experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

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

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

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

fast factorization acceleration of mom compressive domain decomposition domain decomposition dd for integral equation can be achieved by aggregating standard basis functions into specialized basis functions on each sub domain this results in a strong compression of the mom matrix which allows an iteration free e g lu decomposition solution also for electrically large problems fast matrix vector product algorithms can be used in the matrix filling and compression process of the employed aggregate functions approach this hybrid approach has received considerable attention in recent literature in order to quantitatively assess the performance advantages and limitations of this class of methods we start by proposing and demonstrating the use of the adaptive integral method aim fast factorization to accelerate the synthetic function expansion sfx dd approach the method remains iteration free with a significant boost in memory and time performances with analytical predictions of complexity scalings confirmed by numerical results then we address the complexity scaling of both stand alone dd and its combined use with fast mom this is done analytically and discussed with respect to known literature accounts of various implementations of the dd paradigm with nonobvious results that highlight needs and limitations and yielding practical indications

preconditioning technique in the interpolative decomposition multilevel fast multipole algorithm the potential of the interpolative decomposition multilevel fast multipole algorithm id mlfma on developing an effective preconditioning technique for multiscale dynamic electromagnetic problems are analyzed the preconditioner based on multilevel inverse based ilu is developed for id mlfma the proposed preconditioning technique is investigated by numerical experiments on complex targets

ilum a multi elimination ilu preconditioner for general sparse matrices standard preconditioning techniques based on incomplete lu ilu factorizations offer a limited degree of parallelism in general a few of the alternatives advocated so far consist of either using some form of polynomial preconditioning or applying the usual ilu factorization to a matrix obtained from a multicolor ordering in this paper we present an incomplete factorization technique based on independent set orderings and multicoloring we note that in order to improve robustness it is necessary to allow the preconditioner to have an arbitrarily high accuracy as is done with ilus based on threshold techniques the ilum factorization described in this paper is in this category it can be viewed as a multifrontal version of a gaussian elimination procedure with threshold dropping which has a high degree of potential parallelism the emphasis is on methods that deal specifically with general unstructured sparse matrices such as those arising from finite element methods on unstructured meshes

eigenvalues of the moment method matrix and their effect on the convergence of the conjugate gradient algorithm em scattering a theory that relates eigenvalues of a continuous operator to those of the moment method matrix operator is discussed and confirmed by examples this theory suggests reasons for ill conditioning when certain types of basis and testing functions are used in addition the effect of eigenvalue location on the convergence of the conjugate gradient cg method is studied the convergence rate of the cg method is dependent on the eigenvalues of the iteration matrix as well as on the number of eigenvectors of the iteration matrix needed to represent the right side of the equation these findings explain the previously reported convergence behavior of the cg method when applied to electromagnetic scattering problems

a new family of exponential based high order dgtd methods for modeling 3 d transient multiscale electromagnetic problems a new family of exponential based time integration methods are proposed for the time domain maxwell s equations discretized by a high order discontinuous galerkin dg scheme formulated on locally refined unstructured meshes these methods which are developed from the lawson method remove the stiffness on the time explicit integration of the semidiscrete operator associated with the fine part of the mesh and allow for the use of high order time explicit scheme for the coarse part operator they combine excellent stability properties with the ability to obtain very accurate solutions even for very large time step sizes here the explicit time integration of the lawson transformed semidiscrete system relies on a low storage runge kutta lsrk scheme leading to a combined lawson lsrk scheme in addition efficient techniques are presented to further improve the efficiency of this exponential based time integration for the efficient calculation of matrix exponential we employ the krylov subspace method numerical experiments are presented to assess the stability verify the accuracy and numerical convergence of the lawson lsrk scheme they also demonstrate that the dg time domain methods based on the proposed time integration scheme can be much faster than those based on classical fully explicit time stepping schemes with the same accuracy and moderate memory usage increase on locally refined unstructured meshes and are thus very promising for modeling 3 d multiscale electromagnetic problems

software metamaterials transformation media based multiscale techniques for computational electromagnetics this paper presents computational models employing special transformation based media which we call software metamaterials for the purpose of enhancing the ability of numerical modeling methods for solving multi scale electromagnetic boundary value problems involving features with multiple length or frequency scales or both the multi scale problems in general suffer from difficulties in mesh generation and the number of unknowns due to certain meshing requirements dictated by the fine features of the problem the underlying idea in this study is to introduce transformation media into the computational domain of the finite element method or finite difference methods in order to allow uniform and easy to generate meshes by avoiding mesh refinement and therefore to develop efficient and simple computer aided simulation tools in other words a virtual equivalent problem is created which acts like the original problem the techniques proposed in this paper are validated via several finite element simulations in the context of tm z electromagnetic scattering problems

a ribbon hopf algebra approach to the irreducible representations of centralizer algebras the brauer birman wenzl and type a iwahori hecke algebras abstract we show how the ribbon hopf algebra structure on the drinfel d jimbo quantum groups of types a b c and d can be used to derive formulas giving explicit realizations of the irreducible representations of the iwahori hecke algebras of type a and the birman wenzl algebras we use this derivation to give explicit realizations of the irreducible representations of the brauer algebras as well the derivation is accomplished by way of a combination of techniques from operator algebras quantum groups and the theory of 3 manifold invariants although our applications are in the cases of the quantum groups of types a b c and d most of the aspects of our approach apply in the general setting of ribbon hopf algebras

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

incomplete lu preconditioning with the multilevel fast multipole algorithm for electromagnetic scattering iterative solution of large scale scattering problems in computational electromagnetics with the multilevel fast multipole algorithm mlfma requires strong preconditioners especially for the electric field integral equation efie formulation incomplete lu ilu preconditioners are widely used and available in several solver packages however they lack robustness due to potential instability problems in this study we consider various ilu class preconditioners and investigate the parameters that render them safely applicable to common surface integral formulations without increasing the cal o n log n complexity of mlfma we conclude that the no fill ilu 0 preconditioner is an optimal choice for the combined field integral equation cfie for efie we establish the need to resort to methods depending on drop tolerance and apply pivoting for problems with high condition estimate we propose a strategy for the selection of the parameters so that the preconditioner can be used as a black box method robustness and efficiency of the employed preconditioners are demonstrated over several test problems

efie modeling of high definition multiscale structures we propose a method to significantly improve the spectral properties of the efie mom matrix for broad band analysis of structures with fine details non uniform meshes and large overall sizes a multilevel approach allows to overlay fine meshes and quasi nyquist sampled meshes the fine mesh conditioning is solved via hierarchical basis functions and the quasi nyquist sampled part is treated by an algebraic incomplete lu preconditioner numerical results show the effectiveness of the approach for several realistic structures with overall sizes of ten twenty wavelengths and levels of detail from very moderate to extending over the whole structure

wideband fast kernel independent modeling of large multiscale structures via nested equivalent source approximation we propose a wideband fast kernel independent modeling of large multiscale structures we employ a nested equivalent source approximation nesa to compress the dense system matrix the nesa was introduced by these authors for low and moderate frequency problems smaller than a few wavelengths here we introduce a high frequency nesa algorithm and propose a hybrid version with extreme wideband properties the equivalent sources of the wideband nesa wnesa are obtained by an inverse source process enforcing equivalence of radiated fields on suitably defined testing surfaces in the low frequency region the nesa is used unmodified with a complexity of mathcal o n in the high frequency region in order to obtain a fixed rank matrix compression we hierarchically divide the far coupling space into pyramids with angles related to the peer coupling group size and the nesa testing surfaces are defined as the boundaries of the pyramids this results in a directional nested low rank fixed rank approximation with mathcal o n log n computational complexity that is kernel independent overall the approach yields wideband fast solver for the modeling of large structures that inherits the efficiency and accuracy of low frequency nesa for multiscale problems numerical results and discussions demonstrate the validity of the proposed work

incomplete lu preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems we consider the preconditioned iterative solution of large dense linear systems where the coefficient matrix is a complex valued matrix arising from discretizing the integral equation of electromagnetic scattering for some scattering structures this matrix can be poorly conditioned the main purpose of this study is to evaluate the efficiency of a class of incomplete lu ilu factorization preconditioners for solving this type of matrices we solve the electromagnetic wave equations using the bicg method with an ilu preconditioner in the context of a multilevel fast multipole algorithm mlfma the novelty of this work is that the ilu preconditioner as constructed using the near part block diagonal submatrices generated from the mlfma experimental results show that the ilu preconditioner reduces the number of bicg iterations substantially compared to the block diagonal preconditioner the preconditioned iteration scheme also maintains the computational complexity of the mlfma and consequently reduces the total cpu time

exact relationship between the locally corrected nystrom scheme and rwg moment method for the mixed potential integral equation the exact relationship between the rao wilton glisson rwg method of moments mom and the locally corrected nystrom lcn method for the mixed potential mp electric field integral equation efie is presented as an extension to our work where we established analogous exact relationship for solving the efie in its vector potential vp form it is shown that in order to achieve one such relationship for the mp efie the first and zeroth order lcn methods must be respectively used for the discretization of the vp and scalar potential terms of the mp efie the resulting numerical scheme is a point based rwg mom discretization of the mp efie via the nystrom method due to the mp formulation of the efie the proposed method establishes notably higher accuracy compared to either rwg mom or lcn discretizations of the efie in the vp form the increased accuracy is attributed to the analytical cancellation of the line charge contributions in the mp formulation as opposed to numerical cancellation inherent in the vp formulation of the efie the detailed study and explanations of the above cancellations is presented along with their impact on the accuracy of the respective schemes for both canonical and realistic scattering targets at different frequencies

equivalence principle algorithm with body of revolution equivalence surface for the modeling of large multiscale structures we propose a multiscale solver of equivalence principle algorithm with a body of revolution bor equivalence surface es first the whole object is decomposed into subdomains the ess are defined as proper bors e g spheres in this work to enclose each subdomain the rao wilton glisson rwg and bor basis functions are defined on each sphere respectively second the octree is constructed in each nonzero subdomain the multilevel fast multipole algorithm mlfma is employed to solve the equivalent currents on the es individually and then the equivalent currents are projected from rwg onto the bor basis functions the couplings between neighboring subdomains are evaluated directly using mlfma and the separated subdomains are substituted by the es to es couplings and are evaluated efficiently by the bor method of moments bor mom to solve the equation system a hybrid inner and outer iterative solver is employed where the inner iteration is used to solve each local subdomains and the outer iteration is used to update the global solutions by collecting all the local solutions numerical results and discussions demonstrate the validity of the proposed work

a nonconformal domain decomposition scheme for the analysis of multiscale structures we present a domain decomposition dd framework for the analysis of impenetrable structures it allows for the electric field integral equation efie and combined field integral equation cfie and for open closed and open closed structures the dd results in an effective preconditioner for large and complex problems exploiting iterative solution and fast factorizations the dd employs specialized transmission conditions among the domains and the use of discontinuous galerkin dg allows conformal as well as nonconformal discretizations of domain boundaries the nonconformal nature of the decomposition gives considerable flexibility in the meshing the strategy is highly parallelizable as all the operations involving the subdomains can be performed in parallel the proposed scheme is implementation independent and can be easily merged with existing electromagnetic codes

an efficient marching on in degree solution of transient multiscale em scattering problems a marching on in degree mod based time domain domain decomposition method is proposed to efficiently analyze the transient electromagnetic scattering from electrically large multiscale targets the algorithm starts with an octree that divides the whole scattering target into several subdomains then using the equivalence principle algorithm each subdomain is enclosed by an equivalence sphere es where both the rwg and bor spatial basis functions are employed to expand the unknown currents the interactions of the near field subdomains are directly calculated by the method of moments while the far field interactions can be converted into the interactions of corresponding ess this scheme implicitly satisfies the current continuity condition and the convergence can be accelerated as well by harnessing the rotational symmetry of the ess the computational resources are reduced significantly compared with the traditional mod method several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed algorithm

iterative methods for the force based quasicontinuum approximation analysis of a 1d model problem force based atomistic continuum hybrid methods are the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model for this reason and due to their algorithmic simplicity force based coupling methods have become a popular class of atomistic continuum hybrid models as well as other types of multiphysics models however the recently discovered unusual stability properties of the linearized force based quasicontinuum qcf approximation especially its indefiniteness present a challenge to the development of efficient and reliable iterative methods we present analytic and computational results for the generalized minimal residual gmres solution of the linearized qcf equilibrium equations we show that the gmres method accurately reproduces the stability of the force based approximation and conclude that an appropriately preconditioned gmres method results in a reliable and efficient solution method

nested equivalent source approximation for the modeling of multiscale structures we introduce a method to compress the impedance matrix of the method of moments mom for the modeling of high fidelity multiscale structures at low to moderate frequencies we start from a method recently proposed to compress static scalar problems proved to have o n computational complexity we represent far coupling between groups through equivalent source distributions on properly defined automatically generated equivalence surfaces the equivalent sources are obtained via an inverse source process that enforces equivalence of radiated fields on testing surfaces within a prescribed accuracy and are intrinsically multiscale this results in an o n complexity scaling of matrix vector products but reduces multiplicative constants i e reduces time and memory requirements for all structures it affords an improved representation of couplings in multiscale structures resulting in a fast convergence of iterative solvers our approach is green s function independent and easy to be implemented in existing mom codes the present version is based on the electric field integral equation efie numerical results prove the effectiveness of the proposed algorithm for complex multiscale structures

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

a discontinuous galerkin time domain integral equation method for electromagnetic scattering from pec objects a discontinuous galerkin time domain integral equation dg tdie method based on the marching on in time mot scheme is presented to analyze the transient scattering from arbitrarily shaped perfect electrically conducting pec objects in this paper the mono polar rao wilton glisson rwg functions are chosen as the trial and the test spatial functions while the triangular basis functions are for the temporal expansion therefore it is flexible and convenient to analyze objects with nonconformal discretization or targets containing more than three surfaces in contact with each other numerical results are presented to demonstrate the accuracy and the efficiency of the dg tdie method

a low frequency efie mlfma solver based on approximate diagonalization of green s function the low frequency fast multipole algorithm lffma based on the approximate diagonalization of green s function for the electric field integral equation efie is proposed for open and complex structures with high accuracy moreover the multiresolution preconditioner is employed to the mixed potential form of efie with rwg basis functions for stability herein a triangle grouping scheme is adopted for the scalar potential so that the accuracy of the lffma can be guaranteed furthermore the computational resources can be reduced largely since part of the radiation pattern of vector potential can share one of the scalar potentials by taking advantage of some mathematical deductions

a domain decomposition method for boundary integral equations using a transmission condition based on the near zone couplings a domain decomposition method ddm for the solution of boundary integral equations for impenetrable objects is presented the approach uses a transmission condition which is based on the near range couplings in order to obtain close approximations of the global solution on the subdomains the comparatively strong near zone interactions are taken into account to avoid unphysical reflections from the domain interfaces the algorithm can be accelerated by fast integral solvers such as the multilevel fast multipole method mlfmm by creating local subtrees which coincide with the different computing domains this concept is embedded in an inner outer iterative solution algorithm in which the ddm acts as a preconditioner in the inner iterations by introducing small overlap regions the convergence of the iterative solver is further improved this results in a very robust ddm which can be integrated into existing codes with reasonable effort and which is suited for parallelization numerical results for various open and closed objects demonstrate the effectiveness and the excellent performance of the proposed ddm for the solution of electromagnetic radiation and scattering problems

a discontinuous galerkin surface integral equation method for electromagnetic wave scattering from nonpenetrable targets we present a discontinuous galerkin surface integral equation method herein referred to as iedg for time harmonic electromagnetic wave scattering from nonpenetrable targets the proposed iedg algorithm allows the implementation of the combined field integral equation cfie using square integrable trial and test functions without any considerations of continuity requirements across element boundaries due to the local characteristics of basis functions it is possible to employ nonconformal surface discretizations of the targets furthermore it enables the possibility to mix different types of elements and employ different order of basis functions within the same discretization therefore the proposed iedg method is highly flexible to apply adaptation techniques numerical results are included to validate the accuracy and demonstrate the versatility of the proposed iedg method in addition a complex large scale simulation is conducted to illustrate the potential benefits offered by the proposed method for modeling multiscale electrically large targets

a hierarchical fast solver for efie mom analysis of multiscale structures at very low frequencies we present an efie fast solver that is stable down to very low frequencies for very dense meshes and multi scale problems the proposed approach uses a split potentials formulation in association to a hierarchic pre conditioner for the underlying low rank fast factorization potential splitting prevents the numerical cancellation problem that undermines the effectiveness of the low rank factorization negatively impacting on the necessary factorization tolerance the fast method presented in this work can be directly implemented into existing codes