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202
Numerical solution of saddle point problems
 ACTA NUMERICA
, 2005
"... Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has b ..."
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Cited by 179 (30 self)
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Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
Geometric Ergodicity and Hybrid Markov Chains
, 1997
"... Various notions of geometric ergodicity for Markov chains on general state spaces exist. In this paper, we review certain relations and implications among them. We then apply these results to a collection of chains commonly used in Markov chain Monte Carlo simulation algorithms, the socalled hybrid ..."
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Cited by 75 (24 self)
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Various notions of geometric ergodicity for Markov chains on general state spaces exist. In this paper, we review certain relations and implications among them. We then apply these results to a collection of chains commonly used in Markov chain Monte Carlo simulation algorithms, the socalled hybrid chains. We prove that under certain conditions, a hybrid chain will "inherit" the geometric ergodicity of its constituent parts. 1 Introduction A question of increasing importance in the Markov chain Monte Carlo literature (Gelfand and Smith, 1990; Smith and Roberts, 1993) is the issue of geometric ergodicity of Markov chains (Tierney, 1994, Section 3.2; Meyn and Tweedie, 1993, Chapters 15 and 16; Roberts and Tweedie, 1996). However, there are a number of different notions of the phrase "geometrically ergodic", depending on perspective (total variation distance vs. in L 2 ; with reference to a particular V function; etc.). One goal of this paper is to review and clarify the relationship...
The diffusion limit of transport equations derived from velocity jump processes
 Siam J. Appl. Math
, 2000
"... Abstract. In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations c ..."
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Cited by 41 (14 self)
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Abstract. In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropicdiffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak–Keller–Segel–Alt model for chemotaxis.
Generalized functional linear models
 Ann. Statist
, 2005
"... We propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function. A linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function, and the expe ..."
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Cited by 40 (6 self)
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We propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function. A linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function, and the expected value of the response is related to this linear predictor via a link function. If in addition a variance function is specified, this leads to a functional estimating equation which corresponds to maximizing a functional quasilikelihood. This general approach includes the special cases of the functional linear model, as well as functional Poisson regression and functional binomial regression. The latter leads to procedures for classification and discrimination of stochastic processes and functional data. We also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data, using a semiparametric quasilikelihood procedure. An essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated KarhunenLoève expansion. We develop asymptotic inference for the proposed class of generalized regression models. In the proposed asymptotic approach, the truncation parameter increases with sample size, and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics. We establish asymptotic normality for a properly scaled distance
Utility maximization in incomplete markets with random endowment
 Finance & Stochastics
, 2001
"... This paper solves a longstanding open problem in mathematical finance: to find a solution to the problem of maximizing utility from terminal wealth of an agent with a random endowment process, in the general, semimartingale model for incomplete markets, and to characterize it via the associated dua ..."
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Cited by 35 (2 self)
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This paper solves a longstanding open problem in mathematical finance: to find a solution to the problem of maximizing utility from terminal wealth of an agent with a random endowment process, in the general, semimartingale model for incomplete markets, and to characterize it via the associated dual problem. We show that this is indeed possible if the dual problem and its domain are carefully defined. More
GromovHausdorff distance for quantum metric spaces
 Mem. Amer. Math. Soc
"... Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distanc ..."
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Cited by 35 (5 self)
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Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distance. We show that the basic theorems of the classical theory have natural quantum analogues. Our main example involves the quantum tori, Aθ. We show, for consistently defined “metrics”, that if a sequence {θn} of parameters converges to a parameter θ, then the sequence {Aθn} of quantum tori converges in quantum Gromov–Hausdorff distance to Aθ. 1.
Recognition Using Region Correspondences
 International Journal of Computer Vision
, 1995
"... A central problem in object recognition is to determine the transformation that relates the model to the image, given some partial correspondence between the two. This is useful in determining whether an object is present in an image, and if so, determining where the object is. We present a novel me ..."
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Cited by 34 (7 self)
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A central problem in object recognition is to determine the transformation that relates the model to the image, given some partial correspondence between the two. This is useful in determining whether an object is present in an image, and if so, determining where the object is. We present a novel method of solving this problem that uses region information. In our approach the model is divided into volumes, and the image is divided into regions. Given a match between subsets of volumes and regions (without any explicit correspondence between different pieces of the regions) the alignment transformation is computed. The method applies to planar objects under similarity, affine, and projective transformations and to projections of 3D objects undergoing affine and projective transformations. 1 Introduction A fundamental problem in recognition is pose estimation. Given a correspondence between some portions of an object model and some portions of an image, determine the transformation th...
Feedback Control of Quantum State Reduction
, 2004
"... Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for th ..."
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Cited by 32 (2 self)
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Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability.
Group C ∗ algebras as compact quantum metric spaces
 Doc. Math
"... Abstract. Let ℓ be a length function on a group G, and let Mℓ denote the operator of pointwise multiplication by ℓ on ℓ 2 (G). Following Connes, Mℓ can be used as a “Dirac ” operator for C ∗ r (G). It defines a Lipschitz seminorm on C ∗ r (G), which defines a metric on the state space of C ∗ r (G). ..."
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Cited by 23 (0 self)
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Abstract. Let ℓ be a length function on a group G, and let Mℓ denote the operator of pointwise multiplication by ℓ on ℓ 2 (G). Following Connes, Mℓ can be used as a “Dirac ” operator for C ∗ r (G). It defines a Lipschitz seminorm on C ∗ r (G), which defines a metric on the state space of C ∗ r (G). We investigate whether the topology from this metric coincides with the weak ∗ topology (our definition of a “compact quantum metric space”). We give an affirmative answer for G = Zd when ℓ is a wordlength, or the restriction to Zd of a norm on Rd. This works for C ∗ r (G) twisted by a 2cocycle, and thus for noncommutative tori. Our approach involves Connes ’ cosphere algebra, and an interesting compactification of metric spaces which is closely related to geodesic rays. The group C ∗algebras of discrete groups provide a muchstudied class of “compact noncommutative spaces ” (that is, unital C ∗algebras). In [11] Connes showed that the “Dirac ” operator of an unbounded