempted where only local data are given stemming from levelling, gravimetry, classical geodetic network observations, combined with global GPS-like observations. The problem of contemporaneous height datum/geodetic datum determination is shown to be solvable, in ellipsoidal approximation, only if the classical tide gauge and network orientation information is taken into account, specially if the local datum does not refer to very large areas, e.g. of continental size. In fact we will show that when this classical information is missing for the datum problem referring to a fairly small area, it becomes, for instance, impossible to distinguish between a ‘vertical’ shift of the local ellipsoid and a change of the potential of the geoid, i.e. the choice of a slightly different gravity equipotential surface.