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Fast Solvers and Domain Decomposition Preconditioners for Spectral Element Discretizations of Problems in H(curl
, 2001
"... The URLs given were last checked and found valid in November 2001. To Alla, for all her help and care To Lady Mathematics, for all the fun and moments of enlightenment iv Acknowledgments First and foremost I want to thank my advisor and friend, Olof Widlund, for proposing the thesis subject, and for ..."
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The URLs given were last checked and found valid in November 2001. To Alla, for all her help and care To Lady Mathematics, for all the fun and moments of enlightenment iv Acknowledgments First and foremost I want to thank my advisor and friend, Olof Widlund, for proposing the thesis subject, and for all his support and help in the last six years, four of them as his student. I also want to thank Yu Chen and Jonathan Goodman for their willingness to serve as readers on short notice. I thank all the faculty, staff, and students of the Courant Institute who contributed to create a warm and motivating atmosphere. I thank all the professors who transmitted some of their excitement about mathematics to me, I especially want to mention John Rinzel and Bud Mishra. I also want to thank all who fed my neverending interest in all things mathematical and who were as curious as me about mathematics, physics and all that. Thank you, Sávio and Franz. I have had the privilege of becoming very good friends with four fantastic people during my several
Model Reference Adaptive Control of Distributed Parameter Systems
 SIAM J. Control Optim
, 1995
"... A model reference adaptive control law is defined for nonlinear distributed parameter systems. The reference model is assumed to be governed by a strongly coercive linear operator defined with respect to a Gelfand triple of reflexive Banach and Hilbert spaces. The resulting nonlinear closed loop sys ..."
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A model reference adaptive control law is defined for nonlinear distributed parameter systems. The reference model is assumed to be governed by a strongly coercive linear operator defined with respect to a Gelfand triple of reflexive Banach and Hilbert spaces. The resulting nonlinear closed loop system is shown to be well posed. The tracking error is shown to converge to zero, and regularity results for the control input and the output are established. With an additional richness, or persistence of excitation assumption, the parameter error is shown to converge to zero as well. A finite dimensional approximation theory is developed. Examples involving both first (parabolic) and second (hyperbolic) order systems and linear and nonlinear systems are discussed, and numerical simulation results are presented. Supported in part by DFG. y Supported in part by the Air Force Office of Scientific Research under grant AFOSR F496209310198, and in part by NASA under grant NAG11600. z S...
On the Persistence of Excitation in the Adaptive Estimation of Distributed Parameter Systems
 IEEE Trans. Automat. Control
, 1996
"... Persistence of excitation is a sufficient condition for parameter convergence in adaptive identification schemes for dynamical systems. For abstract parabolic and hyperbolic distributed parameter systems, this condition requires that a family of bounded linear functionals be norm bounded away from z ..."
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Persistence of excitation is a sufficient condition for parameter convergence in adaptive identification schemes for dynamical systems. For abstract parabolic and hyperbolic distributed parameter systems, this condition requires that a family of bounded linear functionals be norm bounded away from zero. The level of persistence of excitation of the plant and its implications are considered for a simple parabolic and hyperbolic system. Its effect on the qualitative and quantitative behavior of the estimators is investigated. This research was supported in part by the Air Force Office of Scientific Research under grant AFOSR 910076. 1 Introduction In this short note we consider the persistence of excitation condition in the adaptive, or online, identification of abstract distributed parameter systems. For a certain class of schemes which we have studied in depth in [5] (see also [4]), the persistence of excitation of the plant is a sufficient condition for parameter convergence. T...
Adaptive Identification of Second Order Distributed Parameter Systems
 Inverse Problems
, 1997
"... The adaptive (online) estimation of parameters for a class of second order distributed parameter systems is considered. This class of systems, which includes abstract wave and beam equations with a variety of forms of damping, is frequently used to model the vibration of large flexible structures. ..."
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The adaptive (online) estimation of parameters for a class of second order distributed parameter systems is considered. This class of systems, which includes abstract wave and beam equations with a variety of forms of damping, is frequently used to model the vibration of large flexible structures. A combined state and parameter estimator is constructed as an initial value problem for an infinite dimensional evolution equation in weak or variational form. State convergence is established via a Lyapunovlike estimate. The finite dimensional notion of persistence of excitation is extended to the infinite dimensional case and used to establish parameter convergence. A finite dimensional approximation theory is presented and a convergence result is proven. An example involving the identification of a damped onedimensional wave equation is discussed and results of a numerical study are presented. This research was supported in part by the Air Force Office of Scientific Research under gran...
Error estimates for the discontinuous Galerkin methods for implicit parabolic equations
 No6, 2006, pp 24782499. MR2206444 (2006j:65280
"... Abstract. We analyze the classical discontinuous Galerkin method for a general parabolic equation. Symmetric error estimates for schemes of arbitrary order are presented. The ideas we develop allow us to relax many assumptions freqently required in previous work. For example, we allow different disc ..."
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Abstract. We analyze the classical discontinuous Galerkin method for a general parabolic equation. Symmetric error estimates for schemes of arbitrary order are presented. The ideas we develop allow us to relax many assumptions freqently required in previous work. For example, we allow different discrete spaces to be used at each time step and do not require the spatial operator to be self adjoint or independent of time. Our error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate. These projections are local and enjoy optimal approximation properties when the solution is sufficiently regular. 1. Introduction. We
Singlephase Flow in Composite Poroelastic Media
 Math. Methods Appl. Sci
, 2002
"... . The mathematical formulation and analysis of the BarenblattBiot model of elastic deformation and laminar flow in a heterogeneous porous medium is discussed. This describes consolidation processes in a fluidsaturated doublediffusion model of fractured rock. The model includes various degenerate ..."
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. The mathematical formulation and analysis of the BarenblattBiot model of elastic deformation and laminar flow in a heterogeneous porous medium is discussed. This describes consolidation processes in a fluidsaturated doublediffusion model of fractured rock. The model includes various degenerate cases, such as incompressible constituents or totally fissured components, and it is extended to include boundary conditions arising from partially exposed pores. The quasistatic initialboundary problem is shown to have a unique weak solution, and this solution is strong when the data are smoother. 1. Introduction Any model of fluid flow through a deformable solid matrix must account for the coupling between the mechanical behavior of the matrix and the fluid dynamics. For example, compression of the medium leads to increased pore pressure, if the compression is fast relative to the fluid flow rate. Conversely, an increase in pore pressure induces a dilation of the matrix in response to t...
A GENERALIZATION OF AUBRYMATHER THEORY TO PARTIAL DIFFERENTIAL EQUATIONS AND PSEUDODIFFERENTIAL EQUATIONS
"... Abstract. We discuss an AubryMathertype theory for solutions of nonlinear, possibly degenerate, elliptic PDEs and other pseudodifferential operators. We show that for certain PDEs and ΨDEs with periodic coefficients and a variational structure it is possible to find quasiperiodic solutions for ..."
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Abstract. We discuss an AubryMathertype theory for solutions of nonlinear, possibly degenerate, elliptic PDEs and other pseudodifferential operators. We show that for certain PDEs and ΨDEs with periodic coefficients and a variational structure it is possible to find quasiperiodic solutions for all frequencies. This results also hold under a generalized definition of periodicity that makes it possible to consider problems in universal covers of several manifolds, including manifolds with noncommutative fundamental groups. An abstract result will be provided, from which an AubryMathertype theory for concrete models will be derived.
Convergence of Approximations in Feedback Control of Structures
 North Carolina State University
, 1998
"... Convergence of linear quadratic regulator (LQR) problems in structures is discussed. The abstract formulation of the system using a variational framework based on sesquilinear forms is considered. Since convergence theorems require uniform stabilizability of the finite dimensional approximating syst ..."
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Convergence of linear quadratic regulator (LQR) problems in structures is discussed. The abstract formulation of the system using a variational framework based on sesquilinear forms is considered. Since convergence theorems require uniform stabilizability of the finite dimensional approximating system, we present a detailed proof of a fundamental lemma due to Banks and Ito which can be used to easily verify this condition for many applications. Existing results for the well posedness of the infinite dimensional system and convergence of Galerkin approximations are summarized. Keywords feedback control, approximation, LQR, control convergence 1 Introduction In this paper we discuss in detail the proof of Lemma 6.2 in the electronic and CRSC technical report versions of [1] (stated as Lemma 7.13 in [2]) which allows verification of uniform stabilizability of a family of finite dimensional approximating systems arising in feedback control formulations. This uniform stabilizability condi...
Distributed Capacitance Microstructure In Conductors
"... . A new model for distributed capacitance in a conducting medium is introduced as a system of local RC diffusion equations coupled by a global elliptic equation. This model contains the local geometry of the distributed capacitors on which charge is stored and the exchange of current flux on their i ..."
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. A new model for distributed capacitance in a conducting medium is introduced as a system of local RC diffusion equations coupled by a global elliptic equation. This model contains the local geometry of the distributed capacitors on which charge is stored and the exchange of current flux on their interface with the medium. The resulting degenerate initialboundaryvalue problem is shown to be well posed and certain singular limits are characterized. AMS Subject Classification: 35A05, 35K22, 35K55, 47H05, 47H06, 47H15 Key Words: nonlinear parabolic system, distributed systems, microstructure, maximal monotone operators, evolution equations 1. Introduction. Remarkable progress has been made in the fabrication and understanding of novel materials that do not occur in nature. Investigators have fabricated artificial periodic superlattices, also termed layered synthetic microstructures, consisting of alternating layers of different semiconductors, different metals, or semiconductors and me...
Analysis And Approximation For Inverse Problems In Contaminant Transport And Biodegradation Models
, 1994
"... . In this paper we consider the problem of estimating transport and biodegradation parameters in a contaminant transport model. We develop a convergence theory for parameter identification under approximation that includes the nonlinearities inherent in the biodegradation models. The functional anal ..."
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. In this paper we consider the problem of estimating transport and biodegradation parameters in a contaminant transport model. We develop a convergence theory for parameter identification under approximation that includes the nonlinearities inherent in the biodegradation models. The functional analytic approach provides a means for studying a variety of solute transport and biodegradation problems. *Research supported in part by AFOSR contract number F496209310153. 1. Introduction. Remediation of contaminated groundwater is a major scientific and technological challenge. One promising approach is bioremediation, in which either naturally occurring or externally introduced bacteria are used to decompose contaminants into nontoxic byproducts. Important elements in the design of bioremediation strategies include mathematical models and computational algorithms for accurate prediction of contaminant degradation and transport. The literature on modeling and computation for biodegrad...