On the Hierarchical Preconditioning of the Combined Field Integral Equation

This paper analyzes how hierarchical bases preconditioners constructed for the Electric Field Integral Equation (EFIE) can be effectively applied to the Combined Field Integral Equation (CFIE). For the case where no hierarchical solenoidal basis is available (e.g., on unstructured meshes), a new scheme is proposed: the CFIE is implicitly preconditioned on the solenoidal Helmholtz subspace by using a Helmholtz projector, while a hierarchical non-solenoidal basis is used for the non-solenoidal Helmholtz subspace. This results in a well-conditioned system. Numerical results corroborate the presented theory

A prescription for projectors to compute helicity amplitudes in D dimensions

This article discusses a prescription to compute polarized dimensionally regularized amplitudes, providing a recipe for constructing simple and general polarized amplitude projectors in D dimensions that avoids conventional Lorentz tensor decomposition and avoids also dimensional splitting. Because of the latter, commutation between Lorentz index contraction and loop integration is preserved within this prescription, which entails certain technical advantages. The usage of these D-dimensional polarized amplitude projectors results in helicity amplitudes that can be expressed solely in terms of external momenta, but different from those defined in the existing dimensional regularization schemes. Furthermore, we argue that despite being different from the conventional dimensional regularization scheme (CDR), owing to the amplitude-level factorization of ultraviolet and infrared singularities, our prescription can be used, within an infrared subtraction framework, in a hybrid way without re-calculating the (process-independent) integrated subtraction coefficients, many of which are available in CDR. This hybrid CDR-compatible prescription is shown to be unitary. We include two examples to demonstrate this explicitly and also to illustrate its usage in practice.Comment: Matched to the version to be published in Eur. Phys. J.

Computing diagonal form and Jacobson normal form of a matrix using Gr\"obner bases

In this paper we present two algorithms for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Gr\"obner bases. This can be viewed as the pre-processing for the computation of Jacobson normal form and also used for the computation of Smith normal form in the commutative case. We propose a general framework for handling, among other, operator algebras with rational coefficients. We employ special "polynomial" strategy in Ore localizations of non-commutative $G$-algebras and show its merits. In particular, for a given matrix $M$ we provide an algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds. The polynomial approach allows one to obtain more precise information, than the rational one e. g. about singularities of the system. Our implementation of polynomial strategy shows very impressive performance, compared with methods, which directly use fractions. In particular, we experience quite moderate swell of coefficients and obtain uncomplicated transformation matrices. This shows that this method is well suitable for solving nontrivial practical problems. We present an implementation of algorithms in SINGULAR:PLURAL and compare it with other available systems. We leave questions on the algorithmic complexity of this algorithm open, but we stress the practical applicability of the proposed method to a bigger class of non-commutative algebras

H-Infinity Optimal Interconnections

In this paper, a general ℋ∞ problem for continuous time, linear time invariant systems is formulated and solved in a behavioral framework. This general formulation, which includes standard ℋ∞ optimization as a special case, provides added freedom in the design of sub-optimal compensators, and can in fact be viewed as a means of designing optimal systems. In particular, the formulation presented allows for singular interconnections, which naturally occur when interconnecting first principles models

Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures with Free Body Motion

Negative imaginary (NI) systems play an important role in the robust control of highly resonant flexible structures. In this paper, a generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors, and with free body motion. This definition extends the existing definitions of NI systems. Also, necessary and sufficient conditions are provided for the stability of positive feedback control systems where the plant is NI according to the new definition and the controller is strictly negative imaginary. The stability conditions in this paper are given purely in terms of properties of the plant and controller transfer function matrices, although the proofs rely on state space techniques. Furthermore, the stability conditions given are independent of the plant and controller system order. As an application of these results, a case study involving the control of a flexible robotic arm with a piezo-electric actuator and sensor is presented

Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications

Robust Principal Component Analysis (RPCA) via rank minimization is a powerful tool for recovering underlying low-rank structure of clean data corrupted with sparse noise/outliers. In many low-level vision problems, not only it is known that the underlying structure of clean data is low-rank, but the exact rank of clean data is also known. Yet, when applying conventional rank minimization for those problems, the objective function is formulated in a way that does not fully utilize a priori target rank information about the problems. This observation motivates us to investigate whether there is a better alternative solution when using rank minimization. In this paper, instead of minimizing the nuclear norm, we propose to minimize the partial sum of singular values, which implicitly encourages the target rank constraint. Our experimental analyses show that, when the number of samples is deficient, our approach leads to a higher success rate than conventional rank minimization, while the solutions obtained by the two approaches are almost identical when the number of samples is more than sufficient. We apply our approach to various low-level vision problems, e.g. high dynamic range imaging, motion edge detection, photometric stereo, image alignment and recovery, and show that our results outperform those obtained by the conventional nuclear norm rank minimization method.Comment: Accepted in Transactions on Pattern Analysis and Machine Intelligence (TPAMI). To appea

Joint Unitary Triangularization for MIMO Networks

This work considers communication networks where individual links can be described as MIMO channels. Unlike orthogonal modulation methods (such as the singular-value decomposition), we allow interference between sub-channels, which can be removed by the receivers via successive cancellation. The degrees of freedom earned by this relaxation are used for obtaining a basis which is simultaneously good for more than one link. Specifically, we derive necessary and sufficient conditions for shaping the ratio vector of sub-channel gains of two broadcast-channel receivers. We then apply this to two scenarios: First, in digital multicasting we present a practical capacity-achieving scheme which only uses scalar codes and linear processing. Then, we consider the joint source-channel problem of transmitting a Gaussian source over a two-user MIMO channel, where we show the existence of non-trivial cases, where the optimal distortion pair (which for high signal-to-noise ratios equals the optimal point-to-point distortions of the individual users) may be achieved by employing a hybrid digital-analog scheme over the induced equivalent channel. These scenarios demonstrate the advantage of choosing a modulation basis based upon multiple links in the network, thus we coin the approach "network modulation".Comment: Submitted to IEEE Tran. Signal Processing. Revised versio