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Estimating the Largest Elements of a Matrix
Get PDFWe derive an algorithm for estimating the largest values or for an matrix , along with their locations in the matrix. The matrix is accessed using only matrix--vector or matrix--matrix products. For p = 1 the algorithm estimates the norm or . The algorithm is based on a power method for mixed subordinate matrix norms and iterates on matrices, where is a parameter. For p = t = 1 we show that the algorithm is essentially equivalent to rook pivoting in Gaussian elimination; we also obtain a bound for the expected number of matrix--vector products for random matrices and give a class of counterexamples. Our numerical experiments show that for p = 1 the algorithm usually converges in just two iterations, requiring the equivalent of 4t matrix--vector products, and for t = 2 the algorithm already provides excellent estimates that are usually within a factor 2 of the largest element and frequently exact. For p > 1 we incorporate deflation to improve the performance of the algorithm. Experiments on real-life datasets show that the algorithm is highly effective in practice.
Read More: http://epubs.siam.org/doi/abs/10.1137/15M105364
Characterization of Objects by Fitting the Polarization Tensor
Get PDFThis thesis focuses on some mathematical aspects and a few recent applications of
the polarization tensor (PT). Here, the main concern of the study is to characterize
objects presented in electrical or electromagnetic fields by only using the PT. This is
possible since the PT contains significant information about the object such as shape,
orientation and material properties. Two main applications are considered in the study
and they are electrosensing fish and metal detection. In each application, we present a
mathematical formulation of the PT and briefly discuss its properties.
The PT in the electrosensing fish is actually based on the first order generalized
polarization tensor (GPT) while the GPT itself generalizes the classical PT called as
the P �olya-Szeg �o PT. In order to investigate the role of the PT in electrosensing fish,
we propose two numerical methods to compute the first order PT. The first method is
directly based on the quadrature method of numerical integration while the second
method is an adaptation of some terminologies of the boundary element method (BEM).
A code to use the first method is developed in Matlab while a script in Python is
written as an interface for using the new developed code for BEM called as BEM++.
When comparing the two methods, our numerical results show that the first order PT is
more accurate with faster convergence when computed by BEM++. During this study,
we also give a strategy to determine an ellipsoid from a given first order PT. This is
because we would like to propose an experiment to test whether electrosensing fish can
discriminate a pair of different objects but with the same first order PT such that the
pair could be an ellipsoid and some other object. In addition, the first order PT (or
the P �olya-Szeg �o PT) with complex conductivity (or complex permittivity) which is
similar to the PT for Maxwell�s equations is also investigated.
On the other hand, following recent mathematical foundation of the PT from the
eddy current model, we use the new proposed explicit formula to compute the rank
2 PT for a few metallic targets relevance in metal detection. We show that the PT
for the targets computed from the explicit formula agree to some degree of accuracy
with the PT obtained from metal detectors during experimental works and simulations
conducted by the engineers. This suggests to alternatively use the explicit formula
which depends only on the geometry and material properties of the target as well as
offering lower computational efforts than performing measurements with metal detectors
to obtain the PT. By using the explicit formula of the rank 2 PT, we also numerically
investigate some properties of the rank 2 PT where, the information obtained could be
useful to improve metal detection and also in other potential applications of the eddy
current. In this case, if the target is magnetic but non-conducting, the rank 2 PT of
the target can also be computed by using the explicit formula of the first order PT
Feynman path integrals and Lebesgue-Feynman measures
Get PDFWe define the class of Lebesgue-Feynman Measures (LFM) on any locally convex topological vector space and investigate transformations of the LFM generated by transformations of the domain and also discuss the connections of these transformations of the LFM with so-called quantum anomalies, improving some recent results of the authors and others. We revisit the contradiction between the points of view on quantum anomalies presented in the books of Fujikawa and Suzuki on the one hand, and of Cartier and DeWitt-Morette on the other, coming out in favour of the former
Compressing variable-coefficient exterior Helmholtz problems via RKFIT
Get PDFThe efficient discretization of Helmholtz problems on unbounded domains is a challenging task, in particular, when the wave medium is nonhomogeneous. We present a new numerical approach for compressing finite difference discretizations of such problems, thereby giving rise to efficient perfectly matched layers (PMLs) for nonhomogeneous media. This approach is based on the solution of a nonlinear rational least squares problem using the RKFIT method proposed in [M. Berljafa and S. Guettel, SIAM J. Matrix Anal. Appl., 36(2):894--916, 2015]. We show how the solution of this least squares problem can be converted into an accurate finite difference grid within a rational Krylov framework. Several numerical experiments are included. They indicate that RKFIT computes PMLs more accurate than previous analytic approaches and even works in regimes where the Dirichlet-to-Neumann functions to be approximated are highly irregular. Spectral adaptation effects allow for accurate finite difference grids with point numbers below the Nyquist limit
The Indian Schema Analogy Principles
Get PDFWe investigate the status within Unary Pure Inductive Logic of a family of analogy principles suggested by the Hindu Syllogism from Gotama's Nyāya-Sūtra showing that they all follow from the symmetry principle of Atom Exchangeability
A Daleckii-Krein formula for the Frechet derivative of a generalized matrix function
Get PDFWe state and prove an extension of the Daleckii-Krein theorem, thus obtaining an
explicit formula for the Frechet derivative of generalized matrix functions. Moreover,
we prove the differentiability of generalized matrix functions of real matrices under very
mild assumptions. For complex matrices, we argue that generalized matrix functions
are real differentiable but generally not complex differentiable. Finally, we discuss the
application of our result to the study of the condition number of generalized matrix
functions. Along our way, we also derive generalized matrix functional analogues of
a few classical theorems on polynomial interpolation of classical matrix functions and
their derivatives
Extended affine Weyl groups, the Baum-Connes correspondence and Langlands duality
Get PDFIn this paper we consider the Baum-Connes correspondence for the affine and extended affine Weyl groups of a compact connected semisimple Lie group. We show that the Baum-Connes correspondence in this context arises from Langlands duality for the Lie group
Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms
Get PDFTheoretical and computational aspects of matrix inverse trigonometric and inverse hyperbolic functions are studied. Conditions for existence are given, all possible values are characterized, and principal values \acos, \asin, \acosh, and \asinh are defined and shown to be unique primary matrix functions. Various functional identities are derived, some of which are new even in the scalar case, with care taken to specify precisely the choices of signs and branches. New results include a ``round trip'' formula that relates to and similar formulas for the other inverse functions. Key tools used in the derivations are the matrix unwinding function and the matrix sign function. A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree Pad\'e approximation is derived for computing acos, and it is shown how it can also be used to compute asin, acosh, and asinh. In numerical experiments the algorithm is found to behave in a forward stable fashion and to be superior to computing these functions via logarithmic formulas
A max-plus approach to incomplete Cholesky factorization preconditioners
Get PDFWe present a new method for constructing incomplete Cholesky factorization preconditioners for use in solving large sparse symmetric positive-definite linear systems. This method uses max-plus algebra to predict the positions of the largest entries in the Cholesky factor and then uses these positions as the sparsity pattern for the preconditioner. Our method builds on the max-plus
incomplete LU factorization preconditioner
recently proposed in [J. Hook and F. Tisseur, Incomplete LU preconditioner based on max-plus approximation of LU factorization, MIMS Eprint 2016.47, Manchester, 2016] but applied to symmetric positive-definite matrices, which comprise an important special case for the method and its application. A attractive feature of our approach is that the sparsity pattern of each column of the preconditioner can be computed in parallel. Numerical comparisons are made with other incomplete Cholesky factorization preconditioners using problems from a range of practical applications. We demonstrate that the new preconditioner can outperform traditional level-based preconditioners and offer a parallel alternative to a serial limited-memory based approach
Strong linearizations of rational matrices
Get PDFThis paper defines for the first time strong linearizations of arbitrary rational matrices, studies in depth properties and diferent characterizations
of such linear matrix pencils, and develops infinitely many
examples of strong linearizations that can be explicitly and easily constructed from a minimal state-space realization of the strictly proper
part of the considered rational matrix and the coefficients of the polynomial part. As a consequence, the results in this paper establish a rigorous foundation for the numerical computation of the complete structure of zeros and poles, both finite and at infinity, of any rational matrix by applying any well known backward stable algorithm for generalized eigenvalue problems to any of the strong linearizations explicitly constructed in this work. Since the results of this paper require to use several concepts that are not standard in matrix computations,
a considerable effort has been done to make the paper as self-contained as possible