Boundary Value Problem Approach Research Articles

Racing under uncertainty: Boundary value problem approach

This paper explores the continuous time and continuous space model of racing under uncertainty put forward by Budd, Harris, and Vickers [4] and allows for potentially asymmetric players. To prove the existence of Markov perfect equilibria, I use a boundary value problem formulation which is novel to the dynamic competition literature. In addition, by providing a new and intuitive definition of the pivot of an equilibrium, I show that equilibrium strategies exhibit the discouragement effect similar to that of Harris and Vickers [10] but under a more general class of the cost functions.

Approximations of the GOCE error variance-covariance matrix for least-squares estimation of height datum offsets

AbstractOne main geodetic objective of the European Space Agency’s satellite mission GOCE (gravity field and steady-state ocean circulation explorer) is the contribution to global height system unification. This can be achieved by applying the Geodetic Boundary Value Problem (GBVP) approach. Thereby one estimates the unknown datum offsets between different height networks (datum zones) by comparing the physical (e.g. orthometric) height values H of benchmarks in different datum zones to the corresponding values derived from the difference between ellipsoidal heights h (e.g. determined by means of global navigation satellite systems) and geoid heights N. In the ideal case, i.e. neglecting data errors, the misfit between H and (h − N) is constant inside one datum zone and represents the datum offset. In practise, the accuracy of the offset estimation depends on the accuracy of the three quantitiesH, h andN, where the latter can be computed from the combination of a GOCE-derived Global Potential Model (GPM) for the long to medium wavelength and terrestrial data for the short wavelength content. Providing an optimum estimation of the datum offsets along with realistic error estimates, theoretically, requires propagation of the full error variance and covariance information of the GOCE spherical harmonic coefficients to geoid heights, respectively geoid height differences. From a numerical point of view, this is a very demanding task which cannot simply be run on a single PC. Therefore it is worthwhile to investigate on different levels of approximation of the full variance-covariance matrix (VCM) with the aim of minimizing the numerical effort. In this paper, we compare the estimation error based on three levels of approximation, namely (1) using the full VCM, (2) using only elements of the dominant m-block structure of the VCM and (3) using only the main diagonal of the VCM, i.e. neglecting all error covariances between the spherical harmonic coefficients. We show that the m-block approximation gives almost the same result as provided by the full VCM. The diagonal approximation however over- or underestimates the geoid height error, depending on the geographic location and therefore is not regarded to be a suitable approximation.

Modeling and optimal control of human platform diving with somersaults and twists

The model-based investigation of fast dynamic motions of anthropomorphic systems is an interesting interdisciplinary field combining research efforts from applied mathematics, robotics, biomechanics, computer graphics and sports. Somersaults and twists of professional platform divers represent a particularly fascinating and extremely difficult type of motion. The purpose of this paper is to show how optimal control methods based on whole-body dynamic models of the diver can be very useful in generating natural platform diving motions. We present 3D somersaults with twists as well as pure somersaults in the sagittal plane for a variety of different take-off configurations and positions to be attained in the aerial phase that all have been produced by optimization of criteria related to energy input. By formulating the dive as a problem with several dynamic phases, we are able to treat contact and flight phase simultaneously, and also to split the flight phase in several sub-phases to correctly model requested positions in the air. Divers are modeled as multibody systems with actuators and damper elements at each joint. For the solution of the optimal control problem we use efficient direct multiple shooting methods based on the boundary value problem approach. The optimization results can be used to generate motions in computer graphics or robotics, but also provide useful insights into biological motion, including joint kinematics and the required torques and forces.

Stability analysis for axially-symmetric magnetic field levitation of a superconducting sphere

For a magnetically levitated superconducting sphere, the stability analysis in both axial and radial directions is analyzed theoretically using a direct boundary-value problem approach. The external magnetic field is produced by a system of current-carrying coils with axial symmetry. We also assume complete exclusion of magnetic fields from the interior of the superconductor. Use of the P l 1 associated Legendre functions as a basis set allows for a simple series solution for the magnetic stiffness coefficients as tridiagonal quadratic forms in terms of the expansion coefficients. Analysis shows that stability for this problem is not guaranteed in general. Stability is guaranteed though, if coil windings are designed in such away that all but one of the expansion coefficients are zero. Further cases with two and three non-zero coefficients are also solved. In particular the important 1, 3 and 2, 4 cases are treated in detail, and stability maps have been produced. Finally, the case where levitation is attempted by one pair of discrete current coils has been solved and stability mapping has been thoroughly explored.

A bias-free geodetic boundary value problem approach to height datum unification

A geodetic boundary value problem (GBVP) approach has been formulated which can be used for solving the problem of height datum unification. The developed technique is applied to a test area in Southwest Finland with approximate size of 1.5° × 3° and the bias of the corresponding local height datum (local geoid) with respect to the geoid is computed. For this purpose the bias-free potential difference and gravity difference observations of the test area are used and the offset (bias) of the height datum, i.e., Finnish Height Datum 2000 (N2000) fixed to Normaal Amsterdams Peil (NAP) as origin point, with respect to the geoid is computed. The results of this computation show that potential of the origin point of N2000, i.e., NAP, is (62636857.68 ± 0.5) (m2/s2) and as such is (0.191 ± 0.003) (m) under the geoid defined by W0 = 62636855.8 (m2/s2). As the validity test of our methodology, the test area is divided into two parts and the corresponding potential difference and gravity difference observations are introduced into our GBVP separately and the bias of height datums of the two parts are computed with respect to the geoid. Obtaining approximately the same bias values for the height datums of the two parts being part of one height datum with one origin point proves the validity of our approach. Besides, the latter test shows the capability of our methodology for patch-wise application.

An overdetermined geodetic boundary value problem approach to telluroid and quasi-geoid computations

In this paper an overdetermined Geodetic Boundary Value Problem (GBVP) approach for telluroid and quasi-geoid computations is presented. The presented GBVP approach can solve the problem of potential value computation on the surface of the Earth, which when applied to a mapping scheme, e.g., here minimum distance mapping, provides a point-wise approach to telluroid computation. Besides, we have succeeded in reducing the number of equations and unknowns of the minimum distance telluroid mapping by one. The sufficient condition of minimum distance telluroid mapping is also recapitulated. Since the introduced GBVP approach has the advantage of implementing various gravity observables simultaneously as input boundary data, it can be regarded as a data fusion technique that exploits all available gravity data. The developed GBVP is used for the computation of the quasi-geoid within a test area in Southwest Finland.

Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system

We investigate the organization of mixed-mode oscillations in the self-coupled FitzHugh-Nagumo system. These types of oscillations can be explained as a combination of relaxation oscillations and small-amplitude oscillations controlled by canard solutions that are associated with a folded singularity on a critical manifold. The self-coupled FitzHugh-Nagumo system has a cubic critical manifold for a range of parameters, and an associated folded singularity of node-type. Hence, there exist corresponding attracting and repelling slow manifolds that intersect in canard solutions. We present a general technique for the computation of two-dimensional slow manifolds (smooth surfaces). It is based on a boundary value problem approach where the manifolds are computed as one-parameter families of orbit segments. Visualization of the computed surfaces gives unprecedented insight into the geometry of the system. In particular, our techniques allow us to find and visualize canard solutions as the intersection curves of the attracting and repelling slow manifolds.

Numerical Methods for Parameter Estimation in Nonlinear Differential Algebraic Equations

AbstractNonlinear Differential Algebraic Equations (DAEs) are an important class of models for dynamic processes. To establish models that describe the process behavior in a quantitatvely correct way, often parameters in the model have to be determined from observations or measurements of the process.This paper reviews numerical methods for parameter estimation in DAEs. In particular the so‐called boundary value problem approach and a versatile realisation, the multiple shooting method for parameter estimation, are discussed. Several applications are given to show the numerical performance and the wide applicability of the methods. A difficulty that occurs in practical applications is that the experiments performed to obtain measurements for parameter estimation are expensive, but nevertheless do not guarantee satisfactory parameter accuracy. The optimization of one or more dynamic experiments in order to maximize the accuracy of the results of a parameter estimation subject to cost and further technical inequality constraints leads to very complex non‐standard optimal control problems. Newly developed methods for design of optimal experiments for nonlinear processes are briefly discussed. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A scalar Riemann boundary value problem approach to orthogonal polynomials on the circle

A scalar Riemann boundary value problem defining orthogonal polynomials on the unit circle and the corresponding functions of the second kind is obtained. The Riemann problem is used for the asymptotic analysis of the polynomials orthogonal with respect to an analytical real-valued weight on the circle.

Multi-scale and multi-order singularity approach to non-equilibrium mechanics: Coupling of atomic-micro-macro damage

Offered in this work is the development of a macro/meso/micro model that covers the lineal scale of 10−11 to 100 by application of the volume energy density function. Boundary constraints and defect geometries are shown to play a role at the smaller scale in the same way as those at the macroscopic scale. Different orders of stress (or energy density) singularities are used to describe the defect geometry and prevailing constraint via the boundary conditions in a way similar to singularity adopted in classical fracture mechanics. Two classes of singularities have been identified in addition to classical one without violating the finiteness conditions of the local displacement and energy density. Still the connection of results from the different scales is no small task and is made possible by application of a scale multiplier. It is determined by considering the interactive effects of the parameters at the different scales from the atomic to the macroscopic. Unlike the classical boundary value problem approach, application of the scale multiplier has led to closed-form asymptotic multiscale solutions that otherwise would not have been made possible. The procedure is demonstrated for the anti-plane shear of a macro-micro-atomic model that accounts for imperfection at the different scales

Local geoid determination from airborne vector gravimetry

Methods are illustrated to compute the local geoid using the vertical and horizontal components of the gravity disturbance vector derived from an airborne GPS/inertial navigation system. The data were collected by the University of Calgary in a test area of the Canadian Rocky Mountains and consist of multiple parallel tracks and two crossing tracks of accelerometer and gyro measurements, as well as precise GPS positions. Both the boundary-value problem approach (Hotine’s integral) and the profiling approach (line integral) were applied to compute the disturbing potential at flight altitude. Cross-over adjustments with minimal control were investigated and utilized to remove error biases and trends in the estimated gravity disturbance components. Final estimation of the geoid from the vertical gravity disturbance included downward continuation of the disturbing potential with correction for intervening terrain masses. A comparison of geoid estimates to the Canadian Geoid 2000 (CGG2000) yielded an average standard deviation per track of 14 cm if they were derived from the vertical gravity disturbance (minimally controlled with a cross-over adjustment), and 10 cm if derived from the horizontal components (minimally controlled in part with a simulated cross-over adjustment). Downward continuation improved the estimates slightly by decreasing the average standard deviation by about 0.5 cm. The application of a wave correlation filter to both types of geoid estimates yielded significant improvement by decreasing the average standard deviation per track to 7.6 cm.

Robust Parameter Estimation in Dynamic Systems

In this paper we present a practical method for robust parameter estimation in dynamic systems. In our study we follow the very successful approach for solving optimization problems in dynamic systems, namely the boundary value problem (BVP) approach. The suggested method combines multiple shooting for parameterizing dynamics, a flexible realization of the BVP principle, with a fast Gauss-Newton algorithm for solving the resulting constrained l1 problem. We give an overview of the theoretical background as well as the details of a numerical implementation. We discuss why the Gauss-Newton algorithm, which is known to perform well mainly on well-conditioned problems, is appropriate for parameter estimation problems, while quasi-Newton methods have only limited use for parameter estimation. The method is implemented on the basis of the direct multiple shooting method as implemented in PARFIT, thus inheriting all basic properties of PARFIT such as numerical stability, reliability and efficiency. The new code has been successfully applied to real-life parameter estimation problems in enzyme and chemical kinetics.

INVERSE DESIGN OF THREE-DIMENSIONAL ENCLOSURES WITH TRANSPARENT AND ABSORBING-EMITTING MEDIAL USING AN OPTIMIZATION TECHNIQUE

An inverse analysis is employed to determine the boundary conditions in 3-D enclosures that will provide both the specified radiative heat flux and temperature distribution on the design surface of the enclosure containing transparent or absorbing-emitting media. The inverse radiation problem is posed as a minimization problem of the least square criterion which is solved by the Levenberg-Marquart method. A boundary value problem approach is developed to solve the sensitivity problem. The discrete transfer method is employed to solve the radiative transfer equation. The effect of the spatial and the angular mesh refinements is studied through comparison with the exact solution of a 3-D benchmark problem. The performance of the present technique of inverse analysis is evaluated by numerical experiments in participating and transparent media

An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects

Optimal design and operation of complex chemical processes often require the solution of intricate dynamic optimization problems. A tailored simultaneous solution strategy based on multiple shooting and reduced SQP is presented. This reduced-space boundary value problem (BVP) approach allows an efficient and robust solution of multistage optimal control and design optimization problems for large, sparse DAE process models of index one. The current paper describes the theoretical aspects of the method. Utilizing the natural decomposition of the states into differential and algebraic variables, the structured NLP problem which results from the multiple shooting discretization of the optimization BVP is projected onto the reduced space of differential variables and control parameters. It is shown that this projection can be obtained very efficiently through direct computation of the reduced linearized constraint system via directional sensitivities. Like the original full-space BVP approach, the reduced-space formulation lends itself well to parallel computation. An implementation of the new strategy is provided by the modular optimal control package MUSCOD-II. Software aspects and applications are discussed in a second paper (Part II Software Aspects and Applications, 2002).

Geoid profile determination by direct integration of GPS inertial navigation system vector gravimetry

Analogous to astrogeodetic methods, the geoid undulation may be determined along profiles by integrating the horizontal components of the gravity vector estimated by airborne GPS inertial navigation system. This has the advantage of obviating large regional surveys of the vertical gravity component that are required in the usual boundary value problem approach. Two data sets were tested, one over the Canadian Rocky Mountains and one over the North Atlantic Ocean. Comparisons with the Canadian geoid model and an Arctic geoid model, respectively, showed better than 10 cm (SD) agreement in the relative geoid, with resolution judged to be between 10 and 20 km. The geoid estimation was found to be sensitive to the curvature of the geopotential field in the mountainous region, and, in the ocean area, the estimates clearly revealed the difference between the geoid and the mean sea surface at this resolution and accuracy.

Analysis of fatigue crack growth behaviour in polymers and their composites based on ordinary differential equations parameter estimation

Traditionally, the fatigue behaviour of polymers and polymer composites is analyzed by performing Wöhler experiments, a well-known technique for the investigation of metals. These experiments are very expensive and time consuming. The application of fracture mechanics provides a much faster way to analyze the fatigue crack growth behaviour of polymers and polymer composites. This analysis is based on the Paris–Erdogan equation describing a stable phase of crack growth. With this method the fatigue crack growth diagram is derived simultaneously with the reading of the measurements. Care must be taken however, to minimize error arising from variations in initial crack length, the uncertainty in the determination of the elastic modulus and estimation of the region of stable crack growth. To overcome some of the systematic problems mentioned above, the following procedure is applied: The fracture mechanical parameter estimation problem is mapped onto a nonlinear least squares problem restricted by a differential equation. This problem is numerically solved by a boundary value problem approach which consists of a multiple shooting technique combined with a damped Gauss–Newton procedure. In this procedure, the only data to be measured is the compliance as a function of time or number of cycles. After the problem of least squares is solved and the material parameters determined, the distribution of residuals of the fit is used to estimate the region of stable crack growth. Since the analysis is based on well defined statistical foundations (maximum likelihood estimation), we gain a methodological advantage of the increased objectivity of the analysis. Considering the increasing importance of lifespan prognoses for plastic components based on fatigue crack propagation experiments, this method is preferred to the conventional, rather subjective procedure.

Gap metric problem for MIMO delay systems: Parametrization of all suboptimal controllers

We consider the problem of robustness optimization in the gap metric for MIMO systems with a scalar time delay. We present an algorithm for the parametrization of all suboptimal controllers. In our algorithm the AAK theory plays the central role. Using this approach, suboptimal controllers can be found by computing solutions to certain infinite-rank operator equations. We show that these equations can be solved numerically by modifying the two-point boundary-value problem approach reported earlier for the computation of optimal robustness radius. We present a numerical example to illustrate the procedure.

Nonlinear boundary value problems on semi-infinite intervals

A boundary value problem approach to solving nonlinear differential equations on the half line is discussed. Several examples are presented to illustrate the ideas.

Electromagnetic Wave Propagation in a Magneto-Plasma Filled Coaxial Structure: I, Theoretical

This study is concerned with the problem of electromagnetic wave propagation in a magneto-plasma filled coaxial structure. The problem is formulated using the classical boundary value problem approach. Based on this formulation an extensive numerical investigation is performed. The numerical investigation shows the existence of propagating slow modes, backward modes, a quasi-TEM mode, and waveguide-type modes in a magneto-plasma filled coaxial structure. Dispersion curves for these different modes are presented. The dependence of the different modes on various plasma parameters is considered.

Diffraction of Coherent Light by a System of Lenses

This study concerns a boundary-value problem approach to the action of a lens system upon a coherent light beam. For the sake of simplicity, we assume the lenses to be cylindrical and to be composed of homogeneous perfect dielectric substances. In addition, we assume that the cross sections of the lenses are those of two intersecting circles. The solution to Maxwell's equations is sought by means of Green's identity, which permits us to express the electromagnetic field within and outside of each lens in terms of the field values on the lens surfaces. The application of the continuity relations across these same surfaces then yields a system of coupled Fredholm-type integral equations of the second kind. These are reduced, by means of the numerical quadraturing technique, to a finite-order matrix equation, the unknowns of which represent the sampled field values along the lens surfaces. For a large number of lenses and for a large number of sampling points, the inversion of the matrix equation can be acco...