Performance analysis of error-control B-spline Gaussian collocation software for PDEs

B-spline Gaussian collocation software has been widely used in the numerical solution of boundary value ordinary differential equations (BVODEs) and 1D partial differential equations (PDEs) for several decades. Such packages represent the numerical solution in terms of a piecewise polynomial (B-spline) basis with basis coefficients determined through the use of Gaussian collocation. The software package, BACOL, developed over a decade ago, was the first 1D PDE package of this type to provide both temporal and spatial error control. A recently developed package, BACOLI, improves upon the efficiency of BACOL through the use of new types of spatial error estimation and control. The complexity of the interactions among the component numerical algorithms used by these packages (particularly the spatial and temporal error estimation and control algorithms) implies that extensive testing and analysis of the test results is an essential factor in the ongoing development of these packages In this paper, we investigate the performance of BACOL and BACOLI with respect to several important machine independent algorithmic measures, examine the effectiveness of the new spatial error estimation and control strategies, and investigate the influence of the choice of the degree of the B-spline basis on the performance of the solvers. These results will provide new insights into how to improve BACOLI, potentially lead to improvements in Gaussian collocation BVODE solvers, and guide further development of B-spline Gaussian collocation software with error control for 2D PDEs.     

3D ball skinning using PDEs for generation of smooth tubular surfaces

We present an approach to compute a smooth, interpolating skin of an ordered set of 3D balls. By construction, the skin is constrained to be C¹ continuous, and for each ball, it is tangent to the ball along a circle of contact. Using an energy formulation, we derive differential equations that are designed to minimize the skin’s surface area, mean curvature, or convex combination of both. Given an initial skin, we update the skin’s parametric representation using the differential equations until convergence occurs. We demonstrate the method’s usefulness in generating interpolating skins of balls of different sizes and in various configurations.     

Dual Formulations of Mixed Finite Element Methods with Applications

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the numerical stability of the method. Additionally, we define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods. Examples from magnetostatics and Darcy flow are examined in detail.     

Control of 1D parabolic PDEs with Volterra nonlinearities, Part II: Analysis

For a class of stabilizing boundary controllers for nonlinear 1D parabolic PDEs introduced in a companion paper, we derive bounds for the gain kernels of our nonlinear Volterra controllers, prove the convergence of the series in the feedback laws, and establish the stability properties of the closed-loop system. We show that the state transformation is at least locally invertible and include an explicit construction for computing the inverse of the transformation. Using the inverse, we show L² and H¹ exponential stability and explicitly construct the exponentially decaying closed-loop solutions. We then illustrate the theoretical results on an analytically tractable example.     

Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolic PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains T_n of increasing dimensions n+1 and with a domain shape in the form of a “hyper-pyramid”, 0≤ξ_n≤ξ_n−1?≤ξ₁≤x≤1. We illustrate our design method with several examples. One of the examples is analytical, while in the remaining two examples the controller is numerically approximated. For all the examples we include simulations, showing blow up in open loop, and stabilization for large initial conditions in closed loop. In a companion paper we give a theoretical study of the properties of the transformation, showing global convergence of the transformation and of the control law nonlinear Volterra operators, and explicitly constructing the inverse of the feedback linearizing Volterra transformation; this, in turn, allows us to prove L² and H¹ local exponential stability (with an estimate of the region of attraction where possible) and explicitly construct the exponentially decaying closed loop solutions.     

An Improvement of a Recent Eulerian Method for Solving PDEs on General Geometries

We improve upon a method introduced in Bertalmio et al. [4] for solving evolution PDEs on codimension-one surfaces in As in the original method, by representing the surface as a level set of a smooth function, we use only finite differences on a Cartesian mesh to solve an Eulerian representation of the surface PDE in a neighborhood of the surface. We modify the original method by changing the Eulerian representation to include effects due to surface curvature. This modified PDE has the very useful property that any solution which is initially constant perpendicular to the surface remains so at later times. The change remedies many of problems facing the original method, including a need to frequently extend data off of the surface, uncertain boundary conditions, and terribly degenerate parabolic PDEs. We present numerical examples that include convergence tests in neighborhoods of the surface that shrink with the grid size Work supported by the National Science Foundation     

偏微分方程(PDEs)模型在图像处理中的若干应用

介绍了偏微分方程(PDEs)模型在图像处理与分析中的应用,基本思想,发展历史和解决问题的基本框架。主要阐述了变分方法和形变模型(曲线演化)在图像恢复和图像分割中的应用。理论和实验结果表明,应用偏微分方程模型进行图像处理是一种有效的工具。最后,分析了这种方法的优点和面临的挑战。     

Symmetries of differential behaviors and finite group actions on free modules over a polynomial ring

In this paper we study finite group symmetries of differential behaviors (i.e., kernels of linear constant coefficient partial differential operators). They lead us to study the actions of a finite group on free modules over a polynomial ring. We establish algebraic results which are then used to obtain canonical differential representations of symmetric differential behaviors.     

Modeling and Solution of Some Mechanical Problems on Lie Groups

We apply Munthe-Kaas and Crouch–Grossman methods in the solution of some mechanical problems. These methods are quite new, and they exploit intrinsic properties of the manifolds defined by the mechanical problems, thus ensuring that the numerical solution obey underlying constraints. A brief introduction to the methods is presented, and numerical simulations show some of the properties they possess. We also discuss error estimation and stepsize selection for some of these methods.     

Simflowny: A general-purpose platform for the management of physical models and simulation problems

Simflowny is a software platform which aims to formalize the main elements of a simulation flow. It allows users to manage (i) formal representations of physical models based on Initial Value Problems (hyperbolic, parabolic and mixed-type partial differential equations), (ii) simulation problems based on such models, and (iii) discretization schemes to translate the problem to a finite mesh. Additionally, Simflowny generates automatically code for general-purpose simulation frameworks. This paper first presents an introductory example of such problems. Then, formal representations are explained. Afterwards, it summarizes the platform’s architecture. Finally, validation results are provided.     

A Note on the Numerical Solution of the Heavy Top Equations

In Multibody System Dynamics 2, 71–88, wedescribed the Munthe-Kaas and Crouch–Grossman methods for integratingordinary differential equations numerically on Lie groups. We used theheavy top as a special test problem, and showed that the numericalsolution respects the configuration space TSO(3). We were, however, notable to generate numerical solutions that preserved the first integralsof the top. In this paper, we formulate the heavy top equations on amore natural configuration space, and show that both the Munthe-Kaas andthe Crouch–Grossman methods with suitable coefficient sets can generatenumerical solutions that render first integrals to machine accuracy. Asa partial answer to the comment in Concluding Remarks inMultibody System Dynamics 2, 71–88, we also argue that forHamiltonian systems on the dual space of a Lie algebra, theinfinitesimal generator map describing the differential equation for thecoadjoint action is the functional derivative of the Hamiltonian.     

Computable convergence bounds of series expansions for infinite dimensional linear-analytic systems and application

This paper deals with the convergence of series expansions of trajectories for semi-linear infinite dimensional systems, which are analytic in state and affine in input. A special case of such expansions corresponds to Volterra series which are extensively used for the analysis, the simulation and the control of weakly nonlinear finite dimensional systems. The main results of this paper give computable bounds for both the convergence radius and the truncation error of the series. These results can be used for model simplification and analytic approximation of trajectories with a guaranteed quality. They are available for distributed and boundary control systems. As an illustration, these results are applied to an epidemic population dynamic model. In this example, it is shown that the truncation of the series at order 2 yields an accurate analytic approximation which can be used for time simulation and control issues. The relevance of the method is illustrated by simulations.     

On control and optimization of elastic multilink mechanisms

This paper presents a simple method for control of nonlinear elastic multilink mechanisms. The associated control law consists of open and closed loop components. The open loop component produces the desired overall rigid body motion, while the closed loop component suppresses the elastic motion relative to rigid body motion. Both the control law and the structural dynamics are referred to an inertial coordinate system. As a consequence, the control forces and control moments are easily simulated in a dynamic finite element analysis program. A series of numerical simulations illustrate the generality of this new method.     

Economic model predictive control of parabolic PDE systems: Addressing state estimation and computational efficiency

In a previous work [20], an economic model predictive control (EMPC) system for parabolic partial differential equation (PDE) systems was proposed. Through operating the PDE system in a time-varying fashion, the EMPC system demonstrated improved economic performance over steady-state operation. The EMPC system assumed the knowledge of the complete state spatial profile at each sampling period. From a practical point of view, measurements of the state variables are typically only available at a finite number of spatial positions. Additionally, the basis functions used to construct a reduced-order model (ROM) for the EMPC system were derived using analytical sinusoidal/cosinusoidal eigenfunctions. However, constructing a ROM on the basis of historical data-based empirical eigenfunctions by applying Karhunen-Loève expansion may be more computationally efficient. To address these issues, several EMPC systems are formulated for both output feedback implementation and with ROMs based on analytical sinusoidal/cosinusoidal eigenfunctions and empirical eigenfunctions. The EMPC systems are evaluated using a non-isothermal tubular reactor example, described by two nonlinear parabolic PDEs, where a second-order reaction takes place. The model accuracy, computational time, input and state constraint satisfaction, and closed-loop economic performance of the closed-loop tubular reactor under the different EMPC systems are compared.     

On characteristic roots and stability charts of delay differential equations

This work is devoted to the analytic study of the characteristic roots oftextitscalar autonomous delay differential equations with either real or complex coefficients. The focus is placed on the robust analysis of the position of the roots in the complex plane with respect to the variation of the coefficients, with the final aim of obtaining suitable representations for the relevant stability boundaries and charts. While the real case is almost standard (and known), the investigation of the complex case is not as immediate. Hence, a preliminary shift of the coefficients is proposed, which reduces the number of free parameters. This allows to extend the techniques used for the real case, also allowing for useful graphical visualization of the relevant stability charts. The present research is motivated on the basis of studying the stability of systems with delay. Copyright © 2011 John Wiley & Sons, Ltd.     

On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations

The input-to-state stability of time-invariant systems described by coupled differential and difference equations with multiple noncommensurate and distributed time delays is investigated in this paper. Such equations include neutral functional differential equations in Hale’s form (which model, for instance, partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal, hydraulic and electrical engineering. A general methodology for systematically studying the input-to-state stability, by means of Liapunov-Krasovskii functionals, with respect to measurable and locally essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the functional evaluated at the solution has been studied and solved by introducing the hypothesis that the functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case. It is proved that a linear neutral system in Hale’s form with stable difference operator is input-to-state stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example taken from the literature, concerning an electrical device, is reported, showing the effectiveness of the proposed methodology.     

On the asymptotic stability of coupled delay differential and continuous time difference equations

A two-step Liapunov-Krasovskii methodology for checking the asymptotic stability of nonlinear coupled delay differential and continuous time difference equations is proposed here. The feasibility of such methodology is shown by means of Liapunov-Krasovskii functionals with nonconstant kernels in the integrals, for instance discretized Liapunov-Krasovskii ones. An illustrative example taken from the literature, showing the effectiveness of the proposed method, is reported.     

On approximation of the Laplace–Beltrami operator and the Willmore energy of surfaces

Discrete Laplace–Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace–Beltrami operator are well‐studied, less is known about the strong form. We present a principle for constructing strongly consistent discrete Laplace–Beltrami operators based on the cotan weights. The consistency order we obtain, improves previous results reported for the mesh Laplacian. Furthermore, we prove consistency of the discrete Willmore energies corresponding to the discrete Laplace–Beltrami operators.     

On the complexity of deciding avoidability of sets of partial words

Blanchet-Sadri et al. have shown that Avoidability, or the problem of deciding the avoidability of a finite set of partial words over an alphabet of size k≥2, is NP-hard [F. Blanchet-Sadri, R. Jungers, J. Palumbo, Testing avoidability on sets of partial words is hard, Theoret. Comput. Sci. 410 (2009) 968-972]. Building on their work, we analyze in this paper the complexity of natural variations on the problem. While some of them are NP-hard, others are shown to be efficiently decidable. Using some combinatorial properties of de Bruijn graphs, we establish a correspondence between lengths of cycles in such graphs and periods of avoiding words, resulting in a tight bound for periods of avoiding words. We also prove that Avoidability can be solved in polynomial space, and reduces in polynomial time to the problem of deciding the avoidability of a finite set of partial words of equal length over the binary alphabet. We give a polynomial bound on the period of an infinite avoiding word, in the case of sets of full words, in terms of two parameters: the length and the number of words in the set. We give a polynomial space algorithm to decide if a finite set of partial words is avoided by a non-ultimately periodic infinite word. The same algorithm also decides if the number of words of length n avoiding a given finite set of partial words grows polynomially or exponentially with n.