Results 1  10
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56
Spectral Compression of Mesh Geometry
, 2000
"... We show how spectral methods may be applied to 3D mesh data to obtain compact representations. This is achieved by projecting the mesh geometry onto an orthonormal basis derived from the mesh topology. To reduce complexity, the mesh is partitioned into a number of balanced submeshes with minimal int ..."
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Cited by 180 (6 self)
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We show how spectral methods may be applied to 3D mesh data to obtain compact representations. This is achieved by projecting the mesh geometry onto an orthonormal basis derived from the mesh topology. To reduce complexity, the mesh is partitioned into a number of balanced submeshes with minimal interaction, each of which are compressed independently. Our methods may be used for compression and progressive transmission of 3D content, and are shown to be vastly superior to existing methods using spatial techniques, if slight loss can be tolerated.
Stochastic Power Control for Cellular Radio Systems
 IEEE Trans. Commun
, 1997
"... For wireless communication systems, iterative power control algorithms have been proposed to minimize transmitter powers while maintaining reliable communication between mobiles and base stations. To derive deterministic convergence results, these algorithms require perfect measurements of one or mo ..."
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Cited by 89 (8 self)
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For wireless communication systems, iterative power control algorithms have been proposed to minimize transmitter powers while maintaining reliable communication between mobiles and base stations. To derive deterministic convergence results, these algorithms require perfect measurements of one or more of the following parameters: (i) the mobile's signal to interference ratio (SIR) at the receiver, (ii) the interference experienced by the mobile, and (iii) the bit error rate. However, these quantities are often difficult to measure and deterministic convergence results neglect the effect of stochastic measurements. In this work, we develop distributed iterative power control algorithms that use readily available measurements. Two classes of power control algorithms are proposed. Since the measurements are random, the proposed algorithms evolve stochastically and we define the convergence in terms of the mean squared error (MSE) of the power vector from the optimal power vector that is t...
Structured Pseudospectra For Polynomial Eigenvalue Problems With Applications
 SIAM J. MATRIX ANAL. APPL
, 2001
"... Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. W ..."
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Cited by 46 (9 self)
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Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. We explore connections between structured pseudospectra, structured backward errors, and structured stability radii. Two main approaches for computing pseudospectra are described. One is based on a transfer function and employs a generalized Schur decomposition of the companion form pencil. The other, specific to quadratic polynomials, finds a solvent of the associated quadratic matrLx equation and thereby factorizes the quadratic hmatrLx. Possible approaches for large, sparse problems are also outlined. A collection of examples from vibrating systems, control theory, acoustics, and fluid mechanics is given to illustrate the techniques.
Numerical Computation of Deflating Subspaces of SkewHamiltonian/Hamiltonian Pencils
 SIAM J. Matrix Anal. Appl
, 2002
"... . We discuss the numerical solution of structured generalized eigenvalue problems that arise from linearquadratic optimal control problems, H1 optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires com ..."
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Cited by 42 (25 self)
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. We discuss the numerical solution of structured generalized eigenvalue problems that arise from linearquadratic optimal control problems, H1 optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires computing invariant and deating subspaces of matrices and matrix pencils with Hamiltonian and/or skewHamiltonian structure. We extend the recently developed methods for Hamiltonian matrices to the general case of skewHamiltonian/Hamiltonian pencils. The algorithms circumvent problems with skewHamiltonian/Hamiltonian matrix pencils that lack structured Schur forms by embedding them into matrix pencils that always admit a structured Schur form. The rounding error analysis of the resulting algorithms is favorable. For the embedded matrix pencils, the algorithms use structure preserving unitary matrix computations and are strongly backwards stable, i.e., they compute the exact structured Schur form of a nearby matrix pencil with the same structure. Keywords. eigenvalue problem, deating subspace, Hamiltonian matrix, skewHamiltonian matrix, skewHamiltonian/Hamiltonian matrix pencil. AMS subject classication. 49N10, 65F15, 93B40, 93B36. 1.
The conditioning of linearizations of matrix polynomials
 SIAM J. MATRIX ANAL. APPL
, 2005
"... The standard way of solving the polynomial eigenvalue problem of degree m in n × n matrices is to “linearize” to a pencil in mn × mn matrices and solve the generalized eigenvalue problem. For a given polynomial, P, infinitely many linearizations exist and they can have widely varying eigenvalue con ..."
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Cited by 36 (14 self)
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The standard way of solving the polynomial eigenvalue problem of degree m in n × n matrices is to “linearize” to a pencil in mn × mn matrices and solve the generalized eigenvalue problem. For a given polynomial, P, infinitely many linearizations exist and they can have widely varying eigenvalue condition numbers. We investigate the conditioning of linearizations from a vector space DL(P) of pencils recently identified and studied by Mackey, Mackey, Mehl, and Mehrmann. We look for the best conditioned linearization and compare the conditioning with that of the original polynomial. Two particular pencils are shown always to be almost optimal over linearizations in DL(P) for eigenvalues of modulus greater than or less than 1, respectively, provided that the problem is not too badly scaled and that the pencils are linearizations. Moreover, under this scaling assumption, these pencils are shown to be about as well conditioned as the original polynomial. For quadratic eigenvalue problems that are not too heavily damped, a simple scaling is shown to convert the problem to one that is well scaled. We also analyze the eigenvalue conditioning of the widely used first and second companion linearizations. The conditioning of the first companion linearization relative to that of P is shown to depend on the coefficient matrix norms, the eigenvalue, and the left eigenvectors of the linearization and of P. The companion form is found to be potentially much
Symmetric linearizations for matrix polynomials
 SIAM J. MATRIX ANAL. APPL
, 2006
"... A standard way of treating the polynomial eigenvalue problem P(λ)x = 0 is to convert it into an equivalent matrix pencil—a process known as linearization. Two vector spaces of pencils L1(P) and L2(P), and their intersection DL(P), have recently been defined and studied by Mackey, Mackey, Mehl, and M ..."
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Cited by 31 (12 self)
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A standard way of treating the polynomial eigenvalue problem P(λ)x = 0 is to convert it into an equivalent matrix pencil—a process known as linearization. Two vector spaces of pencils L1(P) and L2(P), and their intersection DL(P), have recently been defined and studied by Mackey, Mackey, Mehl, and Mehrmann. The aim of our work is to gain new insight into these spaces and the extent to which their constituent pencils inherit structure from P. For arbitrary polynomials we show that every pencil in DL(P) is block symmetric and we obtain a convenient basis for DL(P) built from block Hankel matrices. This basis is then exploited to prove that the first deg(P) pencils in a sequence constructed by Lancaster in the 1960s generate DL(P). When P is symmetric, we show that the symmetric pencils in L1(P) comprise DL(P), while for Hermitian P the Hermitian pencils in L1(P) form a proper subset of DL(P) that we explicitly characterize. Almost all pencils in each of these subsets are shown to be linearizations. In addition to obtaining new results, this work provides a selfcontained treatment of some of the key properties of DL(P) together with some new, more concise proofs.
Efficient Numerical Algorithms for Balanced Stochastic Truncation
, 2001
"... We propose an efficient numerical algorithm for relative error model reduction based on balanced stochastic truncation. The method uses fullrank factors of the Gramians to be balanced versus each other and exploits the fact that for largescale systems these Gramians are often of low numerical rank ..."
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Cited by 30 (2 self)
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We propose an efficient numerical algorithm for relative error model reduction based on balanced stochastic truncation. The method uses fullrank factors of the Gramians to be balanced versus each other and exploits the fact that for largescale systems these Gramians are often of low numerical rank. We use the easytoparallelize sign function method as the major computational tool in determining these fullrank factors and demonstrate the numerical performance of the suggested implementation of balanced stochastic truncation model reduction.
Backward error of polynomial eigenproblems solved by linearization
 Manchester Institute for Mathematical Sciences, The University of Manchester
, 2006
"... Abstract. The most widely used approach for solving the polynomial eigenvalue problem P(λ)x = ��m i=0 λi � Ai x =0inn × n matrices Ai is to linearize to produce a larger order pencil L(λ) =λX + Y, whose eigensystem is then found by any method for generalized eigenproblems. For a given polynomial P, ..."
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Cited by 25 (7 self)
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Abstract. The most widely used approach for solving the polynomial eigenvalue problem P(λ)x = ��m i=0 λi � Ai x =0inn × n matrices Ai is to linearize to produce a larger order pencil L(λ) =λX + Y, whose eigensystem is then found by any method for generalized eigenproblems. For a given polynomial P, infinitely many linearizations L exist and approximate eigenpairs of P computed via linearization can have widely varying backward errors. We show that if a certain onesided factorization relating L to P can be found then a simple formula permits recovery of right eigenvectors of P from those of L, and the backward error of an approximate eigenpair of P can be bounded in terms of the backward error for the corresponding approximate eigenpair of L. A similar factorization has the same implications for left eigenvectors. We use this technique to derive backward error bounds depending only on the norms of the Ai for the companion pencils and for the vector space DL(P) of pencils recently identified by Mackey, Mackey, Mehl, and Mehrmann. In all cases, sufficient conditions are identified for an optimal backward error for P. These results are shown to be entirely consistent with those of Higham, Mackey, and Tisseur on the conditioning of linearizations of P. Other contributions of this work are a block scaling of the companion pencils
Nonsymmetric algebraic Riccati equations and WienerHopf factorization for Mmatrices
 SIAM J. Matrix Anal. Appl
, 2001
"... Abstract. We consider the nonsymmetric algebraic Riccati equation for which the four coefficient matrices form an Mmatrix. Nonsymmetric algebraic Riccati equations of this type appear in applied probability and transport theory. The minimal nonnegative solution of these equations can be found by Ne ..."
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Cited by 24 (11 self)
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Abstract. We consider the nonsymmetric algebraic Riccati equation for which the four coefficient matrices form an Mmatrix. Nonsymmetric algebraic Riccati equations of this type appear in applied probability and transport theory. The minimal nonnegative solution of these equations can be found by Newton’s method and basic fixedpoint iterations. The study of these equations is also closely related to the socalled WienerHopf factorization for Mmatrices. We explain how the minimal nonnegative solution can be found by the Schur method and compare the Schur method with Newton’s method and some basic fixedpoint iterations. The development in this paper parallels that for symmetric algebraic Riccati equations arising in linear quadratic control.
On the iterative solution of a class of nonsymmetric algebraic Riccati equations
 SIAM J. Matrix Anal. Appl
"... Abstract. We consider the iterative solution of a class of nonsymmetric algebraic Riccati equations, which includes a class of algebraic Riccati equations arising in transport theory. For any equation in this class, Newton’s method and a class of basic fixedpoint iterations can be used to find its ..."
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Cited by 22 (11 self)
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Abstract. We consider the iterative solution of a class of nonsymmetric algebraic Riccati equations, which includes a class of algebraic Riccati equations arising in transport theory. For any equation in this class, Newton’s method and a class of basic fixedpoint iterations can be used to find its minimal positive solution whenever it has a positive solution. The properties of these iterative methods are studied and some practical issues are addressed. An algorithm is then proposed to find the minimal positive solution efficiently. Numerical results are also given.