Results 1  10
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12
A wildland fire model with data assimilation
, 2006
"... A wildfire model is formulated based on balance equations for energy and fuel, where the fuel loss due to combustion corresponds to the fuel reaction rate. The resulting coupled partial differential equations have coefficients which can be approximated from prior measurements of wildfires. An Ensemb ..."
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Cited by 22 (3 self)
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A wildfire model is formulated based on balance equations for energy and fuel, where the fuel loss due to combustion corresponds to the fuel reaction rate. The resulting coupled partial differential equations have coefficients which can be approximated from prior measurements of wildfires. An Ensemble Kalman Filter technique is then used to assimilate temperatures measured at selected points into running wildfire simulations. The assimilation technique is able to modify the simulations to track the measurements correctly even if the simulations were started with an erroneous ignition location that is quite far away from the correct one.
PredictorCorrector Ensemble Filters for the Assimilation of Sparse Data into HighDimensional Nonlinear Systems
, 2006
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RAYLEIGHRITZ MAJORIZATION ERROR BOUNDS WITH APPLICATIONS TO FEM AND SUBSPACE ITERATIONS
, 2008
"... The RayleighRitz method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is Ainvariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspace ..."
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Cited by 8 (4 self)
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The RayleighRitz method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is Ainvariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is Ainvariant, the absolute changes in the Ritz values of A with respect to X compared to the Ritz values with respect to Y represent the absolute eigenvalue approximation error. A recent paper [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 548559] by M. Argentati et al. bounds the error in terms of the principal angles between X and Y using weak majorization, e.g., a sharp bound is proved if X corresponds to a contiguous set of extreme eigenvalues of A. In this paper, we extend this sharp bound to dimX ≤ dimY and to the general case of an arbitrary Ainvariant subspace X, which was a conjecture in this previous paper. We present our RayleighRitz majorization error bound in the context of the finite element method (FEM), and show how it can improve known FEM eigenvalue error bounds. We derive a new majorizationtype convergence rate bound of subspace iterations and combine it with the previous result to obtain a similar bound for the block Lanczos method.
Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix
 SIAM J. Matrix Anal. Appl
"... Abstract. The RayleighRitz method is widely used for eigenvalue approximation. Given a matrix X with columns that form an orthonormal basis for a subspace X, and a Hermitian matrix A, the eigenvalues of X H AX are called Ritz values of A with respect to X. If the subspace X is Ainvariant then the ..."
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Cited by 5 (4 self)
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Abstract. The RayleighRitz method is widely used for eigenvalue approximation. Given a matrix X with columns that form an orthonormal basis for a subspace X, and a Hermitian matrix A, the eigenvalues of X H AX are called Ritz values of A with respect to X. If the subspace X is Ainvariant then the Ritz values are some of the eigenvalues of A. If the Ainvariant subspace X is perturbed to give rise to another subspace Y, then the vector of absolute values of changes in Ritz values of A represents the absolute eigenvalue approximation error using Y. We bound the error in terms of principal angles between X and Y. We capitalize on ideas from a recent paper [DOI: 10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute values of differences between Ritz values for subspaces X and Y was weakly (sub)majorized by a constant times the sine of the vector of principal angles between X and Y, the constant being the spread of the spectrum of A. In that result no assumption was made on either subspace being Ainvariant. It was conjectured there that if one of the trial subspaces is Ainvariant then an analogous weak majorization bound should be much stronger as it should only involve terms of the order of sine squared. Here we confirm this conjecture. Specifically we prove that the absolute eigenvalue error is weakly majorized by a constant times the sine squared of the vector of principal angles between the subspaces X and Y, where the constant is proportional to the spread of the spectrum of A. For many practical cases we show that the proportionality factor is simply one, and that this bound is sharp. For the general case we can only prove the result with a slightly larger constant, which we believe is artificial. Key words. Hermitian matrices, angles between subspaces, majorization, Lidskii’s eigenvalue theorem, perturbation bounds, Ritz values, RayleighRitz method, invariant subspace.
Constructing a Level Function for Fireline Data Assimilation
, 2006
"... A new method is described for determining the values of the level function from a given interface that is to be represented exactly as the zero level set. The level function is linear on boundary segments of the grid cells, which precludes the definition of the level function on the grid nodes as th ..."
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Cited by 3 (0 self)
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A new method is described for determining the values of the level function from a given interface that is to be represented exactly as the zero level set. The level function is linear on boundary segments of the grid cells, which precludes the definition of the level function on the grid nodes as the signed distance from the interface. Instead, the values of the level function on a strip along the interface are found by solving a constrained least squares problem. The level function is then continued by a nearest neighbor algorithm, or a fast marching method can be used. The method was applied to representation of the burning region and of the fireline from a wildfire simulation code. Unlike other descriptions of interfaces, level set representations can be easily combined by forming linear combinations. This is useful in data assimilation methods, which modify the state of running simulation to match the data.
L.H.: Distance between subspaces of different dimensions. ArXiv eprints
, 2014
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Principal Angles Between Subspaces and Their Tangents
, 2012
"... Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced and used via their cosines. The tangents of PABS have attracted relatively less attention, but are im ..."
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Cited by 1 (0 self)
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Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced and used via their cosines. The tangents of PABS have attracted relatively less attention, but are important for analysis of convergence of subspace iterations for eigenvalue problems. We explicitly construct matrices, such that their singular values are equal to the tangents of PABS, using several approaches: orthonormal and nonorthonormal bases for subspaces, and orthogonal projectors.
MAJORIZATION BOUNDS FOR RITZ VALUES OF HERMITIAN MATRICES
, 2008
"... Given an approximate invariant subspace we discuss the effectiveness of majorization bounds for assessing the accuracy of the resulting RayleighRitz approximations to eigenvalues of Hermitian matrices. We derive a slightly stronger result than previously for the approximation of k extreme eigenvalu ..."
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Given an approximate invariant subspace we discuss the effectiveness of majorization bounds for assessing the accuracy of the resulting RayleighRitz approximations to eigenvalues of Hermitian matrices. We derive a slightly stronger result than previously for the approximation of k extreme eigenvalues, and examine some advantages of these majorization bounds compared with classical bounds. From our results we conclude that the majorization approach appears to be advantageous, and that there is probably much more work to be carried out in this direction.
Angles Between Subspaces and Their Tangents
, 2013
"... Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomp ..."
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Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomposition. We utilize the same idea for the tangents, i.e., explicitly construct matrices, such that their singular values are equal to the tangents of PABS, using several approaches: orthonormal and nonorthonormal bases for subspaces, as well as projectors. Such a construction has applications, e.g., in analysis of convergence of subspace iterations for eigenvalue problems.
SCHUBERT VARIETIES AND DISTANCES BETWEEN SUBSPACES OF DIFFERENT DIMENSIONS
"... Abstract. We resolve two fundamental problems regarding subspace distances that have arisen considerably often in applications: How could one define a notion of distance between (i) two linear subspaces of different dimensions, or (ii) two affine subspaces of the same dimension, in a way that genera ..."
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Abstract. We resolve two fundamental problems regarding subspace distances that have arisen considerably often in applications: How could one define a notion of distance between (i) two linear subspaces of different dimensions, or (ii) two affine subspaces of the same dimension, in a way that generalizes the usual Grassmann distance between equidimensional linear subspaces? We show that (i) is the distance of a point to a Schubert variety, and (ii) is the distance within the Grassmannian of affine subspaces. In our context, a Schubert variety and the Grassmannian of affine subspaces are both regarded as subsets of the usual Grassmannian of linear subspaces. Combining (i) and (ii) yields a notion of distance between (iii) two affine subspaces of different dimensions. Aside from reducing to the usual Grassmann distance when the subspaces in (i) are equidimensional or when the affine subspaces in (ii) are linear subspaces, these distances are intrinsic and do not depend on any embedding of the Grassmannian into a larger ambient space. Furthermore, they can all be written down as concrete expressions involving principal angles, and are efficiently computable in numerically stable ways. We show that our results are largely independent of the Grassmann distance — if desired, it may be substituted by any other common distances between subspaces. Central to our approach to these problems is a concrete algebraic geometric view of the Grassmannian that parallels the differential geometric perspective that is now wellestablished in applied and computational mathematics. A secondary goal of this article is to demonstrate that the basic algebraic geometry of Grassmannian can be just as accessible and useful to practitioners. 1.