an augmented electric field integral equation for layered medium green s function this paper proposes an augmented electric field integral equation a efie for layered medium green s function the newly developed matrix friendly formulation of layered medium green s function is applied in this method by separating charge as extra unknown list and enforcing the current continuity equation the traditional efie can be cast into a generalized saddle point system frequency scaling for the matrix friendly formulation is analyzed when frequency tends to zero rank deficiency and the charge neutrality enforcement of the a efie for layered medium green s function is discussed in detail the electrostatic limit of the a efie is also analyzed without any topological loop searching algorithm electrically small conducting structures embedded in a general layered medium can be simulated by using this new a efie formulation several numerical results are presented to validate this method at the end of this paper

volume delineation by fusion of fuzzy sets obtained from multiplanar tomographic images techniques of three dimensional 3 d volume delineation from tomographic medical imaging are usually based on 2 d contour definition for a given structure several different contours can be obtained depending on the segmentation method used or the user s choice the goal of this work is to develop a new method that reduces the inaccuracies generally observed a minimum volume that is certain to be included in the volume concerned membership degree spl mu 1 and a maximum volume outside which no part of the volume is expected to be found membership degree spl mu 0 are defined semi automatically the intermediate fuzziness region 0 spl mu 1 is processed using the theory of possibility the resulting fuzzy volume is obtained after data fusion from multiplanar slices the influence of the contrast to noise ratio was tested on simulated images the influence of slice thickness as well as the accuracy of the method were studied on phantoms the absolute volume error was less than 2 for phantom volumes of 2 8 cm sup 3 whereas the values obtained with conventional methods were much larger than the actual volumes clinical experiments were conducted and the fuzzy logic method gave a volume lower than that obtained with the conventional method our fuzzy logic method allows volumes to be determined with better accuracy and reproducibility

a novel symmetrical solenoidal basis for the mom analysis of closed surfaces in low frequency or dense mesh problems the electric field integral equation efie discretization requires the extraction of the solenoidal part of the current to avoid severe bad conditioning of the method of moments mom matrix the common approach to do this is using the loop basis functions for the solenoidal part here we show that the conventional loop basis has poor performances when applied to closed bodies we identify the reason of this behavior and propose a new solenoidal basis that is free of this problem in addition the associated generation algorithm also automatically detects topological complexities like handles and produces a basis with significantly improved performances as confirmed by the reported numerical examples the basis generation complexity as well as memory requirements or sparsity for the proposed basis are o n log n

a dc stable well balanced calderon preconditioned time domain electric field integral equation the marching on in time mot solution of the time domain electric field integral equation td efie has traditionally suffered from a number of problems including 1 instability 2 spurious static contributions plaguing the solution 3 low frequency breakdown and 4 dense discretization breakdown the first issue can be resolved by employing proper space time galerkin discretization schemes and accurate quadrature methods the second and the third issue have been resolved by the quasi helmholtz projected td efie qhp tdefie this contribution introduces a multiplicative preconditioner which can be applied to the qhp tdefie without further modifying the original scheme this preconditioner is based on calderon techniques and guarantees that the mot system can be solved efficiently using iterative methods not only for large time step sizes but also for dense spatial discretizations and for both simply and multiply connected geometries

loop star decomposition of basis functions in the discretization of the efie a general and readily applicable scheme is presented for the determination of the basis functions that allow the decomposition of the surface current into a solenoidal part and a nonsolenoidal remainder the proposed approach brings into correspondence these two parts with two scalar functions and generates the known loop and star basis functions the completeness of the loop star basis is discussed employing the presented scheme the issue of the irrotational property of the nonsolenoidal functions is addressed

fully hierarchical divergence conforming basis functions on tetrahedral cells with applications a new set of hierarchical divergence conforming vector basis functions on curvilinear tetrahedrons is presented the basis can model both mixed and full order polynomial spaces to arbitrary order as defined by raviart and thomas and nedelec solenoidal and non solenoidal components are separately represented on the element except in the case of the mixed first order space for which a decomposition procedure on the global mesh wide level is presented therefore the hierarchical aspect of the basis can be made to extend down to zero polynomial order the basis can be used to model divergence conforming quantities such as electromagnetic flux and current density fluid velocity etc within numerical methods such as the finite element method fem or integral equation based methods the basis is ideally suited to p adaptive analysis the paper concludes with two example applications the first is the fem based solution of the linearized acoustic vector wave equation where it is shown how the decomposition into solenoidal components and their complements can be used to stabilize the method at low frequencies the second is the solution of the electric field volume integral equation for electromagnetic scattering analysis where the benefits of the decomposition are again demonstrated

on an integration rule for products of barycentric coordinates over simplexes in rn in finite element computations one often needs to calculate integrals of products of powers of monomials over simplexes in this manuscript we prove a generalisation of the exact integration formula that was reported and proved for two dimensional simplexes by holand bell in 1969 we extend the proof to n dimensional simplexes and to simplexes on d dimensional manifolds in n dimensional space the results are used to develop finite element and boundary element simulation tools the proofs of the theorems are based on mathematical induction and coordinate mappings

analysis and regularization of the td efie low frequency breakdown low frequency breakdown phenomenon plaguing marching on in time solutions of time domain electric field integral equations is analyzed and cured a theoretical analysis of the breakdown identifies deficiencies in the treatments proposed to date a scheme for eliminating low frequency breakdown phenomena that leverages hierarchical regularization is presented it applies to arbitrary meshes and gives rise to linear systems that are better conditioned than those obtained with standard loop star tree bases the scheme s analytical properties and numerical results demonstrate its efficacy in accelerating the iterative solution of the time domain electric field integral equations

on a class of preconditioning methods for dense linear systems from boundary elements in this paper we discuss several sparse matrix based preconditioners suitable for preconditioning dense linear systems of boundary element equations all preconditioners involve only o n nonzeros we provide a framework for constructing operator splitting based preconditioners and use it to analyze a class of sparse preconditioners for singular integral equations a more efficient preconditioner is proposed that has a band 2 structure

high order discrete helmholtz decompositions for the electric field integral equation we develop a differential form based formalism to address the problem of low frequency breakdown of the electric field integral equation efie note in this formalism we approximate the surface magnetic field not the surface current as is conventionally the case a discrete helmholtz decomposition is achieved for both triangular and quadrilateral curvilinear meshes based on a star cotree decomposition these decompositions are based upon the construction of a canonical basis which ab initio possess the required separation into irrotational and nonirrotational spaces this makes the process of construction clear and generally applicable the construction of appropriate bases is demonstrated for a range of interpolation orders the effects of these constructions is demonstrated on a simple flat pec plate problem

efficient preconditioners for boundary element matrices based on grey box algebraic multigrid methods this paper is concerned with the iterative solution of the boundary element equations arising from standard galerkin boundary clement discretizations of first kind boundary integral operators of positive and negative order we construct efficient preconditioners on the basis of so called grey box algebraic multigrid methods that are well adapted to the treatment of boundary element matrices in particular the coarsening is based on an auxiliary matrix that represents the underlying topology in a certain sense this auxiliary matrix is additionally used for the construction of the smoothers and the transfer operators finally we present the results of some numerical studies that show the efficiency of the proposed algebraic multigrid preconditioners

fast calderon preconditioning for helmholtz boundary integral equations abstract calderon multiplicative preconditioners are an effective way to improve the condition number of first kind boundary integral equations yielding provable mesh independent bounds however when discretizing by local low order basis functions as in standard galerkin boundary element methods their computational performance worsens as meshes are refined this stems from the barycentric mesh refinement used to construct dual basis functions that guarantee the discrete stability of l 2 pairings based on coarser quadrature rules over dual cells and h matrix compression we propose a family of fast preconditioners that significantly reduce assembly and computation times when compared to standard versions of calderon preconditioning for the three dimensional helmholtz weakly and hyper singular boundary integral operators several numerical experiments validate our claims and point towards further enhancements

a new method for volume segmentation of pet images based on possibility theory 18f fluorodeoxyglucose positron emission tomography 18fdg pet has become an essential technique in oncology accurate segmentation and uptake quantification are crucial in order to enable objective follow up the optimization of radiotherapy planning and therapeutic evaluation we have designed and evaluated a new nearly automatic and operator independent segmentation approach this incorporated possibility theory in order to take into account the uncertainty and inaccuracy inherent in the image the approach remained independent of pet facilities since it did not require any preliminary calibration good results were obtained from phantom images percent error 18 38 mean 9 72 standard deviation results on simulated and anatomopathological data sets were quantified using different similarity measures and showed the method was efficient simulated images dice index 82 18 13 53 for suv 2 5 the approach could therefore be an efficient and robust tool for uptake volume segmentation and lead to new indicators for measuring volume of interest activity

on the efficiency of nested gmres preconditioners for 3d acoustic and elastodynamic h matrix accelerated boundary element methods abstract this article is concerned with the derivation of fast boundary element methods for 3d acoustic and elastodynamic problems in particular we are interested in the acceleration of hierarchical matrix h matrix based iterative solvers while h matrix representations allow to reduce the storage requirements and the cost of a matrix vector product the number of iterations for an iterative solver as the frequency or the problem size increases remains an issue we consider an inner outer preconditioning strategy i e the preconditioner is applied through an iterative solver at the inner level the preconditioner is defined as a h matrix representation of the system matrix with a given accuracy we investigate the influence of various parameters of the preconditioner i e the h matrix accuracy the gmres threshold and the maximum number of iterations of the inner solver different numerical results are presented to compare the efficiency of the preconditioner with respect to the unpreconditioned reference system finally we propose a way to define the optimal setting for this preconditioner

broadband muller mlfma for electromagnetic scattering by dielectric objects the multilevel fast multipole algorithm mlfma is very efficient for solving large scale electromagnetic scattering problems however at low frequencies or when the discretization is small compared with the wavelength both the mlfma and the underlying integral equation formulation typically suffer from a subwavelength breakdown for the electromagnetic scattering from a homogeneous dielectric object we obtain a stable and well conditioned surface integral formulation using a variant of the classical muumlller formulation and linear basis functions to overcome the subwavelength breakdown of the mlfma we use both propagating and evanescent plane waves to represent the fields the implementation is based on a combination of the spectral representation of the green s function and rokhlin s translation formula we also present a new interpolation scheme for the evanescent part which significantly improves the error controllability of the mlfma implementation several numerical results verify both the error controllability and scalability of the proposed algorithm

two level hierarchical fem method for modeling passive microwave devices in recent years multigrid methods have been proven to be very efficient for solving large systems of linear equations resulting from the discretization of positive definite differential equations by either the finite difference method or theh version of the finite element method in this paper an iterative method of the multiple level type is proposed for solving systems of algebraic equations which arise from thep version of the finite element analysis applied to indefinite problems a two levelv cycle algorithm has been implemented and studied with a gauss seidel iterative scheme used as a smoother the convergence of the method has been investigated and numerical results for a number of numerical examples are presented

preconditioned electric field integral equation using calderon identities and dual loop star basis functions an analytic preconditioner for the electric field integral equation based on the calderon identities is considered it is shown based on physical reasoning that rwg elements are not suitable for discretizing the electric field integral operator appearing in the preconditioner instead the geometrically dual basis functions proposed by buffa and christiansen are used however it is found that this preconditioner is vulnerable to roundoff errors at low frequencies a loop star decomposition of the buffa christiansen basis functions is presented along with numerical results demonstrating its effectiveness

on a generalized l_2 projection and some related stability estimates in sobolev spaces in this paper we investigate a stability estimate needed in hybrid finite and boundary element methods especially in hybrid coupled domain decomposition methods including mortar finite elements this stability estimate is equivalent to the stability of a generalized l 2 projection in certain sobolev spaces using piecewise linear trial spaces and appropriate piecewise constant test spaces the stability of the generalized l 2 projection is proved assuming some mesh conditions locally

spectral properties of the efie mom matrix for dense meshes with different types of bases the relationship between the properties of the basis functions employed to discretize the electric field integral equation efie and the eigenvalue spectrum of the resulting method of moments mom matrix is studied we concentrate on dense meshes i e on complex geometries with large number of unknowns per wavelength and or disparate mesh cells sizes and or low frequency these are typical occurrences in antennas packaging and microwave circuits we show that if the basis functions have separated fourier spectra the diagonal mom matrix entries are close to the eigenvalues and explain it in terms of the fourier spectrum of the green s function for such a basis diagonal preconditioning drastically reduces the condition number this does not happen with sub domain basis functions like the rao wilton glisson rwg or with their linear combination into loop tree or loop star bases that are employed to solve low frequency problems finally we analyze the properties of a multiresolution mr basis formed by linear combinations of rwg but whose functions possess some degree of fourier spectral resolution we show that there is still correspondence between matrix diagonal green s function fourier spectrum and matrix eigenvalues diagonal preconditioning of the mr mom matrix causes the eigenvalues to cluster around 1 as it happens with the mom matrix for the magnetic field integral equation this strongly impacts on the efie matrix condition number and the speed of convergence of iterative solvers

a discontinuous galerkin augmented electric field integral equation for multiscale electromagnetic scattering problems a discontinuous galerkin dg augmented electric field integral equation method based on the domain decomposition is proposed in this paper for full wave solution of multiscale targets the conventional surface integral equation based dg method allowing both conformal and nonconformal discretizations for multiscale structures suffers from low frequency breakdown by augmenting the dg efie with current continuity equation the proposed scheme can alleviate the low frequency breakdown in the augmented system the electric field integral equation and the current continuity equation are discretized by using hybrid basis functions including rao wilton glisson rwg and half rwg basis functions since the half rwg basis is not divergence conforming line charge degrees of freedom on the adjoining edges are introduced in this paper it is observed that the resulting linear system is well conditioned at low frequencies which leads to a rapid convergence over wide frequency band numerical examples demonstrate the accuracy and efficiency of the augmented system

integral equation based domain decomposition method for solving electromagnetic wave scattering from non penetrable objects the integral equation ie method is commonly utilized to model time harmonic electromagnetic em problems one of the greatest challenges in its applications arises in the solution of the resulting ill conditioned matrix equation we introduce a new domain decomposition method ddm for the ie solution of em wave scattering from non penetrable objects the proposed method is a non overlapping non conformal ddm and it provides a computationally efficient and effective preconditioner for the ie matrix equations moreover the proposed approach is very suitable for dealing with multi scale electromagnetic problems since each sub domain has its own characteristics length and will be meshed independently furthermore for each sub domain we are free to choose the most effective ie sub domain solver based on its local geometrical features and electromagnetic characteristics additionally the multilevel fast multi pole algorithm mlfma is utilized to accelerate the computations of couplings between sub domains numerical results demonstrate that the proposed method yields rapid convergence in the outer krylov iterative solution process finally simulations of several large scale examples testify to the effectiveness and robustness of the proposed ie based ddm

markus yamabe conjecture for non autonomous dynamical systems abstract the aim of this paper is the study of the problem of global asymptotic stability of trivial solutions of non autonomous dynamical systems both with continuous and discrete time we study this problem in the framework of general non autonomous dynamical systems cocycles in particular we present some new results for a non autonomous version of the markus yamabe conjecture

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

on efficient preconditioners for iterative solution of a galerkin boundary element equation for the three dimensional exterior helmholtz problem the paper presents a galerkin numerical method for solving the hyper singular boundary integral equations for the exterior helmholtz problem in three dimensions with a neumann s boundary condition previous work in the topic has often dealt with the collocation method with a piecewise constant approximation because high order collocation and galerkin methods are not available due to the presence of a hypersingular integral operator this paper proposes a high order galerkin method by using singularity subtraction technique to reduce the hyper singular operator to a weakly singular one moreover we show here how to extend the previous work j appl numer math 36 4 2001 475 489 on sparse preconditioners to the galerkin case leading to fast convergence of two iterative solvers the conjugate gradient normal method and the generalised minimal residual method a comparison with the collocation method is also presented for the helmholtz problem with several wavenumbers

the decoupled potential integral equation for time harmonic electromagnetic scattering this is the peer reviewed version of the following article vico f ferrando m greengard l and gimbutas z 2016 the decoupled potential integral equation for time harmonic electromagnetic scattering commun pur appl math 69 771 812 which has been published in final form at http dx doi org 10 1002 cpa 21585 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

a fuzzy locally adaptive bayesian segmentation approach for volume determination in pet accurate volume estimation in positron emission tomography pet is crucial for different oncology applications the objective of our study was to develop a new fuzzy locally adaptive bayesian flab segmentation for automatic lesion volume delineation flab was compared with a threshold approach as well as the previously proposed fuzzy hidden markov chains fhmc and the fuzzy c means fcm algorithms the performance of the algorithms was assessed on acquired datasets of the iec phantom covering a range of spherical lesion sizes 10 37 mm contrast ratios 4 1 and 8 1 noise levels 1 2 and 5 min acquisitions and voxel sizes 8 and 64 mm3 in addition the performance of the flab model was assessed on realistic nonuniform and nonspherical volumes simulated from patient lesions results show that flab performs better than the other methodologies particularly for smaller objects the volume error was 5 15 for the different sphere sizes down to 13 mm contrast and image qualities considered with a high reproducibility variation 4 by comparison the thresholding results were greatly dependent on image contrast and noise whereas fcm results were less dependent on noise but consistently failed to segment lesions 2 cm in addition flab performed consistently better for lesions 2 cm in comparison to the fhmc algorithm finally the flab model provided errors less than 10 for nonspherical lesions with inhomogeneous activity distributions future developments will concentrate on an extension of flab in order to allow the segmentation of separate activity distribution regions within the same functional volume as well as a robustness study with respect to different scanners and reconstruction algorithms

scalable preconditioned conjugate gradient inversion of vector finite element mass matrices mass matrices arise in the numerical solution of time dependent partial differential equations by the galerkin method since these systems must be inverted at each time step rapid inversion algorithms for these systems are important when nodal finite elements are used as basis functions it is known that the mass matrices can be consistently approximated by a diagonal matrix or solved by a scalable conjugate gradient method this may not be the case for other basis functions in this paper we show that the preconditioned conjugate gradient method is scalable when used to invert mass matrices that arise from vector finite element basis functions these basis functions are particularly important for solving maxwell s equations on unstructured grids by the galerkin method

a high order algorithm for obstacle scattering in three dimensions in this work we describe implement and analyse in detail a high order fully discrete spectral algorithm for solving the helmholtz equation exterior to a bounded sound soft sound hard or absorbing obstacle in three space dimensions with dirichlet neumann or robin impedance boundary conditions our algorithm may be thought of as a discrete glerkin method but it is also equivalent to a nystrom method after a simple transformation we test our algorithm with extensive computational experiments on a variety of three dimensional smooth and non smooth obstacles with conical singularities our tests include the computation of scattered and far fields induced by incident plane waves our method is shown to be very accurate for scattering from surfaces which are globally parameterised by spherical coordinates and tests show that it performs very much better than several of the well established fast algorithms for obstacle sattering on a range of such surfaces even some which are non smooth further we prove superalgebraic convergence of the scattered and far fields obtained using our algorithm in the case of smooth scatterers

a discretization method with the h_ mathrm div inner product for electric field integral equations a discretization method with the h mathrm div inner product for the electric field integral equation efie is proposed the efie with the conventional galerkin discretization shows bad accuracy for problems with a small frequency a problem known as the low frequency breakdown the discretization method proposed in this paper utilizes the h mathrm div scalar product with a scalar coefficient for the galerkin discretization and overcomes the low frequency problem with an appropriately chosen coefficient as regards the preconditioning we find that a naive use of the widely used calderon preconditioning is not efficient for reducing the computational time with the new discretization we therefore propose a new preconditioning which can accelerate the computation successfully the efficiency of the proposed discretization and preconditioning is verified through some numerical examples

a generalized calderon formula for open arc diffraction problems theoretical considerations we deal with the general problem of scattering by open arcs in two dimensional space we show that this problem can be solved by means of certain second kind integral equations of the form ns where n and s are first kind integral operators whose composition gives rise to a generalized calderon formula of the form ns j 0 k in a weighted periodized sobolev space here j 0 is a continuous and continuously invertible operator and k is a compact operator the ns formulation provides for the first time a second kind integral equation for the open arc scattering problem with neumann boundary conditions numerical experiments show that for both the dirichlet and neumann boundary conditions our second kind integral equations have spectra that are bounded away from zero and infinity as k to the authors knowledge these are the first integral equations for these problems that possess this desirable property this situation is in stark contrast with that arising from the related classical open surface hypersingular and single layer operators n and s whose composition ns maps for example the function 1 into a function that is not even square integrable our proofs rely on three main elements algebraic manipulations enabled by the presence of integral weights use of the classical result of continuity of the cesaro operator and explicit characterization of the point spectrum of j 0 which interestingly can be decomposed into the union of a countable set and an open set both of which are tightly clustered around 1 4 as shown in a separate contribution the new approach can be used to construct simple spectrally accurate numerical solvers and when used in conjunction with krylov subspace iterative solvers such as the generalized minimal residual method it gives rise to a dramatic reduction in the number of iterations compared with those required by other approaches

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

hierarchical bases for nonhierarchic 3 d triangular meshes we describe a novel basis of hierarchical multiscale functions that are linear combinations of standard rao wilton glisson rwg functions when the basis is used for discretizing the electric field integral equation efie for pec objects it gives rise to a linear system immune from low frequency breakdown and well conditioned for dense meshes the proposed scheme can be applied to any mesh with triangular facets and therefore it can be used as if it were an algebraic preconditioner the properties of the new system are confirmed by numerical results that show fast convergence rates of iterative solvers significantly better than those for the loop tree basis as a byproduct of the basis generation a generalization of the rwg functions to nonsimplex cells is introduced

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

a helmholtz stable fast solution of the electric field integral equation a new fast matrix vector multiplication scheme for the solution of the electric field integral equation is presented in this work similarly to other fast methods our approach reduces the matrix vector multiplication cost from o n2 to o n logn differently from other fast solvers however the effectiveness of efie preconditioning techniques such as quasi helmholtz decompositions or calderon approaches is maintained by our method even for very high matrix compression rates this is thanks to the fact that in the scheme we are proposing the contribution from the scalar potential when applied to or tested with solenoidal functions is always zero independent of the compression error in addition the new method will take advantage of the redundancies of the efie matrix in the low frequency dense discretization regime and it will further decrease both the memory storage and the multiplication cost with respect to currently available fast solvers numerical results will show the effectiveness of our approach and its impact on the solution of realistic problems

on a low frequency and refinement stable pmchwt integral equation leveraging the quasi helmholtz projectors classical poggio miller chan harrington wu tsai pmchwt formulations for modeling radiation and scattering from penetrable objects suffer from ill conditioning when the frequency is low or when the mesh density is high the most effective techniques to solve these problems unfortunately either require the explicit detection of the so called global loops of the structure or suffer from numerical cancellation at extremely low frequency in this contribution a novel regularization method for the pmchwt equation is proposed which is based on the quasi helmholtz projectors this method not only solves both the low frequency and the dense mesh ill conditioning problems of the pmchwt but it is immune from low frequency numerical cancellations and it does not require the detection of global loops this is obtained by projecting the range space of the pmchwt operator onto a dual basis by rescaling the resulting quasi helmholtz components by replicating the strategy in the dual space and finally by combining the primal and the dual equations in a calderon like fashion implementation related treatments and details alternate the theoretical developments in order to maximize impact and practical applicability of the approach finally numerical results corroborate the theory and show the effectiveness of the new schemes in real case scenarios

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

note on the markus yamabe conjecture for gradient dynamical systems abstract let v r n r n be a c 1 vector field which has a singular point o and its linearization is asymptotically stable at every point of r n we say that the vector field v satisfies the markus yamabe conjecture if the critical point o is a global attractor of the dynamical system x v x in this note we prove that if v is a gradient vector field i e v f f c 2 then the basin of attraction of the critical point o is the whole r n thus implying the markus yamabe conjecture for this class of vector fields an analogous result for discrete dynamical systems of the form x m 1 f x m is proved

high order calderon preconditioned time domain integral equation solvers two high order accurate calderon preconditioned time domain electric field integral equation tdefie solvers are presented in contrast to existing calderon preconditioned time domain solvers the proposed preconditioner allows for high order surface representations and current expansions by using a novel set of fully localized high order div and quasi curl conforming dqcc basis functions numerical results demonstrate that the linear systems of equations obtained using the proposed basis functions converge rapidly regardless of the mesh density and of the order of the current expansion

optimal additive schwarz preconditioning for adaptive 2d iga boundary element methods abstract we define and analyze local multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions hypersingular and weakly singular integral equations are considered we prove that the condition number of the preconditioned systems of linear equations is independent of the mesh size and the refinement level therefore the computational complexity when using appropriate iterative solvers is optimal our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results

a polynomial counterexample to the markus yamabe conjecture abstract we give a polynomial counterexample to both the markus yamabe conjecture and the discrete markus yamabe problem for all dimensions 3

embedding calderon multiplicative preconditioners in multilevel fast multipole algorithms calderon preconditioners have recently been demonstrated to be very successful in stabilizing the electric field integral equation efie for perfect electric conductors at lower frequencies previous authors have shown that by using a dense matrix preconditioner based on the calderon identities the low frequency instability is removed while still maintaining the inherent accuracy of the efie it was also demonstrated that the spectral properties of the caldero n preconditioner are conserved during discretization if the efie operator is discretized with rao wilton glisson expansion functions and the preconditioner with buffa christiansen expansion functions in this article we will show how the calderon multiplicative preconditioner cmp can be combined with fast multipole methods to accelerate the numerical solution leading to an overall complexity of o n logn for the entire iterative solution at low frequencies where the cmp is most useful the traditional multilevel fast multipole algorithm mlfma is unstable and we apply the nondirectional stable plane wave mlfma nspwmlfma that resolves the low frequency breakdown of the mlfma the combined algorithm will be called the cmp nspwmlfma applying the cmp nspwmlfma at open surfaces or very low frequencies leads to certain problems which will be discussed in this article

on the stability of the l_2 projection in fractional sobolev spaces in this paper we prove the stability of the l 2 projection onto the finite element trial space of piecewise polynomial in particular piecewise linear basis functions in h s omega for s in 0 1 we formulate explicit and computable local mesh conditions to be satisfied which depend on the sobolev index s in conclusion we prove a stability condition needed in the numerical analysis of mixed and hybrid boundary element methods as well as in the construction of efficient preconditioners in adaptive boundary and finite element methods

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

a hypersingular boundary integral method for two dimensional screen and crack problems we analyze hypersingular integral equations on a curved open smooth arc in r 2 that model either curved cracks in an elastic medium or the scattering of acoustic and elastic waves at a hard screen by using the mellin transformation we obtain sharp regularity results for the solution of these equations in sobolev spaces in the form of singular expansions in particular we show that the expansions do not contain logarithmic singularities

a multiresolution system of rao wilton glisson functions a multiresolution mr basis is described for the method of moments mom analysis of a generic 3 d conductor discretized with triangular cells the mr basis functions are constructed as linear combination of rao wilton glisson rwg basis functions thus allowing direct applicability on existing mom codes with respect to previous work by the authors the generation of the mr functions is significantly simplified yet keeping similar applicability and performance in terms of conditioning of the mom matrix convergence of the iterative solvers and possibility of sparsifying the mom matrix by clipping moreover with the proposed basis the basis change matrix from rwg to mr is highly sparse at all levels this allows a more efficient generation of the mr mom matrix

the k operator and the galerkin method for strongly elliptic equations on smooth curves local estimates superconvergence in the l2 norm for the galerkin approximation of the integral equation lu f is studied where i is a strongly elliptic pseudodifferential operator on a smooth closed or open curve let uf be the galerkin approximation to u by using the operator an operator that averages the values of uh we will construct a better approximation than uh itself that better approximation is a legacy of the highest order of convergence in negative norms for symm s equation on a slit the same order of convergence can be recovered if the mesh is suitably graded in this paper we shall study a way of increasing the order of local convergence in the l2 norm for the galerkin approximation to the solution of strongly ellip tic pseudodifferential equations on a smooth closed or open curve in r2 this kind of integral equation is of importance in solving interior or exterior bound ary value problems of potential theory the most common example is symm s first kind integral equation with logarithmic kernel see 6 7 12 other appli cations are hypersingular integral equations and singular integral equations of cauchy type which occur e g in elasticity see 18 for general petrov galerkin methods when smoothest splines are used as trial and test functions local error estimates were proved in 11 for smooth closed curves and in 16 for smooth open curves consider for example symm s equation with piecewise constant functions used as trial and test functions it was proved that the local l2 error converges with order 0 h in the case of smooth closed curves 11 and with order 0 hxi2 in the case of smooth open curves 16 however it is well known that the highest orders of global convergence achieved in negative norms are 0 h3 for the closed smooth case 5 and 0 h for the open smooth case 4 13 the purpose of this article is to construct from the galerkin solution a better approximate solution which inherits the highest possible orders of global convergence to give best local con vergence in the l2 norm e g in the example mentioned above order 0 hy for the closed case and 0 h for the open case can be achieved locally in the l2 norm that better approximation is constructed by averaging the values of the galerkin solution using the k operator

a broadband volume integral formulation based on edge elements for full wave analysis of lossy interconnects a new numerical fully three dimensional 3 d volume integral formulation for the electromagnetic analysis from static to microwave frequencies of penetrable materials dielectric eventually lossy and conductors with finite conductivity is here discussed its key feature is the introduction of a volumetric loop star decomposition for treating piecewise homogeneous materials the associated shape functions have been determined to decompose the volume current density in a solenoidal and a nonsolenoidal part in analogy to the surface loop star shape functions used for modeling surface current densities on perfect electric conductors the possibility of modeling volumetric ohmic and polarization current densities allows to compute in an accurate way the electromagnetic field in complex 3 d geometries such as high speed interconnects on a broad range of frequencies

a single level low rank ie qr algorithm for pec scattering problems using efie formulation this paper presents a single level matrix compression algorithm termed ie qr based on a low rank approximation to speed up the electric field integral equation efie formulation it is shown with the number of groups chosen to be proportional to n sup 1 2 where n is the number of unknowns the memory and cpu time for the resulting algorithm are both o n sup 1 5 the unique features of the algorithm are a the ie qr algorithm is based on the near rank deficiency property for well separated groups this near rank deficiency assumption holds true for many integral equation methods such as laplacian radiation and scattering problems in electromagnetics em the same algorithm can be adapted to other applications outside em with few or no modifications and b the rank estimation is achieved by a dual rank process which ranks the transmitting and receiving groups respectively thus the ie qr algorithm can achieve matrix compression without assembling the entire system matrix also a geometric neighboring preconditioner is presented in this paper this geometric neighboring preconditioner when used in conjunction with gmres is proven to be both efficient and effective for solving the compressed matrix equations

a memory saving augmented efie with modified basis functions for low frequency problems a new augmented electric field integral equation a efie formulation is discussed in the low frequency in the new formulation both the electric current and charge are considered as unknowns different from the conventional a efie a constraint on the normal component of the magnetic field is added to modifying the formulation of a efie for improving the accuracy and a kind of constant nodal functions is introduced for expanding the electric charge with the proposed method besides avoiding the low frequency breakdown the storage of impedance matrix is reduced over 35 compared with the conventional a efie which is owing to the application of constant nodal functions associated with vertices numerical examples illuminate the validity and performance of the new a efie with constant nodal functions

a multiresolution curvilinear rao wilton glisson basis function for fast analysis of electromagnetic scattering a new set of multiresolution curvilinear rao wilton glisson mr crwg basis functions is proposed for the method of moments mom solution of integral equations for three dimensional 3 d electromagnetic em problems the mr crwg basis functions are constructed as linear combinations of curvilinear rao wilton glisson crwg basis functions which are defined over curvilinear triangular patches thus allowing direct application on the existing mom codes that using crwg basis the multiresolution property of the mr crwg basis can lead to the fast convergence of iterative solvers merely by a simple diagonal preconditioning to the corresponding mom matrices moreover the convergence of iterative solvers can be further improved by introducing a perturbation from the principle value term of the magnetic field integral equation mfie operator to construct diagonal preconditioners for efficient iterative solution of the electric field integral equation efie another important property of the mr crwg basis is that the mom matrices using the mr crwg basis can be highly sparsified without loss of accuracy the mr crwg basis has been applied to the 3 d electromagnetic scattering problems and the numerical results indicate that the mr crwg basis performs much better than the crwg basis

a high order spectral algorithm for elastic obstacle scattering in three dimensions in this paper we describe a high order spectral algorithm for solving the time harmonic navier equations in the exterior of a bounded obstacle in three space dimensions with dirichlet or neumann boundary conditions our approach is based on combined field boundary integral equation cfie reformulations of the navier equations we extend the spectral method developed by ganesh and hawkins 20 for solving second kind boundary integral equations in electromagnetism to linear elasticity for solving cfies that commonly involve integral operators with a strongly singular or hypersingular kernel the numerical scheme applies to boundaries which are globally parametrised by spherical coordinates the algorithm has the interesting feature that it leads to solve linear systems with substantially fewer unknowns than with other existing fast methods the computational performances of the proposed spectral algorithm are demonstrated using numerical examples for a variety of three dimensional convex and non convex smooth obstacles

fast analysis of large finite arrays with a combined multiresolution sm aim approach we present a synthesis of the sparse matrix adaptive integral method sm aim and the multiresolution mr approach for the analysis of electrically large finite arrays with planar or 3 d radiating elements the two methods were separately introduced previously the use of the mr has the effect of a preconditioner and speeds up the convergence rate of the sm aim of almost two orders of magnitude with a total reduction of the numerical complexity with respect to the standard mom of almost three orders of magnitude

loop star basis functions and a robust preconditioner for efie scattering problems an electric field integral equation efie formulation using the loop star basis functions has been developed for modeling plane wave scattering from perfect conducting objects a stability analysis at the dc limit shows that the use of the rao wilton glisson rwg basis functions results in a singular matrix operator however the use of the loop star basis functions results in a well conditioned matrix moreover a preconditioner constructed from a two step process based on near interactions and an incomplete factorization with a heuristic drop strategy has been proposed in conjunction with the conjugate gradient method to solve the resulting matrix equation the approach is shown to be effective for resolving both the low frequency instability and the bad conditioning of the efie method the computational complexity of the proposed approach is shown to be o n sup 2

electromagnetic scattering from rough surface using single integral equation and adaptive integral method an efficient algorithm for electromagnetic wave scattering from rough dielectric surfaces is developed the algorithm is based on the single magnetic field integral equation smfie and the surface is discretized using rao wilton glisson rwg triangular basis functions the new feature of the algorithm is the application of the adaptive integral method aim with smfie for speeding up the calculation the developed new method utilizes the flexibility of rwg functions to model arbitrary rough surface and the speed of fft enabling accurate simulations over large surfaces

loop star and loop tree decompositions analysis and efficient algorithms a new analysis of the spectral properties of loop star and loop tree decompositions is presented in this work the analysis will shed light on the behavior of these decompositions when used with regular operators such as the magnetic field and the calderon preconditioned electric field integral operators this work will explain the ill conditioning problems reported in literature and will provide a family of efficient algorithms to solve the ill conditioning and regularizing several loop star tree decomposed equations of interest for applications the theory will be corroborated by numerical results that will show the practical impact of the theoretical developments

time domain calderon identities and their application to the integral equation analysis of scattering by pec objects part i preconditioning time domain electric field integral equations often are used to analyze transient scattering from perfect electrically conducting objects when discretized using marching on in time recipes they give rise to linear systems of equations that can be solved for the induced currents for all time steps unfortunately when the scatterer is approximated by increasingly dense meshes the condition number of these systems grows rapidly slowing down the convergence of iterative solvers here time domain calderon identities are derived and subsequently used to construct a calderon preconditioned time domain electric field integral equation that can be discretized even with dense meshes using buffa christiansen basis functions numerical results that demonstrate the effectiveness and accuracy of the proposed method are presented

automatic basis function regularization for integral equations in the presence of ill shaped mesh elements bad meshes are a common problem in using computational electromagnetics codes often much of an engineer s or analyst s time ids wasted improving a mesh until the solver finally converges we investigate the problem as it comes up in integral equation methods and observe that its effects can be detected by analyzing the singular values of the electric field integral equation matrix armed with this information we automatically remove the undesirable degrees of freedom caused by the ill shaped mesh elements so that the iterative solver behaves as if they were not present all without any extra effort by the user an a posteriori error estimator is developed which verifies the improved accuracy for test problems and also suggests potential future applications in adaptive mesh refinement for integral equation methods further numerical examples demonstrate the improved solution convergence

inverse inequalities on non quasi uniform meshes and application to the mortar element method we present a range of mesh dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape regular but possibly non quasi uniform meshes these inequalities involve norms of the form h u w s p for positive and negative s and where h is a function which reflects the local mesh diameter in an appropriate way the only global parameter involved is n the total number of degrees of freedom in the finite element space and we avoid estimates involving either the global maximum or minimum mesh diameter our inequalities include new variants of inverse inequalities as well as trace and extension theorems they can be used in several areas of finite element analysis to extend results previously known only for quasi uniform meshes to the locally refined case here we describe applications to i the theory of nonlinear approximation and ii the stability of the mortar element method for locally refined meshes

random walk and graph cut for co segmentation of lung tumor on pet ct images accurate lung tumor delineation plays an important role in radiotherapy treatment planning since the lung tumor has poor boundary in positron emission tomography pet images and low contrast in computed tomography ct images segmentation of tumor in the pet and ct images is a challenging task in this paper we effectively integrate the two modalities by making fully use of the superior contrast of pet images and superior spatial resolution of ct images random walk and graph cut method is integrated to solve the segmentation problem in which random walk is utilized as an initialization tool to provide object seeds for graph cut segmentation on the pet and ct images the co segmentation problem is formulated as an energy minimization problem which is solved by max flow min cut method a graph including two sub graphs and a special link is constructed in which one sub graph is for the pet and another is for ct and the special link encodes a context term which penalizes the difference of the tumor segmentation on the two modalities to fully utilize the characteristics of pet and ct images a novel energy representation is devised for the pet a downhill cost and a 3d derivative cost are proposed for the ct a shape penalty cost is integrated into the energy function which helps to constrain the tumor region during the segmentation we validate our algorithm on a data set which consists of 18 pet ct images the experimental results indicate that the proposed method is superior to the graph cut method solely using the pet or ct is more accurate compared with the random walk method random walk co segmentation method and non improved graph cut method

recycling krylov subspaces for sequences of linear systems many problems in science and engineering require the solution of a long sequence of slowly changing linear systems we propose and analyze two methods that significantly reduce the total number of matrix vector products required to solve all systems we consider the general case where both the matrix and right hand side change and we make no assumptions regarding the change in the right hand sides furthermore we consider general nonsingular matrices and we do not assume that all matrices are pairwise close or that the sequence of matrices converges to a particular matrix our methods work well under these general assumptions and hence form a significant advancement with respect to related work in this area we can reduce the cost of solving subsequent systems in the sequence by recycling selected subspaces generated for previous systems we consider two approaches that allow for the continuous improvement of the recycled subspace at low cost we consider both hermitian and non hermitian problems and we analyze our algorithms both theoretically and numerically to illustrate the effects of subspace recycling we also demonstrate the effectiveness of our algorithms for a range of applications from computational mechanics materials science and computational physics

on a well conditioned electric field integral operator for multiply connected geometries all known integral equation techniques for simulating scattering and radiation from arbitrarily shaped perfect electrically conducting objects suffer from one or more of the following shortcomings i they give rise to ill conditioned systems when the frequency is low ii and or when the discretization density is high iii their applicability is limited to the quasi static regime iv they require a search for global topological loops v they suffer from numerical cancellations in the solution when the frequency is very low this work presents an equation that does not suffer from any of the above drawbacks when applied to smooth and closed objects the new formulation is obtained starting from a helmholtz decomposition of two discretizations of the electric field integral operator obtained by using rwgs and dual bases respectively the new decomposition does not leverage loop and star tree basis functions but projectors that derive from them following the decomposition the two discretizations are combined in a calderon like fashion resulting in a new overall equation that is shown to exhibit self regularizing properties without suffering from the limitations of existing formulations numerical results show the usefulness of the proposed method both for closed and open structures

fast solution methods in electromagnetics various methods for efficiently solving electromagnetic problems are presented electromagnetic scattering problems can be roughly classified into surface and volume problems while fast methods are either differential or integral equation based the resultant systems of linear equations are either solved directly or iteratively a review of various differential equation solvers their complexities and memory requirements is given the issues of grid dispersion and hybridization with integral equation solvers are discussed several fast integral equation solvers for surface and volume scatterers are presented these solvers have reduced computational complexities and memory requirements

computation of scattering by db objects with surface integral equation method electromagnetic scattering by objects with a db boundary condition normal components of the electric and magnetic flux densities d and b vanish at the boundary surface is investigated scattering of arbitrary shaped three dimensional objects is analyzed with the surface integral equation method a significant difference compared to the more conventional boundary conditions is that the uniqueness of the solution to the scattering problem with db boundary depends on the connectivity of the object for a simply connected object the solution is unique but for a multiply connected object some additional conditions are required it appears that these additional conditions are satisfied if divergence free basis functions are used to expand the unknown surface current densities the developed method is applied to calculate polarizabilities scattering efficiencies and radar cross sections rcs of the db sphere and cube and the results are compared with those of the pec cube and sphere

a numerical implementation of a modified form of the electric field integral equation the details of a galerkin discretization scheme for a modified form of the electric field integral equation are outlined for smooth three dimensional perfectly conducting scatterers limitations of the divergence conforming finite element bases in preserving the self stabilizing properties of the electric field integral equation operator are indicated a numerically efficient alternative is outlined which relies on an operator based helmholtz decomposition the condition number of the resulting matrix equation is demonstrated to be frequency independent for scattering from a perfectly conducting sphere at various frequencies

well conditioned muller formulation for electromagnetic scattering by dielectric objects numerical solution of electromagnetic scattering by homogeneous dielectric objects with the method of moments mom and rao wilton glisson rwg basis functions is discussed it is shown that the low frequency breakdown associated to the mom solution of scattering by dielectric objects can be avoided by the classical muller formulation without the loop tree or loop star basis functions two variations of the muller formulation t mu spl uml ller and n muller are considered it is demonstrated that only the n muller formulation with the galerkin method and rwg functions gives stable solution discretization of the n muller formulation leads to a well conditioned matrix equation and rapidly converging iterative solutions on a wide frequency range from very low frequencies to microwave frequencies at zero frequency the n muller formulation decouples into the electrostatic and magnetostatic integral equations

a single level dual rank ie qr algorithm to model large microstrip antenna arrays a single level dual rank ie qr algorithm is introduced so that the resulting dense method of moments mom matrix is efficiently compressed for a system of n equations an amount of work of the order o n sup 2 has traditionally been required for both matrix assembly and matrix vector multiplication in an iterative matrix solver the algorithm of the present paper reduces the memory requirement and cpu time for both matrix assembly and matrix vector multiplication to o n sup 3 2 making it practical to solve for large antenna arrays with full wave approach in conjunction with a geometric neighboring preconditioner for matrix solution using gmres the current approach solves problems involving large antenna arrays using only a fraction of what are needed by conventional mom both in term of memory and total cpu time

computational scales of sobolev norms with application to preconditioning this paper provides a framework for developing computationally efficient multilevel preconditioners and representations for sobolev norms specifically given a hilbert space v and a nested sequence of subspaces v 1 c v 2 c c v we construct operators which are spectrally equivalent to those of the form a k k q k q k 1 here k k 1 2 are positive numbers and q k is the orthogonal projector onto v k with q 0 0 we first present abstract results which show when a is spectrally equivalent to a similarly constructed operator a defined in terms of an approximation q k of q k for k 1 2 we show that these results lead to efficient preconditioners for discretizations of differential and pscudo differential operators of positive and negative order these results extend to sums of operators for example singularly perturbed problems such as i e can be preconditioned uniformly independently of the parameter e we also show how to precondition an operator which results from tikhonov regularization of a problem with noisy data finally we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition

mesh independent operator preconditioning for boundary elements on open curves boundary value problems for the poisson equation in the exterior of an open bounded lipschitz curve mathcal c can be recast as first kind boundary integral equations featuring weakly singular or hypersingular boundary integral operators bios based on the recent discovery in c jerez hanckes and j nedelec siam j math anal 44 2012 pp 2666 2694 of inverses of these bios for mathcal c 1 1 we pursue operator preconditioning of the linear systems of equations arising from galerkin petrov discretization by means of zeroth and first order boundary elements the preconditioners rely on boundary element spaces defined on dual meshes and they can be shown to perform uniformly well independently of the number of degrees of freedom even for families of locally refined meshes

split formulation of the charge and current integral equations for arbitrarily shaped dielectrics the low frequency instability in the method of moments can be resolved by reformulating the underlying integral equations to explicitly include charge as an unknown a new technique based on this charge and current formulation that reduces the computational complexity both in time and memory will be introduced considerations when using a charge and current technique include the practical problems of basis functions and the evaluation of the singular surface integrals well as more theoretical problems such as whether the continuity equation still holds the technique is implemented using a simple set of basis and testing functions and has been verified quantitatively checking the charge and current distributions against analytic solutions where existing as well as qualitatively against the boundary conditions for the scattered field these results show the method to be both stable and accurate

some observations on babu vs ka and brezzi theories summary some observations are made on abstract error estimates for galerkin approximations based on babuska brezzi conditions a basic error estimate due to babuska is sharpened by means of an identity that p i p for any nontrivial idempotent operator p some remarks are also made on the brezzi s theory for mixed variational problems and their galerkin approximations

uniform preconditioners for problems of positive order uniform preconditioners for operators of negative order discretized by dis continuous piecewise polynomials of any order are constructed from a boundedly invertible operator of opposite order discretized by continuous piecewise linears besides the cost of the application of the latter discretized operator the other cost of the preconditioner scales linearly with the number of mesh cells compared to earlier proposals the preconditioner has the following advantages it does not require the inverse of a non diagonal matrix it applies without any mildly grading assumption on the mesh and it does not require a barycentric refinement of the mesh underlying the trial space

zeroth order complete discretizations of integral equation formulations involving conducting or dielectric objects at very low frequencies we present the self loop basis functions a divergence conforming set with zero charge density these basis functions allow a rearrangement of the linear linear basis functions set to overcome the low frequency breakdown we show the accuracy and stability at very low frequencies of the linear linear discretizations of i electric field integral equation for perfectly conducting objects and ii the poggio miller chang harrington wu tsai formulation for dielectric bodies

a surface integral formulation of maxwell equations for topologically complex conducting domains a general and effective method is presented to numerically solve the electric field integral equation efie for topologically complex conducting domains by the finite element method a new technique is proposed to decompose the surface current density into a solenoidal part and a nonsolenoidal remainder to avoid the low frequency breakdown the surface current density field is approximated through div conforming facet elements the solenoidal part is represented through the null space of the discrete approximation d of the surface divergence operator in the subspace spanned by the facet elements whereas the nonsolenoidal remainder is represented through its complement the basis functions of the null space and its complement are evaluated respectively by the null and pseudo inverse of the matrix d the completeness of the null pinv basis functions is studied unlike the loop star and loop tree basis functions the null pinv basis functions allow to deal with topologically complex conducting domains in a general and readily applicable way a topological interpretation of the null pinv decomposition is given and a general and simple method to evaluate the null and pseudo inverse of d is proposed the computational complexity of the proposed method is discussed

on the existence of branch points in the eigenvalues of the electric field integral equation operator in the complex frequency plane it is demonstrated by specific examples that eigenvalues of the electric field integral equation efie operator for finite extent objects in lossless media can have branch points in the complex frequency s plane specifically it is shown from the analysis of the spheroidal wave equation in the s plane that branch points are present in the eigenvalues of the integral operator associated with the scalar scattering problem from a perfectly conducting spheroid similarly it is concluded from the analysis of the mathieu s differential equation that branch points besides that associated with the infinite extent of the object exist in the s plane behavior of the eigenvalues of the efie operator for a perfectly conducting infinite elliptic cylinder a proof is given that branch point singularities cannot occur in the eigenvalues of efie operators for structures in which geometrical symmetry completely determines the eigenfunctions and hence they are frequency independent it is conjectured that branch points may always be present when sufficient object symmetry is lacking this conjecture is supported by the fact that branch points appear when a sphere is deformed into a spheroid or when a circular cylinder is deformed into an elliptic one an analogous phenomenon has been observed in circuit and transmission line problems for example it is shown that when the taper parameter of an exponential transmission line goes to zero the line becomes uniform and thus a symmetrical structure the branch points in the eigenvalues of the impedance matrix move away to infinity