On a Frequency-Stabilized Single Current Inverse Source Formulation

Several strategies are available for solving the inverse source problem in electromagnetics. Among them, many have been focusing in retrieving Love currents by solving, after regularization, for Love’s electric and magnetic currents. In this work we present a dual-element discretization, analysis, and stabilization of an inverse source formulation providing Love data by solving for only one current. This results in substantial savings and allows for an effective quasi-Helmholtz projector stabilization of the resulting operator. Theoretical considerations are complemented by numerical tests showing effectiveness and efficiency of the newly proposed method

A Fast Quasi-Conformal Mapping Preconditioner for Electromagnetic Integral Equations

Boundary Element Methods (BEMs) are efficient strategies to numerically solve electromagnetic radiation and scattering problems. Unfortunately, however, classical BEM formulations suffer from ill-conditioning when the frequency is low, or the discretization density is high. In the past, several remedies have been presented for these ill-conditioning problems including preconditioners based on Calderón identities, hierarchical bases, and current decompositions. While effective, these strategies however require ad-hoc procedures including mesh-refinements, new basis function definitions, and adapted fast methods that, if not implemented properly, can become computationally cumbersome

On preconditioning electromagnetic integral equations in the high frequency regime via helmholtz operators and quasi-helmholtz projectors

Fast and accurate resolution of electromagnetic problems via the boundary element method (BEM) is oftentimes challenged by conditioning issues occurring in three distinct regimes: (i) when the frequency decreases and the discretization density remains constant, (ii) when the frequency is kept constant while the discretization is refined and (iii) when the frequency increases along with the discretization density. While satisfactory remedies to the problems arising in regimes (i) and (ii), respectively based on Helmholtz decompositions and Calderon-like techniques have been presented, the last regime is still challenging. In fact, this last regime is plagued by both spurious resonances and ill-conditioning, the former can be tackled via combined field strategies and is not the topic of this work. In this contribution new symmetric scalar and vectorial electric type formulations that remain well-conditioned in all of the aforementioned regimes and that do not require barycentric discretization of the dense electromagnetic potential operators are presented along with a spherical harmonics analysis illustrating their key properties

On a Low-Frequency and Contrast Stabilized Full-Wave Volume Integral Equation Solver for Lossy Media

In this article, we present a new regularized electric flux volume integral equation (D-VIE) for modeling high-contrast conductive dielectric objects in a broad frequency range. This new formulation is particularly suitable for modeling biological tissues at low frequencies, as it is required by brain epileptogenic area imaging, but also at higher ones, as it is required by several applications, including, but not limited to, deep brain stimulation (DBS). When modeling inhomogeneous objects with high complex permittivities at low frequencies, the traditional D-VIE is ill-conditioned and suffers from numerical instabilities that result in slower convergence and less accurate solutions. In this work, we address these shortcomings by leveraging a new set of volume quasi-Helmholtz projectors. Their scaling by the material permittivity matrix allows for the rebalancing of the equation when applied to inhomogeneous scatterers and, thereby, makes the proposed method accurate and stable even for high complex permittivity objects until arbitrarily low frequencies. Numerical results, canonical and realistic, corroborate the theory and confirm the stability and the accuracy of this new method both in the quasi-static regime and at higher frequencies

Magnetic and Combined Field Integral Equations Based on the Quasi-Helmholtz Projectors

Boundary integral equation methods for analyzing electromagnetic scattering phenomena typically suffer from several of the following shortcomings: 1) ill-conditioning when the frequency is low; 2) ill-conditioning when the discretization density is high; 3) ill-conditioning when the structure contains global loops (which are computationally expensive to detect); 4) incorrect solution at low frequencies due to a loss of significant digits; and 5) the presence of spurious resonances. In this article, quasi-Helmholtz projectors are leveraged to obtain magnetic field integral equation (MFIE) that is immune to drawbacks 1)-4). Moreover, when this new MFIE is combined with a regularized electric field integral equation (EFIE), a new quasi-Helmholtz projector-combined field integral equation (CFIE) is obtained that also is immune to 5). The numerical results corroborate the theory and show the practical impact of the newly proposed formulations

Single-crystal diffraction at the high-pressure Indo-Italian beamline Xpress at Elettra, Trieste

In this study the first in situ high-pressure single-crystal X-ray diffraction experiments at Xpress, the Indo-Italian beamline of the Elettra synchrotron, Trieste (Italy), are reported. A description of the beamline experimental setup and of the procedures for single-crystal centring, data collection and processing, using diamond anvil cells, are provided. High-pressure experiments on a synthetic crystal of clinoenstatite (MgSiO3), CaCO3 polymorphs and a natural sample of leucophoenicite [Mn7Si3O12(OH)2] validated the suitability of the beamline experimental setup to: (i) locate and characterize pressure-induced phase transitions; (ii) solve ab initio the crystal structure of high-pressure polymorphs; (iii) perform fine structural analyses at the atomic scale as a function of pressure; (iv) disclose complex symmetry and structural features undetected using conventional X-ray sources

On a Fast Solution Strategy for a Surface-Wire Integral Formulation of the Anisotropic Forward Problem in Electroencephalography

This work focuses on a quasi-linear-in-complexity strategy for a hybrid surface-wire integral equation solver for the electroencephalography forward problem. The scheme exploits a block diagonally dominant structure of the wire self block— that models the neuronal fibers self interactions—and of the surface self block—modeling interface potentials. This structure leads to two Neumann iteration schemes further accelerated with adaptive integral methods. The resulting algorithm is linear up to logarithmic factors. Numerical results confirm the performance of the method in biomedically relevant scenarios

High-pressure behavior and P-induced phase transition of CaB3O4(OH)3·H2O (colemanite)

Colemanite (ideally CaB3O4(OH)3\ub7H2O, space group P21/a, unit-cell parameters: a ~ 8.74, b ~ 11.26, c ~ 6.10 \uc5, \u3b2 ~ 110.1\ub0) is one of the principal mineralogical components of borate deposits and the most important mineral commodity of boron. Its high-pressure behavior is here described, for the first time, by means of in situ single-crystal synchrotron X-ray diffraction with a diamond anvil cell up to 24 GPa (and 293 K). Colemanite is stable, in its ambient-conditions polymorph, up to 13.95 GPa. Between 13.95 and 14.91 GPa, an iso-symmetric first-order single-crystal to single-crystal phase transition (reconstructive in character) toward a denser polymorph (colemanite-II) occurs, with: aCOL-II=3\ub7aCOL, bCOL-II=bCOL, and cCOL-II=2\ub7cCOL. Up to 13.95 GPa, the bulk compression of colemanite is accommodated by the Ca-polyhedron compression and the tilting of the rigid three-membered rings of boron polyhedra. The phase transition leads to an increase in the average coordination number of both the B and Ca sites. A detailed description of the crystal structure of the high-P polymorph, compared to the ambient-conditions colemanite, is given. The elastic behaviors of colemanite and of its high-P polymorph are described by means of III- and II-order Birch-Murnaghan equations of state, respectively, yielding the following refined parameters: KV0=67(4) GPa and KV\u2032=5.5(7) [\u3b2V0=0.0149(9) GPa-1] for colemanite; KV0=50(8) GPa [\u3b2V0=0.020(3) GPa-1] for its high-P polymorph

A New Refinement-Free Preconditioner for the Symmetric Formulation in Electroencephalography

Widely employed for the accurate solution of the electroencephalography forward problem, the symmetric formulation gives rise to a first kind, ill-conditioned operator illsuited for complex modelling scenarios. This work presents a novel preconditioning strategy based on an accurate spectral analysis of the operators involved which, differently from other Calderón-based approaches, does not necessitate the barycentric refinement of the primal mesh (i.e., no dual matrix is required). The discretization of the new formulation gives rise to a well-conditioned, symmetric, positive-definite system matrix, which can be efficiently solved via fast iterative techniques. Numerical results for both canonical and realistic head models validate the effectiveness of the proposed formulation

Enumeration of simple random walks and tridiagonal matrices

We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the $n$-th power of a tridiagonal matrix and the enumeration of weighted paths of $n$ steps allows an easier combinatorial enumeration of the paths. It also seems promising for the theory of tridiagonal random matrices .Comment: several ref.and comments added, misprints correcte