129 research outputs found

    Cubature rules for unitary Jacobi ensembles

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    We present Chebyshev type cubature rules for the exact integration of rational symmetric functions with poles on prescribed coordinate hyperplanes. Here the integration is with respect to the densities of unitary Jacobi ensembles stemming from the Haar measures of the orthogonal and the compact symplectic Lie groups.Comment: 10 pages, 1 tabl

    Costate Convergence with Legendre-Lobatto Collocation for Trajectory Optimization

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    This paper introduces a new method of discretization that collocates both endpoints of the domain and enables the complete convergence of the costate variables associated with the Hamilton boundary-value problem. This is achieved through the inclusion of an \emph{exceptional sample} to the roots of the Legendre-Lobatto polynomial, thus promoting the associated differentiation matrix to be full-rank. We study the location of the new sample such that the differentiation matrix is the most robust to perturbations and we prove that this location is also the choice that mitigates the Runge phenomenon associated with polynomial interpolation. Two benchmark problems are successfully implemented in support of our theoretical findings. The new method is observed to converge exponentially with the number of discretization points used

    Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra

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    We present an extension of the summation-by-parts (SBP) framework to tensor-product spectral-element operators in collapsed coordinates. The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms. Specifically, a methodology is developed for constructing triangular and tetrahedral spectral-element operators of any order which possess the SBP property (i.e. satisfying a discrete analogue of integration by parts) as well as a tensor-product decomposition. Such operators are then employed within the context of discontinuous spectral-element methods based on nodal expansions collocated at the tensor-product quadrature nodes as well as modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter approach resolving the time step limitation associated with the singularity of the collapsed coordinate transformation. Energy-stable formulations for curvilinear meshes are obtained using a skew-symmetric splitting of the metric terms, and a weight-adjusted approximation is used to efficiently invert the curvilinear modal mass matrix. The proposed schemes are compared to those using non-tensorial multidimensional SBP operators, and are found to offer comparable accuracy to such schemes in the context of smooth linear advection problems on curved meshes, but at a reduced computational cost for higher polynomial degrees.Comment: 26 pages, 5 figure

    Implementations of the Universal Birkhoff Theory for Fast Trajectory Optimization

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    This is part II of a two-part paper. Part I presented a universal Birkhoff theory for fast and accurate trajectory optimization. The theory rested on two main hypotheses. In this paper, it is shown that if the computational grid is selected from any one of the Legendre and Chebyshev family of node points, be it Lobatto, Radau or Gauss, then, the resulting collection of trajectory optimization methods satisfy the hypotheses required for the universal Birkhoff theory to hold. All of these grid points can be generated at an O(1)\mathcal{O}(1) computational speed. Furthermore, all Birkhoff-generated solutions can be tested for optimality by a joint application of Pontryagin's- and Covector-Mapping Principles, where the latter was developed in Part~I. More importantly, the optimality checks can be performed without resorting to an indirect method or even explicitly producing the full differential-algebraic boundary value problem that results from an application of Pontryagin's Principle. Numerical problems are solved to illustrate all these ideas. The examples are chosen to particularly highlight three practically useful features of Birkhoff methods: (1) bang-bang optimal controls can be produced without suffering any Gibbs phenomenon, (2) discontinuous and even Dirac delta covector trajectories can be well approximated, and (3) extremal solutions over dense grids can be computed in a stable and efficient manner

    Autonomous Trajectory Planning and Guidance Control for Launch Vehicles

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    This open access book highlights the autonomous and intelligent flight control of future launch vehicles for improving flight autonomy to plan ascent and descent trajectories onboard, and autonomously handle unexpected events or failures during the flight. Since the beginning of the twenty-first century, space launch activities worldwide have grown vigorously. Meanwhile, commercial launches also account for the booming trend. Unfortunately, the risk of space launches still exists and is gradually increasing in line with the rapidly rising launch activities and commercial rockets. In the history of space launches, propulsion and control systems are the two main contributors to launch failures. With the development of information technologies, the increase of the functional density of hardware products, the application of redundant or fault-tolerant solutions, and the improvement of the testability of avionics, the launch losses caused by control systems exhibit a downward trend, and the failures induced by propulsion systems become the focus of attention. Under these failures, the autonomous planning and guidance control may save the missions. This book focuses on the latest progress of relevant projects and academic studies of autonomous guidance, especially on some advanced methods which can be potentially real-time implemented in the future control system of launch vehicles. In Chapter 1, the prospect and technical challenges are summarized by reviewing the development of launch vehicles. Chapters 2 to 4 mainly focus on the flight in the ascent phase, in which the autonomous guidance is mainly reflected in the online planning. Chapters 5 and 6 mainly discuss the powered descent guidance technologies. Finally, since aerodynamic uncertainties exert a significant impact on the performance of the ascent / landing guidance control systems, the estimation of aerodynamic parameters, which are helpful to improve flight autonomy, is discussed in Chapter 7. The book serves as a valuable reference for researchers and engineers working on launch vehicles. It is also a timely source of information for graduate students interested in the subject

    Models of Delay Differential Equations

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    This book gathers a number of selected contributions aimed at providing a balanced picture of the main research lines in the realm of delay differential equations and their applications to mathematical modelling. The contributions have been carefully selected so that they cover interesting theoretical and practical analysis performed in the deterministic and the stochastic settings. The reader will find a complete overview of recent advances in ordinary and partial delay differential equations with applications in other multidisciplinary areas such as Finance, Epidemiology or Engineerin

    The Theory of Functional Connections: A Journey from Theory to Application

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    The Theory of Functional Connections (TFC) is a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The functionals derived from this method, called "constrained expressions," analytically satisfy the imposed constraints and can be leveraged to transform constrained optimization problems to unconstrained ones. By simplifying the optimization problem, this technique has been shown to produce a numerical scheme that is faster, more accurate, and robust to poor initialization. The content of this dissertation details the complete development of the Theory of Functional Connections. First, the seminal paper on the Theory of Functional Connections is discussed and motivates the discovery of a more general formulation of the constrained expressions. Leveraging this formulation, a rigorous structure of the constrained expression is produced with associated mathematical definitions, claims, and proofs. Furthermore, the second part of this dissertation explains how this technique can be used to solve ordinary differential equations providing a wide variety of examples compared to the state-of-the-art. The final part of this work focuses on unitizing the techniques and algorithms produced in the prior sections to explore the feasibility of using the Theory of Functional Connections to solve real-time optimal control problems, namely optimal landing problems

    Randomized matrix-free quadrature for spectrum and spectral sum approximation

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    We study randomized matrix-free quadrature algorithms for spectrum and spectral sum approximation. The algorithms studied are characterized by the use of a Krylov subspace method to approximate independent and identically distributed samples of vHf[A]v\mathbf{v}^{\sf H}f[\mathbf{A}]\mathbf{v}, where v\mathbf{v} is an isotropic random vector, A\mathbf{A} is a Hermitian matrix, and f[A]f[\mathbf{A}] is a matrix function. This class of algorithms includes the kernel polynomial method and stochastic Lanczos quadrature, two widely used methods for approximating spectra and spectral sums. Our analysis, discussion, and numerical examples provide a unified framework for understanding randomized matrix-free quadrature and shed light on the commonalities and tradeoffs between them. Moreover, this framework provides new insights into the practical implementation and use of these algorithms, particularly with regards to parameter selection in the kernel polynomial method

    On a Family of Variational Time Discretization Methods

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    We consider a family of variational time discretizations that generalizes discontinuous Galerkin (dG) and continuous Galerkin-Petrov (cGP) methods. In addition to variational conditions the methods also contain collocation conditions in the time mesh points. The single family members are characterized by two parameters that represent the local polynomial ansatz order and the number of non-variational conditions, which is also related to the global temporal regularity of the numerical solution. Moreover, with respect to Dahlquist’s stability problem the variational time discretization (VTD) methods either share their stability properties with the dG or the cGP method and, hence, are at least A-stable. With this thesis, we present the first comprehensive theoretical study of the family of VTD methods in the context of non-stiff and stiff initial value problems as well as, in combination with a finite element method for spatial approximation, in the context of parabolic problems. Here, we mainly focus on the error analysis for the discretizations. More concrete, for initial value problems the pointwise error is bounded, while for parabolic problems we rather derive error estimates in various typical integral-based (semi-)norms. Furthermore, we show superconvergence results in the time mesh points. In addition, some important concepts and key properties of the VTD methods are discussed and often exploited in the error analysis. These include, in particular, the associated quadrature formulas, a beneficial postprocessing, the idea of cascadic interpolation, connections between the different VTD schemes, and connections to other classes of methods (collocation methods, Runge-Kutta-like methods). Numerical experiments for simple academic test examples are used to highlight various properties of the methods and to verify the optimality of the proven convergence orders.:List of Symbols and Abbreviations Introduction I Variational Time Discretization Methods for Initial Value Problems 1 Formulation, Analysis for Non-Stiff Systems, and Further Properties 1.1 Formulation of the methods 1.1.1 Global formulation 1.1.2 Another formulation 1.2 Existence, uniqueness, and error estimates 1.2.1 Unique solvability 1.2.2 Pointwise error estimates 1.2.3 Superconvergence in time mesh points 1.2.4 Numerical results 1.3 Associated quadrature formulas and their advantages 1.3.1 Special quadrature formulas 1.3.2 Postprocessing 1.3.3 Connections to collocation methods 1.3.4 Shortcut to error estimates 1.3.5 Numerical results 1.4 Results for affine linear problems 1.4.1 A slight modification of the method 1.4.2 Postprocessing for the modified method 1.4.3 Interpolation cascade 1.4.4 Derivatives of solutions 1.4.5 Numerical results 2 Error Analysis for Stiff Systems 2.1 Runge-Kutta-like discretization framework 2.1.1 Connection between collocation and Runge-Kutta methods and its extension 2.1.2 A Runge-Kutta-like scheme 2.1.3 Existence and uniqueness 2.1.4 Stability properties 2.2 VTD methods as Runge-Kutta-like discretizations 2.2.1 Block structure of A VTD 2.2.2 Eigenvalue structure of A VTD 2.2.3 Solvability and stability 2.3 (Stiff) Error analysis 2.3.1 Recursion scheme for the global error 2.3.2 Error estimates 2.3.3 Numerical results II Variational Time Discretization Methods for Parabolic Problems 3 Introduction to Parabolic Problems 3.1 Regularity of solutions 3.2 Semi-discretization in space 3.2.1 Reformulation as ode system 3.2.2 Differentiability with respect to time 3.2.3 Error estimates for the semi-discrete approximation 3.3 Full discretization in space and time 3.3.1 Formulation of the methods 3.3.2 Reformulation and solvability 4 Error Analysis for VTD Methods 4.1 Error estimates for the l th derivative 4.1.1 Projection operators 4.1.2 Global L2-error in the H-norm 4.1.3 Global L2-error in the V-norm 4.1.4 Global (locally weighted) L2-error of the time derivative in the H-norm 4.1.5 Pointwise error in the H-norm 4.1.6 Supercloseness and its consequences 4.2 Error estimates in the time (mesh) points 4.2.1 Exploiting the collocation conditions 4.2.2 What about superconvergence!? 4.2.3 Satisfactory order convergence avoiding superconvergence 4.3 Final error estimate 4.4 Numerical results Summary and Outlook Appendix A Miscellaneous Results A.1 Discrete Gronwall inequality A.2 Something about Jacobi-polynomials B Abstract Projection Operators for Banach Space-Valued Functions B.1 Abstract definition and commutation properties B.2 Projection error estimates B.3 Literature references on basics of Banach space-valued functions C Operators for Interpolation and Projection in Time C.1 Interpolation operators C.2 Projection operators C.3 Some commutation properties C.4 Some stability results D Norm Equivalences for Hilbert Space-Valued Polynomials D.1 Norm equivalence used for the cGP-like case D.2 Norm equivalence used for final error estimate Bibliograph

    Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

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    This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions
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