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We investigate several fundamental aspects of the theory of linear distributed systems with spatially periodic coefficients. We develop a spatial-frequency domain representation analogous to the lifted or frequency response operator representation for linear time periodic systems. Using this representation, we introduce the notion of the [H.sup.2] norm for this class of systems and provide algorithms for its computation. A stochastic interpretation of the [H.sup.2] norm is given in terms of spatially cyclostationary random fields and spectral-correlation density operators. When the periodic coefficients are viewed as feedback modifications of spatially invariant systems, we show how they can stabilize or destabilize the original systems in a manner analogous to vibrational control or parametric resonance in time periodic systems. Two examples from physics are provided to illustrate the main results. Index Terms--Cyclostationary random fields, frequency domain lifting, frequency response operators, [H.sup.2] norm, partial differential equation (PDE) with periodic coefficients, spatially periodic systems.

This paper presents a new method of missing point estimation (MPE) to derive efficient reduced-order models for large-scale parameter-varying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projection-based model reduction framework is used where projection spaces are inferred from proper orthogonal decompositions of data-dependent correlation operators. The key contribution of the MPE method is to perform online computations efficiently by computing Galerkin projections over a restricted subset of the spatial domain. Quantitative criteria for optimally selecting such a spatial subset are proposed and the resulting optimization problem is solved using an efficient heuristic method. The effectiveness of the MPE method is demonstrated by applying it to a nonlinear computational fluid dynamic model of an industrial glass furnace. For this example, the Galerkin projection can be computed using only 25% of the spatial grid points without compromising the accuracy of the reduced model. Index Terms--Model reduction, parameter-varying systems, proper orthogonal decomposition, time-varying systems.

Most of the myriad (order of [10.sup.9]) interconnected coronary vessels interact nonlinearly with their embedding contracting myocardium. Their dynamic flow can be simulated based on a nonlinear distributive segmental flow model involving highly nonlinear partial differential equations. Such network flow analysis, although of high accuracy, is computationally excessively complex. On the other hand, a corresponding nonlinear lumped analysis is significantly less demanding since it involves ordinary differential equations. This is, however, at the detriment of accuracy. In the present technical report, a nonlinear lumped representation of coronary segmental flow is presented and tested against predictions of the corresponding distributive analysis. The results suggest that under physiological conditions, the proposed lumped model achieves similar accuracy to the distributive one, yet with considerably higher computational speed.

In this paper, the problem of driving a collection of mobile robots to a given target destination is studied. In particular, we are interested in achieving this transfer in an orderly manner so as to ensure that the agents remain in the convex polytope spanned by the leader-agents, while the remaining agents, only employ local interaction rules. To this aim we exploit the theory of partial difference equations and propose hybrid control schemes based on stop-go rules for the leader-agents. Non-Zenoness, liveness and convergence of the resulting system are also analyzed. Index Terms--Containment problems, decentralized control, graph theory, leader-following, multi-agent systems, partial difference equations.

A fast algorithm for multiscale representation and segmentation of hyperspectral imagery is introduced in this paper. The multiscale/scale-space representation is obtained by solving a nonlinear diffusion partial differential equation (PDE) for vector-valued images. We use algebraic multigrid techniques to obtain a fast and scalable solution of the PDE and to segment the hyperspectral image following the intrinsic multigrid structure. We test our algorithm on four standard hyperspectral images that represent different environments commonly found in remote sensing applications: agricultural, urban, mining, and marine. The experimental results show that the segmented images lead to better classification than using the original data directly, in spite of the use of simple similarity metrics and piecewise constant approximations obtained from the segmentation maps. Index Terms--Geometric partial differential equations (PDEs), hyperspectral images, multigrid, multiscale, segmentation.

We develop adaptive controllers for parabolic partial differential equations (PDEs) controlled from a boundary and containing unknown destabilizing parameters affecting the interior of the domain. These are the first adaptive controllers for unstable PDEs without relative degree limitations, open-loop stability assumptions, or domain-wide actuation. It is the first necessary step towards developing adaptive controllers for physical systems such as fluid, thermal, and chemical dynamics, where actuation can be only applied non-intrusively, the dynamics are unstable, and the parameters, such as the Reynolds, Rayleigh, Prandtl, or Peclet numbers are unknown because they vary with operating conditions. Our method builds upon our explicitly parametrized control formulae in [27] to avoid solving Riccati or Bezout equations at each time step. Most of the designs we present are state feedback but we present two benchmark designs with output feedback which have infinite relative degree. Index Terms--Adaptive control, backstepping, boundary control, distributed parameter systems.

We stabilize the unstable "shock-like" equilibrium profiles of the viscous Burgers equation using control at the boundaries. These equilibria are not stabilizable (even locally) using the standard "radiation feedback boundary conditions." Using a nonlinear spatially-scaled transformation (that employs three ingredients, of which one is the Hopf--Cole non-linear integral transformation) and linear backstepping, we design an explicit nonlinear full-state control law that achieves exponential stability, with a region of attraction for which we give an estimate. The region of attraction is not the entire state space since the Burgers PDE is known not to be globally controllable. Index Terms--Burgers PDE.

We present an algorithm to reconstruct helical cone beam computed tomography (CT) data acquired at variable pitch. The algorithm extracts a halfscan segment of projections using an extended version of the advanced single slice rebinning (ASSR) algorithm. ASSR rebins constant pitch cone beam data to fan beam projections that approximately lie on a plane that is tilted to optimally fit the source helix. For variable pitch, the error between the tilted plane chosen by ASSR and the source helix increases, resulting in increased image artifacts. To reduce the artifacts, we choose a reconstruction plane, which is tilted and shifted relative to the source trajectory. We then correct rebinned fan beam data using John's equation to virtually move the source into the tilted and shifted reconstruction plane. Results obtained from simulated phantom images and scanner images demonstrate the applicability of the proposed algorithm. Index Terms--Advanced single slice rebinning (ASSR), computed tomography (CT), helical, John's equation, NSR, spiral, variable pitch.

A wide range of computer vision applications require an accurate solution of a particular Hamilton-Jacobi (HJ) equation known as the Eikonal equation. In this paper, we propose an improved version of the fast marching method (FMM) that is highly accurate for both 2D and 3D Cartesian domains. The new method is called multistencils fast marching (MSFM), which computes the solution at each grid point by solving the Eikonal equation along several stencils and then picks the solution that satisfies the upwind condition. The stencils are centered at each grid point and cover its entire nearest neighbors. In 2D space, two stencils cover 8-neighbors of the point, whereas in 3D space, six stencils cover its 26-neighbors. For those stencils that are not aligned with the natural coordinate system, the Eikonal equation is derived using directional derivatives and then solved using higher order finite difference schemes. The accuracy of the proposed method over the state-of-the-art FMM-based techniques has been demonstrated through comprehensive numerical experiments. Index Terms--Multistencils fast marching methods, monotonically advancing fronts, fast marching methods, level set methods, Eikonal equation.

An overview of the software design and data abstraction decisions chosen for deal.II, a general purpose finite element library written in C++, is given. The library uses advanced object-oriented and data encapsulation techniques to break finite element implementations into smaller blocks that can be arranged to fit users requirements. Through this approach, deal.II supports a large number of different applications covering a wide range of scientific areas, programming methodologies, and application-specific algorithms, without imposing a rigid framework into which they have to fit. A judicious use of programming techniques allows us to avoid the computational costs frequently associated with abstract object-oriented class libraries. The paper presents a detailed description of the abstractions chosen for defining geometric information of meshes and the handling of degrees of freedom associated with finite element spaces, as well as of linear algebra, input/output capabilities and of interfaces to other software, such as visualization tools. Finally, some results obtained with applications built atop deal.II are shown to demonstrate the powerful capabilities of this toolbox. Categories and Subject Descriptors: G.4 [Mathematical Software]; G. 1.8 [Numerical Analysis]: Partial Differential Equations--Finite element methods General Terms: Algorithms, Design, Documentation Additional Key Words and Phrases: Object-orientation, software design

hpGEM, a novel framework for the implementation of discontinuous Galerkin finite element methods (FEMs), is described. We present data structures and methods that are common for many (discontinuous) FEMs and show how we have implemented the components as an object-oriented framework. This framework facilitates and accelerates the implementation of finite element programs, the assessment of algorithms, and their application to real-world problems. The article documents the status of the framework, exemplifies aspects of its philosophy and design, and demonstrates the feasibility of the approach with several application examples. Categories and Subject Descriptors: D.3.2 [Programming Languages]: Language Classifications; G.1.8 [Numerical Analysis]: Partial Differential Equations Finite element methods; G.4 [Mathematical Software]:--Algorithm design and analysis General Terms: Algorithms, Design Additional Key Words and Phrases: Discontinuous Galerkin methods, PDE, unstructured mesh, object-oriented programming

It is proved that for every recursive total continuous functional there is a PCF-definable representative in the hierarchy of partial continuous functionals, where PCF is Plotkin's programming language for computable functionals. In addition, various notions of totality in connection with PCF are discussed.

Constant coefficient one-dimensional linear hyperbolic systems of partial differential equations (PDEs) are often used for description of fluid-structure interaction (FSI) phenomena during transient conditions in piping systems. In the past, these systems of equations have been numerically solved with method of characteristics (MOCs). The MOC method is actually the most efficient and accurate method for description of the single-phase transient in the cold liquid where the constant coefficient mathematical model describes phenomenon with sufficient accuracy. In energy production systems where hot pressurized liquid is used for heat transfer between the heat source and the steam generator, more complex and nonlinear mathematical models are needed to describe transient flow and these models cannot be solved with MOC method because the models are not constant. In addition, the MOC method can be used for pipes having discontinuities like elbows, geometrical changes, material properties changes, etc., but only with some extra numerical modeling. An interesting alternative is explicit characteristic upwind numerical method, known as Godunov's method that is frequently used for nonlinear systems or systems where properties change with position. In the present study, applicability of the Godunov's method for the FSI analyses is tested with eight first order PDEs mathematical model. The conventional linear mathematical model is improved with convective term that makes the system nonlinear and additional terms that enable simulations of the FSI in arbitrarily shaped piping systems located in a plane. Two PDEs describe pressure waves in the single-phase fluid and six PDEs describe axial, lateral, and rotational stress waves in the pipe. The applied system of equations has stiff source terms. This numerical problem is solved introducing implicit iterations. The proposed model is verified with a rod impact experiment that is carried out on single-elbow pipe hanging on wires. Godunov's method is found as a very promising numerical method for simulations of the FSI problems. [DOI: 10.1115/1.2937758] Keywords: fluid-structure interaction, single-phase flow, water hammer, nonlinear system, Godunov's method

In 1970, F. Y. M. Wan derived a single, complex-valued ordinary, differential equation for an elastically isotropic right circular conical shell ('On the Equations of the Linear Theory of Elastic Conical Shells,' Studies Appl. Math., 49, pp. 69-83). The unknown was the nth Fourier component of a complex combination of the midsurface normal displacement and its static-geometric dual, a stress function. However, an attempt to formally replace the Fourier index n by a partial derivative in the circumferential angle [theta] results in a partial differential equation, which is eighth order in [theta]. The present paper takes as unknowns the traces of the bending strain and stress resultant tensors, respectively, and derives static-geometric dual partial differential equations of fourth order in both the axial and circumferential variables. Because of the explicit appearance of Poisson ratios of bending and stretching, these two equations cannot be combined into a single complex-valued equation. Reduced equations for beamlike (axisymmetric and lateral) deformations are also derived.