Results 1  10
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23
Nearoptimal Columnbased Matrix Reconstruction
, 2011
"... We consider lowrank reconstruction of a matrix using a subset of its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are: (i) the use of fast approximate SVDlike decompositions for ..."
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Cited by 32 (3 self)
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We consider lowrank reconstruction of a matrix using a subset of its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are: (i) the use of fast approximate SVDlike decompositions for columnbased matrix reconstruction, and (ii) two deterministic algorithms for selecting rows from matrices with orthonormal columns, building upon the sparse representation theorem for decompositions of the identity that appeared in [1].
Should one compute the Temporal Difference fix point or minimize the Bellman Residual? The unified oblique projection view
"... We investigate projection methods, for evaluating a linear approximation of the value function of a policy in a Markov Decision Process context. We consider two popular approaches, the onestep Temporal Difference fixpoint computation (TD(0)) and the Bellman Residual (BR) minimization. We describe ..."
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Cited by 30 (5 self)
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We investigate projection methods, for evaluating a linear approximation of the value function of a policy in a Markov Decision Process context. We consider two popular approaches, the onestep Temporal Difference fixpoint computation (TD(0)) and the Bellman Residual (BR) minimization. We describe examples, where each method outperforms the other. We highlight a simple relation between the objective function they minimize, and show that while BR enjoys a performance guarantee, TD(0) does not in general. We then propose a unified view in terms of oblique projections of the Bellman equation, which substantially simplifies and extends the characterization of Schoknecht (2002) and the recent analysis of Yu & Bertsekas (2008). Eventually, we describe some simulations that suggest that if the TD(0) solution is usually slightly better than the BR solution, its inherent numerical instability makes it very bad in some cases, and thus worse on average.
Discrete empirical interpolation for nonlinear model reduction, in: Decision and Control, 2009 held jointly with the 2009
 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on, 2009
"... A dimension reduction method called Discrete Empirical Interpolation (DEIM) is proposed and shown to dramatically reduce the computational complexity of the popular Proper Orthogonal Decomposition (POD) method for constructing reducedorder models for unsteady and/or parametrized nonlinear partial ..."
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Cited by 19 (0 self)
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A dimension reduction method called Discrete Empirical Interpolation (DEIM) is proposed and shown to dramatically reduce the computational complexity of the popular Proper Orthogonal Decomposition (POD) method for constructing reducedorder models for unsteady and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard PODGalerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. Here we describe DEIM as a modification of POD that reduces the complexity as well as the dimension of general nonlinear systems of ordinary differential equations (ODEs). It is, in particular, applicable to ODEs arising from finite difference discretization of unsteady time dependent PDE and/or parametrically dependent steady state problems. Our contribution is a greatly simplified description of Empirical Interpolation in a finite dimensional setting. The method possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the 1D FitzHughNagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a longtime integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results. 1
New Error Bounds for Approximations from Projected Linear Equations
, 2008
"... We consider linear fixed point equations and their approximations by projection on a low dimensional subspace. We derive new bounds on the approximation error of the solution, which are expressed in terms of low dimensional matrices and can be computed by simulation. When the fixed point mapping is ..."
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Cited by 18 (8 self)
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We consider linear fixed point equations and their approximations by projection on a low dimensional subspace. We derive new bounds on the approximation error of the solution, which are expressed in terms of low dimensional matrices and can be computed by simulation. When the fixed point mapping is a contraction, as is typically the case in Markovian decision processes (MDP), one of our bounds is always sharper than the standard worst case bounds, and another one is often sharper. Our bounds also apply to the noncontraction case, including policy evaluation in MDP with nonstandard projections that enhance exploration. There are no error
ON THE FIELD OF VALUES OF OBLIQUE PROJECTIONS ∗
, 2010
"... Abstract. We highlight some properties of the field of values (or numerical range) W (P) of an oblique projector P on a Hilbert space, i.e., of an operator satisfying P 2 = P. If P is neither null nor the identity, we present a direct proof showing that W (P) = W (I − P), i.e., the field of values ..."
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Cited by 3 (0 self)
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Abstract. We highlight some properties of the field of values (or numerical range) W (P) of an oblique projector P on a Hilbert space, i.e., of an operator satisfying P 2 = P. If P is neither null nor the identity, we present a direct proof showing that W (P) = W (I − P), i.e., the field of values of an oblique projection coincides with that of its complementary projection. We also show that W (P) is an elliptical disk with foci at 0 and 1 and eccentricity 1/‖P ‖. These two results combined provide a new proof of the identity ‖P ‖ = ‖I−P ‖. We discuss the relation between the minimal canonical angle between the range and the null space of P and the shape of W (P). In the finite dimensional case, we show a relation between the eigenvalues of matrices related to these complementary projections and present a second proof to the fact that W (P) is an elliptical disk. Key words. Idempotent operators. Oblique Projections. Field of Values. Numerical Range.
Constructively wellposed approximation method with unity infsup and continuity constants for partial differential equations
 Math. Comp
"... Abstract. Starting from the generalized LaxMilgram theorem and from the fact that the approximation error is minimized when the continuity and inf– sup constants are unity, we develop a theory that provably delivers wellposed approximation methods with unity continuity and inf–sup constants for nu ..."
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Cited by 2 (2 self)
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Abstract. Starting from the generalized LaxMilgram theorem and from the fact that the approximation error is minimized when the continuity and inf– sup constants are unity, we develop a theory that provably delivers wellposed approximation methods with unity continuity and inf–sup constants for numerical solution of linear partial differential equations. We demonstrate our singleframework theory on scalar hyperbolic equations to constructively derive two different hp finite element methods. The first one coincides with a least squares discontinuous Galerkin method, and the other appears to be new. Both methods are proven to be trivially wellposed, with optimal hpconvergence rates. The numerical results show that our new discontinuous finite element method, namely a discontinuous PetrovGalerkin method, is more accurate, has optimal convergence rate, and does not seem to have nonphysical diffusion compared to the upwind discontinuous Galerkin method. 1.
A Stabilized Mixed Finite Element Method for Thin Plate Splines Based on Biorthogonal Systems
, 2009
"... The thin plate spline is a popular tool for the interpolation and smoothing of scattered data. In this paper we propose a novel stabilized mixed finite element method for the discretization of thin plate splines. The mixed formulation is obtained by introducing the gradient of the smoother as an add ..."
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The thin plate spline is a popular tool for the interpolation and smoothing of scattered data. In this paper we propose a novel stabilized mixed finite element method for the discretization of thin plate splines. The mixed formulation is obtained by introducing the gradient of the smoother as an additional unknown. Working with a pair of bases for the gradient of the smoother and the Lagrange multiplier which forms a biorthogonal system, we can easily eliminate these two variables (gradient of the smoother and Lagrange multiplier) leading to a positive definite formulation. The optimal a priori estimate is proved by using a superconvergence property of a gradient recovery operator. Key words: Thin plate splines, scattered data smoothing, mixed finite element method, saddle point problem, biorthogonal system, a priori estimate AMS subject classification: 65D10, 65D15, 65L60, 41A15 1
A note on the norm of oblique projections
, 2013
"... Abstract The purpose of this note is to give a somewhat simplified version of ..."
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Abstract The purpose of this note is to give a somewhat simplified version of
A GRADIENT RECOVERY OPERATOR BASED ON AN OBLIQUE PROJECTION ∗
"... Abstract. We present a construction of a gradient recovery operator based on an oblique projection, where the basis functions of two involved spaces satisfy a condition of biorthogonality. The biorthogonality condition guarantees that the recovery operator is local. Key words. gradient recovery, a p ..."
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Abstract. We present a construction of a gradient recovery operator based on an oblique projection, where the basis functions of two involved spaces satisfy a condition of biorthogonality. The biorthogonality condition guarantees that the recovery operator is local. Key words. gradient recovery, a posteriori error estimate, biorthogonal system AMS subject classifications. 65N30, 65N15, 65N50