Planning rigid body motions using elastic curves

This paper tackles the problem of computing smooth, optimal trajectories on the Euclidean group of motions SE(3). The problem is formulated as an optimal control problem where the cost function to be minimized is equal to the integral of the classical curvature squared. This problem is analogous to the elastic problem from differential geometry and thus the resulting rigid body motions will trace elastic curves. An application of the Maximum Principle to this optimal control problem shifts the emphasis to the language of symplectic geometry and to the associated Hamiltonian formalism. This results in a system of first order differential equations that yield coordinate free necessary conditions for optimality for these curves. From these necessary conditions we identify an integrable case and these particular set of curves are solved analytically. These analytic solutions provide interpolating curves between an initial given position and orientation and a desired position and orientation that would be useful in motion planning for systems such as robotic manipulators and autonomous-oriented vehicles.

A similarity solution for a shock reflection problem in isentropic gas dynamics

A one-dimensional shock-reflection test problem in the case of slab, cylindrical, or spherical symmetry is discussed. The differential equations for a similarity solution are derived and solved numerically in conjunction with the Rankie-Hugoniot shock relations.

Similarity solutions for multicomponent flows

A one-dimensional shock-reflection test problem in the case of slab, cylindrical or spherical symmetry is discussed for multi-component flows. The differential equations for a similarity solution are derived and then solved numerically in conjunction with the Rankine-Hugoniot shock relations.

On a class of non-self-adjoint periodic eigenproblems with boundary and interior singularities

We prove that all the eigenvalues of a certain highly non-self-adjoint Sturm–Liouville differential operator are real. The results presented are motivated by and extend those recently found by various authors (Benilov et al. (2003) [3], Davies (2007) [7] and Weir (2008) [18]) on the stability of a model describing small oscillations of a thin layer of fluid inside a rotating cylinder.

On a class of nonselfadjoint periodic boundary value problems with discrete real spectrum

In a previous paper (J. of Differential Equations, Vol. 249 (2010), 3081-3098) we examined a family of periodic Sturm-Liouville problems with boundary and interior singularities which are highly non-self-adjoint but have only real eigenvalues. We now establish Schatten class properties of the associated resolvent operator.

Solving complex optimal control problems at no cost with PSOPT

This paper introduces PSOPT, an open source optimal control solver written in C++. PSOPT uses pseudospectral and local discretizations, sparse nonlinear programming, automatic differentiation, and it incorporates automatic scaling and mesh refinement facilities. The software is able to solve complex optimal control problems including multiple phases, delayed differential equations, nonlinear path constraints, interior point constraints, integral constraints, and free initial and/or final times. The software does not require any non-free platform to run, not even the operating system, as it is able to run under Linux. Additionally, the software generates plots as well as LATEX code so that its results can easily be included in publications. An illustrative example is provided.

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number $k$, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is a circle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping' obstacles. In this paper we prove further bounds, sharpening and extending our previous results. In particular we show that there exist trapping obstacles for which the condition numbers grow as fast as $\exp(\gamma k)$, for some $\gamma>0$, as $k\to\infty$ through some sequence. This result depends on exponential localisation bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low $k$. In the second part of the paper we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.

A model for exchange rates with crawling bands: an application to the Colombian peso

This paper builds upon previous research on currency bands, and provides a model for the Colombian peso. Stochastic differential equations are combined with information related to the Colombian currency band to estimate competing models of the behaviour of the Colombian peso within the limits of the currency band. The resulting moments of the density function for the simulated returns describe adequately most of the characteristics of the sample returns data. The factor included to account for the intra-marginal intervention performed to drive the rate towards the Central Parity accounts only for 6.5% of the daily change, which supports the argument that intervention, if performed by the Central Bank, it is not directed to push the currency towards the limits. Moreover, the credibility of the Colombian Central Bank, Banco de la República’s ability to defend the band seems low.

Wave scattering by an axisymmetric ice floe of varying thickness

The problem of water wave scattering by a circular ice floe, floating in fluid of finite depth, is formulated and solved numerically. Unlike previous investigations of such situations, here we allow the thickness of the floe (and the fluid depth) to vary axisymmetrically and also incorporate a realistic non-zero draught. A numerical approximation to the solution of this problem is obtained to an arbitrary degree of accuracy by combining a Rayleigh–Ritz approximation of the vertical motion with an appropriate variational principle. This numerical solution procedure builds upon the work of Bennets et al. (2007, J. Fluid Mech., 579, 413–443). As part of the numerical formulation, we utilize a Fourier cosine expansion of the azimuthal motion, resulting in a system of ordinary differential equations to solve in the radial coordinate for each azimuthal mode. The displayed results concentrate on the response of the floe rather than the scattered wave field and show that the effects of introducing the new features of varying floe thickness and a realistic draught are significant.

Vekua theory for the Helmholtz operator

Vekua operators map harmonic functions defined on domain in \mathbb R2R2 to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves.

Advances in sequential data assimilation and numerical weather forecasting: an Ensemble Transform Kalman-Bucy Filter, a study on clustering in deterministic Ensemble Square Root Filters, and a test of a new time stepping scheme in an atmospheric model

This dissertation deals with aspects of sequential data assimilation (in particular ensemble Kalman filtering) and numerical weather forecasting. In the first part, the recently formulated Ensemble Kalman-Bucy (EnKBF) filter is revisited. It is shown that the previously used numerical integration scheme fails when the magnitude of the background error covariance grows beyond that of the observational error covariance in the forecast window. Therefore, we present a suitable integration scheme that handles the stiffening of the differential equations involved and doesn’t represent further computational expense. Moreover, a transform-based alternative to the EnKBF is developed: under this scheme, the operations are performed in the ensemble space instead of in the state space. Advantages of this formulation are explained. For the first time, the EnKBF is implemented in an atmospheric model. The second part of this work deals with ensemble clustering, a phenomenon that arises when performing data assimilation using of deterministic ensemble square root filters in highly nonlinear forecast models. Namely, an M-member ensemble detaches into an outlier and a cluster of M-1 members. Previous works may suggest that this issue represents a failure of EnSRFs; this work dispels that notion. It is shown that ensemble clustering can be reverted also due to nonlinear processes, in particular the alternation between nonlinear expansion and compression of the ensemble for different regions of the attractor. Some EnSRFs that use random rotations have been developed to overcome this issue; these formulations are analyzed and their advantages and disadvantages with respect to common EnSRFs are discussed. The third and last part contains the implementation of the Robert-Asselin-Williams (RAW) filter in an atmospheric model. The RAW filter is an improvement to the widely popular Robert-Asselin filter that successfully suppresses spurious computational waves while avoiding any distortion in the mean value of the function. Using statistical significance tests both at the local and field level, it is shown that the climatology of the SPEEDY model is not modified by the changed time stepping scheme; hence, no retuning of the parameterizations is required. It is found the accuracy of the medium-term forecasts is increased by using the RAW filter.

Realistic mean field forward predictions for the integration of co-registered EEG/fMRI

Brain activity can be measured non-invasively with functional imaging techniques. Each pixel in such an image represents a neural mass of about 105 to 107 neurons. Mean field models (MFMs) approximate their activity by averaging out neural variability while retaining salient underlying features, like neurotransmitter kinetics. However, MFMs incorporating the regional variability, realistic geometry and connectivity of cortex have so far appeared intractable. This lack of biological realism has led to a focus on gross temporal features of the EEG. We address these impediments and showcase a "proof of principle" forward prediction of co-registered EEG/fMRI for a full-size human cortex in a realistic head model with anatomical connectivity, see figure 1. MFMs usually assume homogeneous neural masses, isotropic long-range connectivity and simplistic signal expression to allow rapid computation with partial differential equations. But these approximations are insufficient in particular for the high spatial resolution obtained with fMRI, since different cortical areas vary in their architectonic and dynamical properties, have complex connectivity, and can contribute non-trivially to the measured signal. Our code instead supports the local variation of model parameters and freely chosen connectivity for many thousand triangulation nodes spanning a cortical surface extracted from structural MRI. This allows the introduction of realistic anatomical and physiological parameters for cortical areas and their connectivity, including both intra- and inter-area connections. Proper cortical folding and conduction through a realistic head model is then added to obtain accurate signal expression for a comparison to experimental data. To showcase the synergy of these computational developments, we predict simultaneously EEG and fMRI BOLD responses by adding an established model for neurovascular coupling and convolving "Balloon-Windkessel" hemodynamics. We also incorporate regional connectivity extracted from the CoCoMac database [1]. Importantly, these extensions can be easily adapted according to future insights and data. Furthermore, while our own simulation is based on one specific MFM [2], the computational framework is general and can be applied to models favored by the user. Finally, we provide a brief outlook on improving the integration of multi-modal imaging data through iterative fits of a single underlying MFM in this realistic simulation framework.

Maximum principles for vectorial approximate minimisers on non-convex functionals

We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property. The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradient term under the Orthogonal Group. Maximum Principles are implied for the relaxed solution in the case of non-existence of minimizers and for minimizing solutions of the Euler–Lagrange system of PDE.