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The perturbation of electromagnetic fields at distances that are large compared to the object�s size
No full textWe rigorously derive the leading order terms in asymptotic expansions for the scattered electric and magnetic fields in the presence of a small object at distances that are large compared to its size. Our expansions hold for fixed wavenumber when the scatterer is a (lossy) homogeneous dielectric object with constant material parameters or a perfect conductor. We also derive the corresponding leading order terms in expansions for the fields for a low frequency problem when the scatterer is a non�lossy homogeneous dielectric object with constant material parameters or a perfect conductor. In each case we express our results in terms of polarisation tensors
Characterising the shape and material properties of hidden targets from magnetic induction data
No full textThe purpose of this paper is to clear up a major mystery in metal detection and confirm that the engineering prediction of HT .M.HM, for the sensitivity of measurements of the perturbed magnetic field to the presence of a general conducting object placed in a low frequency background field, is correct. Explicitly, HT is the background field generated by the transmitter coil, HM is the background field generated by the receiving coil as if it were used as a transmitter and Mind is a rank 2 polarisation tensor, which describes the shape and material properties of the object. To show this, we apply a recently derived asymptotic formula for the perturbed magnetic field due to the presence of a conducting object, which is expressed in terms of a new class of rank 4 polarisation tensors (H. Ammari, J. Chen, Z. Chen, J. Garnier and D. Volkov Target detection and characterization from electromagnetic induction data, Journal de Mathe �matiques Pures et Applique �es (2013) http://dx.doi.org/10.1016/j.matpur.2013.05.002). At first sight this appears to contradict the engineering prediction, however, contrary to this, we show that at most 9 rather than 81 coefficients are required to describe the rank 4 tensor for a conducting object and a further 9 are required if the object is magnetic. We then show that the rank 4 tensor does in fact reduce to a rank 2 tensor, thus providing a solid theoretical foundation for the engineering prediction. Furthermore, by combining the reduced conductivity and permeability tensors, we obtain a symmetric rank 2 tensor, which describes a general conducting object in terms of just 6 complex independent coefficients. For objects with rotational and mirror symmetries we show that the number of coefficients is still smaller. We include numerical examples to demonstrate that the new polarisation tensors can be accurately computed by solving a vector valued transmission problem by hp�finite elements and include evidence to confirm that the asymptotic formula describing the perturbed fields agrees with the numerically predictions
Duality of matrix pencils and linearizations
No full textWe consider two theoretical tools that have been introduced decades ago but whose usage is not widespread in modern literature on matrix pencils. One is \emph{dual pencils}, a pair of pencils with the same regular part and related singular structures. They were introduced by V.~Kublanovskaya in the 1980s. The other is \emph{Wong chains}, families of subspaces, associated with (possibly singular) matrix pencils, that generalize Jordan chains. They were introduced by K.T.~Wong in the 1970s. Together, dual pencils and Wong chains form a powerful theoretical framework to treat elegantly singular pencils in applications, especially in the context of linearizations of matrix polynomials.
We first give a self-contained introduction to these two concepts, using modern language and extending them to a more general form; we describe the relation between them and show how they act on the Kronecker form of a pencil and on spectral and singular structures (eigenvalues, eigenvectors and minimal bases). Then we present several new applications of these results to more recent topics in matrix pencil theory, including: constraints on the minimal indices of singular Hamiltonian and symplectic pencils, new sufficient conditions under which pencils in , linearization spaces are strong linearizations, a new perspective on Fiedler pencils, and a link between the Möller-Stetter theorem and some linearizations of matrix polynomials
The Maximal Subgroups of E_7(2)
No full textHere we determine up to conjugacy all the maximal subgroups of the finite exceptional group of Lie-type E_7(2)
Multilevel communication optimal LU and QR factorizations for hierarchical platforms
Get PDFThis study focuses on the performance of two classical dense linear algebra algorithms, the LU and the QR factorizations, on multilevel hierarchical platforms. We first introduce a new model called Hi- erarchical Cluster Platform (HCP), encapsulating the characteristics of such platforms. The focus is set on reducing the communication requirements of studied algorithms at each level of the hierarchy. Lower bounds on communications are therefore extended with respect to the HCP model. We then introduce multilevel LU and QR algorithms tailored for those platforms, and provide a detailed performance anal- ysis. We also provide a set of numerical experiments and performance predictions demonstrating the need for such algorithms on large platforms
Living on the edge: A geometric theory of phase transitions in convex optimization
Get PDFRecent empirical research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerges in the minimization method for identifying a sparse vector from random linear samples. Indeed, this approach succeeds with high probability when the number of samples exceeds a threshold that depends on the sparsity level; otherwise, it fails with high probability.
This paper provides the first rigorous analysis that explains why phase transitions are ubiquitous in random convex optimization problems. It also describes tools for making reliable predictions about the quantitative aspects of the transition, including the location and the width of the transition region. These techniques apply to regularized linear inverse problems with random measurements, to demixing problems under a random incoherence model, and also to cone programs with random affine constraints.
These applications depend on foundational research in conic geometry. This paper introduces a new summary parameter, called the statistical dimension, that canonically extends the dimension of a linear subspace to the class of convex cones. The main technical result demonstrates that the sequence of conic intrinsic volumes of a convex cone concentrates sharply near the statistical dimension. This fact leads to an approximate version of the conic kinematic formula that gives bounds on the probability that a randomly oriented cone shares a ray with a fixed cone
The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential
Get PDFA new matrix function corresponding to the scalar unwinding number of
Corless, Hare, and Jeffrey is introduced.
This matrix unwinding function, , is shown to
be a valuable tool for deriving
identities involving the matrix logarithm and
fractional matrix powers,
revealing, for example, the
precise relation between and .
The unwinding function is also shown to be closely connected with the matrix
sign function.
An algorithm for computing the unwinding function based on the
Schur--Parlett method with a special reordering is proposed.
It is shown that matrix argument reduction using the function
,
which has eigenvalues with imaginary parts in the interval
and for which \e^A = \e^{\mathrm{mod}(A)},
can give significant computational savings in the evaluation
of the exponential by scaling and squaring algorithms
A Hyperbolic Augmented Elasto-Plastic Model for Pressure-Dependent Yield
Get PDFA three-dimensional elasto-plastic model for the deformation and flow of granular materials which generalises the plastic potential model and contains an additional term analogous to that appearing in the double shearing model is presented. It is shown that for planar flows the resulting system of first order partial differential equations is hyperbolic. This is in distinct contrast to the non-associated plastic potential model rule and the double shearing model, both of which fail to be hyperbolic. The ill-posedness of the planar double shearing model is due to the presence of the rotation-rate of the principal axes of stress. The ill-posedness of the non-associated plastic potential model is due to distinct quasi-static spatial stress and velocity characteristics. The present model attains well-posedness by replacing the planar rotation-rate of the principal stress axes by the vector intrinsic spin of a Cosserat continuum and using it to ensure identical spatial stress and velocity characteristics. Flows in which the intrinsic spin vector is constant in both space and time correspond to flows in an ordinary continuum. The model governing such flows is embedded into a Cosserat model in such a way that the characteristic structure is preserved
LU FACTORIZATION WITH PANEL RANK REVEALING PIVOTING AND ITS COMMUNICATION AVOIDING VERSION
Get PDFWe present the block LU factorization with panel rank revealing
pivoting (block LU_PRRP), a decomposition algorithm based on strong
rank revealing QR panel factorization.
Block LU_PRRP is more stable than Gaussian elimination with partial
pivoting (GEPP), with a theoretical upper bound of the growth factor
of , where is the size of the panel used
during the block factorization, is a parameter of the strong
rank revealing QR factorization, and is the number of columns of
the matrix. For example, if the size of the panel is , and
, then , where
is the upper bound of the growth factor of GEPP. Our
extensive numerical experiments show that the new factorization scheme
is as numerically stable as GEPP in practice, but it is more resistant
to pathological cases. The block LU_PRRP factorization does only
additional floating point operations compared to GEPP.
We also present block CALU_PRRP, a communication avoiding version of block
LU_PRRP that minimizes communication. Block CALU_PRRP is based on
tournament pivoting, with the selection of the pivots at each step of
the tournament being performed via strong rank revealing QR
factorization. Block CALU_PRRP is more stable than CALU, the communication avoiding
version of GEPP, with a theoretical upper bound of the growth factor
of , where is the height of the reduction tree used during tournament
pivoting. The upper bound of the growth factor of CALU is
. Block CALU_PRRP is also more stable in practice and
is resistant to pathological cases on which GEPP and CALU fail
Free centre-by-metabelian Lie algebras in characteristic 2
Get PDFWe study free centre-by-metabelian Lie algebras over a field of characteristic 2. By using homological methods, we determine the dimensions of the fine homogeneous components of the second derived algebra. In conjunction with earlier results by Mansuro\v{g}lu and the second author, this leads to a complete description of the additive structure of the second derived ideal in the free centre-by-metabelian Lie ring