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

the mixed order bcgs fft method for the scattering of three dimensional inhomogeneous anisotropic magnetodielectric objects this paper presents the first fft based fast volume integral equation solver for inhomogeneous anisotropic magnetodielectric objects the volume integral equations are formulated by employing mixed order basis functions that expand the flux densities and vector potentials in the coupled field integral equations in terms of different sets of basis functions with different orders volumetric roof top basis functions are used for flux densities whereas second order curl conforming basis functions are used for vector potentials a fast volume integral equations solver namely the bcgs fft method is then applied to accelerate the solution of this mixed order weak form formulation examples show that the mixed order bcgs fft method has high accuracy compared to both analytical and numerical solutions examples also show the mixed order bcgs fft method has high computational efficiency compared to commercial software

the multifrontal method for sparse matrix solution theory and practice this paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems the method is formulated in terms of frontal matrices update matrices and an assembly tree formal definitions of these notions are given based on the sparse matrix structure various advances to the basic method are surveyed they include the role of matrix reorderings the use of supernodes and other implementatjon techniques the use of the method in different computational environments is also described

regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

maximally orthogonalized higher order bases over generalized wires quadrilaterals and hexahedra this paper presents a general theory of maximally orthogonalized div and curl conforming higher order basis functions hobfs over generalized wires quadrilaterals and hexahedra in particular all elements of such bases necessary for fast and easy implementation are listed up to order n 8 numerical results given for div conforming bases applied in an iterative method of moments solution of integral equations show that the condition number and the number of iterations are a much lower than in the case of other hobfs of polynomial type and b practically not dependent on the applied expansion order

hierarchical divergence conforming nedelec elements for volumetric cells a new family of hierarchical divergence conforming vector bases is proposed for tetrahedral prism and brick cells these functions span the divergence conforming reduced curl spaces of nedelec the bases are constructed from orthogonal scalar polynomials to enhance their linear independence which is a simpler process than an orthogonalization applied to the final vector functions specific functions are tabulated to order 6 5 these basis are intended for use with numerical solutions of volume integral equations or differential equations containing divergence operators

the stable topology of self dual moduli spaces soit m une 4 variete de riemann compacte connexe orientee et soit g un groupe de lie connexe simple on fixe un entier q m q alors l application inclusion i induit un epimorphisme sur les groupes respectifs d homologie et d homotopie a toutes les dimensions inferieures ou egales a q

solving the volume integral equations of electromagnetic scattering time harmonic electromagnetic scattering by inhomogeneous three dimensional structures within a free space environment can be described by electric and magnetic field volume integral equations involving the free space green function a comprehensive set of galerkin projection formulations also known as moment methods for the numerical solution of these equations is presented together with comparative numerical results such formulations are widely used for particle scattering analysis optical near field calculation etc results are obtained with higher order divergence and curl conforming basis functions on iso parametric tetrahedral meshes the results demonstrate that all formulations converge with similar accuracy in the case of an analytically solvable test problem when modeling flux densities as solution variables it is argued that solenoidal function spaces should be used rather than the standard divergence conforming function spaces this assertion is supported by the results some of the formulations involve solving for curl conforming fields such fields can be discretized with fewer unknowns than divergence conforming ones implying lower computational costs additionally some formulations yield system matrices which are approximately halfway sparse meaning that computational costs will be down by a factor of 2 when iterative solvers are employed which is the case for the widely used fast methods

an efficient surface integral equation solution to em scattering by chiral objects above a lossy half space in this paper an extension of the electric magnetic current combined field integral equation jmcfie is presented for the efficient analysis of em scattering by an arbitrary shaped homogenous chiral object located above a lossy half space this formulation can lead to a well tested equation system using the method of moments mom solution with the vector triangular basis functions and galerkin s method the multilevel fast multipole algorithm mlfma is then employed to reduce the memory requirement and computational complexity of the mom solution the inner outer flexible generalized minimal residual fgmres method is used to further speed up the convergence the accuracy and efficiency are confirmed with a couple of numerical examples

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

a hybrid fem mom method for 3 d electromagnetic scattering in layered medium accurate and efficient prediction of electromagnetic scattering from inhomogeneous objects in layered medium is one of the most challenging issues in engineering applications this paper presents the first 3 d higher order hybrid finite element method fem and method of moments mom for the accurate modeling of inhomogeneous dielectric objects in multilayered medium the main challenges of this paper include 1 the integration of these algorithms for layered medium and 2 the higher order computational approach involved in layered medium for high efficiency in the proposed method the mom with the layered medium dyadic green s function is used as the exact radiation boundary condition in an inhomogeneous background and the fem is applied to model the inhomogeneous objects furthermore the higher order maximally orthogonal basis functions with curl conforming and divergence conforming properties are used in the fem and mom respectively to improve the modeling capability of this algorithm for 3 d inhomogeneous objects scattering in multilayered medium this new method requires a much more tightly truncated simulation domain than the traditional fem and provides much higher flexibility than the pure surface integral equation method finally some numerical results are provided to validate the accuracy efficiency and flexibility of this method

volume surface integral equation method based on higher order hierarchical vector basis functions for em scattering and radiation from composite metallic and dielectric structures this paper presents a novel galerkin type method of moments solution of the volume surface integral equation vsie which is developed for the analysis of electromagnetic scattering and radiation from composite metallic and dielectric structures our new scheme utilizes curved tetrahedral elements and curved triangular patches for geometric modeling and the associated higher order hierarchical vector hohv basis functions for volume surface current modeling in contrast to the previous studies on higher order vsie our scheme which is reported for the first time in vsie is much more flexible in comparison with the conventional low order scheme our scheme requires much less memory and computational time for the same accuracy level numerical results are provided to demonstrate the accuracy efficiency and flexibility of our scheme

free discontinuities in optimal transport we prove a nonsmooth implicit function theorem applicable to the zero set of the difference of convex functions this theorem is explicit and global it gives a formula representing this zero set as a difference of convex functions which holds throughout the entire domain of the original functions as applications we prove results on the stability of singularities of envelopes of semi convex functions and solutions to optimal transport problems under appropriate perturbations along with global structure theorems on certain discontinuities arising in optimal transport maps for the bilinear cost c x bar x langle x bar x rangle for x bar x in bf r n for targets whose components satisfy additional convexity separation multiplicity and affine independence assumptions we show that these discontinuities occur on submanifolds of the appropriate codimension which are parameterized locally as differences of convex functions dc hence c 2 rectifiable and depending on the precise assumptions c 1 alpha smooth under these hypotheses any n 1 affinely independent components of the target measure select at most one point from the source measure where the transport divides between all n 1 specified target components

hierarchal vector boundary elements and p adaption for 3 d electromagnetic scattering this paper describes the development of new vector boundary elements for solving electromagnetic em scattering problems the new elements are suitable for the magnetic field integral equation mfie electrical field integral equation efie or the combination field integral equation cfie the basis functions are assigned to the edges of an element rather than to its nodes the new element guarantees the continuity of the normal component of the surface current across element edges furthermore the basis functions are hierarchical from linear to higher order which enables one to use the new elements in a p adaption scheme

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

a novel grid robust higher order vector basis function for the method of moments a set of novel grid robust higher order vector basis functions is proposed for the method of moments mom solution of integral equations for three dimensional 3 d electromagnetic em problems these basis functions are defined over curvilinear triangular patches and represent the unknown electric current density within each patch using the lagrange interpolation polynomials the highlight of these basis functions is that the lagrange interpolation points are chosen to be the same as the nodes of the well developed gaussian quadratures as a result the evaluation of the integrals in the mom is greatly simplified additionally the surface of an object to be analyzed can be easily meshed because the new basis functions do not require the side of a triangular patch to be entirely shared by another triangular patch which is a very stringent requirement for traditional vector basis functions the proposed basis functions are implemented with point matching for the mom solution of the electric field integral equation the magnetic field integral equation and the combined field integral equation numerical examples are presented to demonstrate the higher order convergence and the grid robustness for defective meshes using the new basis functions

a highly effective preconditioner for solving the finite element boundary integral matrix equation of 3 d scattering a highly effective preconditioner is presented for solving the system of equations obtained from the application of the hybrid finite element boundary integral fe bi method to three dimensional 3 d electromagnetic scattering problems different from widely used algebraic preconditioners the proposed one is based on a physical approximation and is constructed from the finite element method fem using an absorbing boundary condition abc on the truncation boundary it is shown that the large eigenvalues of the finite element fe abc system are similar to those of the fe bi system hence the preconditioned system has a spectrum distribution clustered around 1 in the complex plane consequently when a krylov subspace based method is employed to solve the preconditioned system the convergence can be greatly accelerated numerical results show that the proposed preconditioner can improve the convergence of an iterative solution by approximately two orders of magnitude for large problems

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

curl conforming hierarchical vector bases for triangles and tetrahedra a new family of hierarchical vector bases is proposed for triangles and tetrahedra these functions span the curl conforming reduced gradient spaces of nedelec the bases are constructed from orthogonal scalar polynomials to enhance their linear independence which is a simpler process than an orthogonalization applied to the final vector functions specific functions are tabulated to order 6 5 preliminary results confirm that the new bases produce reasonably well conditioned matrices

hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements new vector finite elements are proposed for electromagnetics the new elements are triangular or tetrahedral edge elements tangential vector elements of arbitrary polynomial order they are hierarchal so that different orders can be used together in the same mesh and p adaption is possible they provide separate representation of the gradient and rotational parts of the vector field explicit formulas are presented for generating the basis functions to arbitrary order the basis functions can be used directly or after a further stage of partial orthogonalization to improve the matrix conditioning matrix assembly for the frequency domain curl curl equation is conveniently carried out by means of universal matrices application of the new elements to the solution of a parallel plate waveguide problem demonstrates the expected convergence rate of the phase of the reflection coefficient for tetrahedral elements to order 4 in particular the full order elements have only the same asymptotic convergence rate as elements with a reduced gradient space such as the whitney element however further tests reveal that the optimum balance of the gradient and rotational components is problem dependent

efficient surface integral equation using hierarchical vector bases for complex em scattering problems a set of hierarchical divergence conforming vector basis functions based on curved triangular patches is presented for method of moment mom solutions of surface integral equations in this paper the higher order method of combined field integral equation cfie solves perfect electric conductor electromagnetic scattering problems the higher order electric magnetic current combined field integral equation jmcfie formulations solves dielectric objects and their combination is able to efficiently analyzes the scattering of hybrid pec dielectric objects the expressions of the divergence conforming hierarchical basis functions up to order 3 5 are reported in this paper the multilevel fast multipole algorithm mlfma is then employed to reduce the memory requirements and computational complexity numerical experiments indicate that the proposed hierarchical vector basis functions can provide well conditioned linear system for iterative solution

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

on the regularity of solutions of optimal transportation problems we give a necessary and sufficient condition on the cost function so that the map solution of monge s optimal transportation problem is continuous for arbitrary smooth positive data this condition was first introduced by ma trudinger and wang 24 30 for a priori estimates of the corresponding monge ampere equation it is expressed by a socalled cost sectional curvature being non negative we show that when the cost function is the squared distance of a riemannian manifold the cost sectional curvature yields the sectional curvature as a consequence if the manifold does not have non negative sectional curvature everywhere the optimal transport map cannot be continuous for arbitrary smooth positive data the non negativity of the cost sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost convex function the proper generalization of a convex function and any of its supporting functions when the cost sectional curvature is uniformly positive we obtain that optimal maps are continuous or holder continuous under quite weak assumptions on the data compared to what is needed in the euclidean case this case includes the quadratic cost on the round sphere

fdtd algorithm in curvilinear coordinates em scattering the finite difference time domain fdtd algorithm for the solution of electromagnetic scattering problems is formulated in generalized coordinates in two dimensions and implemented in a code with the lowest order bayliss turkel radiation boundary condition expressed in cylindrical coordinates it is shown that for a perfect conductor such a formulation leads to a stable well posed algorithm and that in regions where the curvature of the coordinate lines is not great the dispersion and anisotropy effects are negligible such effects become more pronounced in regions of high curvature leading to unphysical phase shifts the magnitude of such shifts and the amount of wavefront distortion is studied via numerical experiments using a cylindrical mesh near field results are given for two canonical shapes for each polarization the circular cylinder and cylinders of square and rectangular cross sections these results are compared with those obtained by exact eigenfunction expansion techniques with method of moments mm solutions and with solutions obtained from an alternate fdtd approach in each case agreement is excellent the propagation of a plane wave through a polar space in the absence of a scatterer is also examined and it is shown that the fdtd algorithm is capable of tracking the incident wave closely

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

the geometry of the yang mills moduli space for definite manifolds on etudie la geometrie riemannienne de l espace des modules des connexions auto duales su 2 k 1 sur une classe tres generale de 4 varietes des 4 varietes compactes orientees simplement connexe m a forme d intersection definie positive

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

numerical evaluation of hierarchical vector finite elements on curvilinear domains in 2 d when using the finite element method computational efficiency may be improved by using higher order elements typical 2 d triangular rectilinear elements cannot exactly model curved geometries negating the advantage of higher order elements when such geometries are modeled curvilinear elements can be applied to such problems and have significantly better computational efficiency than rectilinear elements when higher order elements are used results are shown for elemental basis orders ranging from ct ln through full fifth order in rectilinear and curvilinear domains

double higher order large domain volume surface integral equation method for analysis of composite wire plate dielectric antennas and scatterers a novel double higher order large domain galerkin type method of moments based on higher order geometrical modeling and higher order current modeling is proposed for analysis of composite dielectric and metallic radiation scattering structures combining the volume integral equation vie approach for dielectric parts and the surface integral equation sie approach for metallic parts of the structure the technique employs lagrange type interpolation generalized hexahedra and quadrilaterals of arbitrary geometrical mapping orders for the approximation of geometry and hierarchical divergence conforming polynomial vector basis functions of arbitrary expansion orders for the approximation of currents within the elements the double higher order vsie vie sie method is extensively validated and evaluated against the analytical solutions and the numerical results obtained by alternative higher order methods

higher order hybrid fem mom technique for analysis of antennas and scatterers a novel higher order large domain hybrid computational electromagnetic technique based on the finite element method fem and method of moments mom is proposed for three dimensional analysis of antennas and scatterers in the frequency domain the geometry of the structure is modeled using generalized curved parametric hexahedral and quadrilateral elements of arbitrary geometrical orders the fields and currents on elements are modeled using curl and divergence conforming hierarchical polynomial vector basis functions of arbitrary approximation orders and the galerkin method is used for testing the elements can be as large as about two wavelengths in each dimension as multiple mom objects are possible in a global exterior region the mom part provides much greater modeling versatility and potential for applications especially in antenna problems than just as a boundary integral closure to the fem part the examples demonstrate excellent accuracy convergence efficiency and versatility of the new fem mom technique and very effective large domain meshes that consist of a very small number of large flat and curved fem and mom elements with p refined field and current distributions of high approximation orders the reduction in the number of unknowns is by two orders of magnitude when compared to available data for low order fem mom modeling

analysis of probe fed conformal microstrip antennas on finite grounded substrate a method based on the volume surface wire integral equation and precorrected fast fourier transform fft method is presented for accurate analysis of probe fed microstrip antennas on arbitrarily shaped finite sized ground plane and substrate the method of moments mom is used to solve the integral equation three triangular type basis functions are used to represent the unknown currents in the substrates on the conducting surfaces and probes respectively the connection of a vertical probe feed to the patch is rigorously modeled by an attachment mode at the junction the precorrected fft method is applied to reduce the memory requirement and computational cost of the traditional mom to facilitate analysis of large antenna arrays

study of mixed order basis functions for the locally corrected nystro spl uml m method a high order locally corrected nystro spl uml m lcn method employing the mixed order basis functions proposed by c 80al spl iota s 80kan and peterson is presented for the electromagnetic scattering by targets composed of both dielectric and conducting bodies an integral operator based on a combined field formulation for conducting surfaces and a mu spl uml ller formulation for dielectric surfaces is used it is found that for general scattering objects mixed order basis functions accelerate the convergence of the lcn solution can eliminate spurious charges and can significantly reduce the condition number of the impedance matrix

removable singularities in yang mills fields we show that a field satisfying the yang mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite we obtain the result that every yang mills field overr4 with bounded functional l2 norm may be obtained from a field ons4 r4 hodge or coulomb gauges are constructed for general small fields in arbitrary dimensions including 4

higher order frequency domain computational electromagnetics a review of the higher order computational electromagnetics cem for antenna wireless and microwave engineering applications is presented higher order cem techniques use current field basis functions of higher orders defined on large e g on the order of a wavelength in each dimension curvilinear geometrical elements which greatly reduces the number of unknowns for a given problem the paper reviews all major surface volume integral and differential equation electromagnetic formulations within a higher order computational framework focusing on frequency domain solutions with a systematic and unified review of generalized curved parametric quadrilateral triangular hexahedral and tetrahedral elements and various types of higher order hierarchical and interpolatory vector basis functions in both divergence and curl conforming arrangements a large number of actual higher order techniques representing various combinations of formulations elements bases and solution procedures are identified and discussed the examples demonstrate the accuracy efficiency and versatility of higher order techniques and their advantages over low order discretizations the most important one being a much faster higher order convergence of the solution it is demonstrated that both components of the higher order modeling namely higher order geometrical modeling and higher order current field modeling are essential for accurate and efficient cem analysis of general antenna scattering and microwave structures

an efficient solution of the volume surface integral equation for electromagnetic scattering and radiation of the composite dielectric conductor objects with reduced number of unknowns an efficient method of moments mom solution of the volume surface integral equation vsie is presented for the analysis of electromagnetic em scattering and radiation of composite objects comprising arbitrarily contacted dielectrics and conductors in contrast to the conventional schaubert wilton glisson swg and rao wilton glisson rwg basis functions which were widely used in the mom solutions of the vsie a new hybrid pulse rooftop basis function is proposed to explicitly enforce the boundary condition of vanishing tangential electric field at the dielectric conductor interface as a consequence the displacement currents in the dielectrics can be efficiently modeled by a reduced number of volume bases on the same set of meshes particularly when thin dielectric slabs are involved in the composite structures the numerical results of em scattering and radiation of several composite objects are shown to illustrate the accuracy and efficiency of the proposed scheme since the new hybrid basis function can be easily implemented into the existing fast algorithms the vsie solution presented in this paper is promising for em problems of composite targets with large size

cfvie formulation for em scattering on inhomogeneous anisotropic metallic objects the coupled field volume integral equation dini series expansion cfvie dse method is employed for electromagnetic scattering on inhomogeneous anisotropic metallic objects the anisotropic coating is both of gyroelectric and gyromagnetic type unlike traditional methods that describe the metallic core through a surface integral a full wave method is developed based on vies only in conjunction with modified tensorial green s functions taking into account the electric boundary condition bc on core s perfect electric conducting surface cfvie dse is then solved using entire domain basis functions of dini type new orthogonal sets in the domain of anisotropy are developed for the proper satisfaction of the physical bcs along with the reduction of the cfvie to algebraic form the method is fully validated and numerical results regarding radar cross section rcs computations for various anisotropic configurations are presented

precorrected fft solution of the volume integral equation for 3 d inhomogeneous dielectric objects this work presents a fast solution to the volume integral equation for electromagnetic scattering from three dimensional inhomogeneous dielectric bodies by using the precorrected fast fourier transform fft method the object is modeled using tetrahedral volume elements and the basis functions proposed by schaubert et al are employed to expand the unknown electric flux density the basis functions are then projected onto a fictitious uniform grid surrounding the nonuniform mesh enabling the fft to be used to speed up the matrix vector multiplies in the iterative solution of the matrix equation the resultant method greatly reduces the memory requirement to o n and the computational complexity to o nlogn where n is the number of unknowns as a result this method is capable of computing electromagnetic scattering from large complex dielectric objects

multilevel fast multipole method for higher order discretizations the multi level fast multipole method mlfmm for a higher order ho discretization is demonstrate do n high frequency hf problems illustrating for the first time how an efficient mlfmm for ho can be achieved even for very large groups applying several novel ideas beneficial to both lower order and higher order discretizations results from a low memory high speed mlfmm implementation of a ho hierarchical discretization are shown these results challenge the general view that the benefits of ho and hf mlfmm cannot be combined

an accurate broad band method of moments using higher order basis functions and tree loop decomposition a method is presented that combines the low frequency accuracy of tree loop decomposition with higher order basis functions to achieve an electric field integral equation based method of moments that is accurate over a wide band of frequencies in general the use of higher order divergence conforming bases introduces more solenoidal degrees of freedom than the standard rao wilton glisson zeroth order bases these solenoidal degrees of freedom are counted and identified and an efficient method to construct the required higher order loops is presented numerical results demonstrate that the technique allows the use of a single mesh to acquire scattering results over a wide band of frequencies using arbitrary order basis functions

a fast volume surface integral equation solver for scattering from composite conducting dielectric objects this paper presents a fast hybrid volume surface integral equation approach for the computation of electromagnetic scattering from objects comprising both conductors and dielectric materials the volume electric field integral equation is applied to the material region and the surface electric field integral equation is applied on the conducting surface the method of moments mom is used to convert the integral equation into a matrix equation and the precorrected fft p fft method is employed to reduce the memory requirement and cpu time for the matrix solution the present approach is sufficiently versatile in handling problems with either open or closed conductors and dielectric materials of arbitrary inhomogeneity due to the combination of the surface and volume electric field integral equations the application of the precorrected fft method facilitates the solving of much larger problems than can be handled by the conventional mom

a wideband electromagnetic scattering analysis using mlfma with higher order hierarchical vector basis functions a method of the multilevel fast multipole algorithm mlfma with the higher order hierarchical vector basis functions is introduced to solve the wideband electromagnetic scattering problems in frequency domain instead of employing different meshing grids at different frequencies this new method requires only one coarse mesh generated at the lowest frequency of the given bandwidth the scattering at other higher frequency sample points can be calculated simply by adjusting the order of the bases this approach saves much work in geometrical modeling and leads to less unknowns than using the traditional bases for wideband scattering problem some important parameters in the numerical calculation for example the patch size the order of the basis and the cube size in the finest layer of mlfma etc and their coordination related to this new method are also discussed in detail an improved interpolation method is introduced to get scattering results without the direct calculation at the points beyond the frequency samples some numerical examples are presented in this paper to show the advantages of the method in the wideband scattering analysis

volume integral equation with higher order hierarchical basis functions for analysis of dielectric electromagnetic scattering an efficient method of moments mom solution of volume electric field integral equation v efie with a kind of higher order basis functions is presented to model scattering from a dielectric object with arbitrary shapes and inhomogenuity this method is based on both higher order geometrical modeling and higher order current modeling which is realized using a family of higher order hierarchical vector hhov basis functions to expand the equivalent volume electric currents within curvilinear volume elements this higher order scheme of mom results in a significant reduction in meshing density and number of unknowns leading to a remarkable reduction of memory and cpu time in comparison with conventional low order mom while the accuracy of the solution remains at a comparable level numerical results agree with analytical solutions or with simulation results obtained by commercial software demonstrating the accuracy convergence efficiency flexibility and adaptability of the new approach

the geometry of the moduli space of cp2 instantons the moduli space of self dual connections instantons of instanton numberk 1 overcp2 is topologically a cone oncp2 the complement of the vertex of this cone carries a natural riemannian metric the l2 metric we find an explicit formula for this metric and deduce certain consequences in particular i the boundary is a totally geodesic submanifold of the completion ii is positively curved at and iii is negatively curved in certain directions near the vertex of the cone

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

the riemannian geometry of the yang mills moduli space the moduli space of self dual connections over a riemannian 4 manifold has a natural riemannian metric inherited from thel 2 metric on the space of connections we give a formula for the curvature of this metric in terms of the relevant green operators we then examine in great detail the moduli space 1 ofk 1 instantons on the 4 sphere and obtain an explicit formula for the metric in this case in particular we prove that 1 is rotationally symmetric and has finite geometry it is an incomplete 5 manifold with finite diameter and finite volume

mlace mlfma combined with reduced basis method for efficient wideband electromagnetic scattering from metallic targets in this paper a fast interpolation algorithm multilevel accelerated cartesian expansion mlace multilevel fast multipole algorithm mlfma reduced basis method rbm is proposed for the first time by leveraging the benefits of the mlace the classical mlfma and the rbm furthermore a novel nonblind sampling method nbsm is proposed to approximate an orthogonal subspace for the mlace mlfma rbm by integrating the empirical interpolation method eim and the error based compensation method ebcm into the nbsm we first efficiently establish a robust and precision controllable orthogonal subspace next with the accurate orthogonal subspace the mlace mlfma rbm can simplify the iterative solution process of the mlace mlfma or the mlfma into n text g matrix vector multiplications mvms and the solution of a reduced n text g dimensional linear equation n text g is the number of orthonormal vectors note that only the mvms between the far region impedance matrix and the orthogonal vector bases are calculated in comparison with the classical mlace mlfma and the mlace mlfma rbm based on different sampling methods for wideband electromagnetic em responses our proposed algorithms require less memory and cpu time and integrate higher flexibility the numerical results are provided to demonstrate the accuracy efficiency and flexibility

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

development and application of a novel class of hierarchical tangential vector finite elements for electromagnetics tangential vector finite elements tvfes overcome most of the shortcomings of node based finite elements for electromagnetic simulations for a triangular element this paper proposes a class of hierarchical tvpes that differ from traditional tvfes the hierarchical nature of the proposed tvfes makes them ideally suited for employing an efficient selective field expansion the lowest order tvfe employed within part of the computational domain and a higher order tvfe employed within the remaining part of the computational domain this is an attractive feature not shared by nonhierarchical tvfes for which a more traditional approach the same tvfe employed throughout the computational domain must be applied for determining the scattering by composite cylinders this paper argues that the performance in terms of accuracy memory and in most cases cpu time of the proposed class of hierarchical tvfes when applying selective field expansion is superior to that of the lowest order tvfe and a traditional nonhierarchical tvfe this is the case when an artificial absorber as well as a boundary integral is used for truncating the computational domain a guideline is given as to how lowest and higher order tvfes shall be combined for optimal performance of the proposed class of hierarchical tvfes

higher order hierarchical legendre basis functions for electromagnetic modeling this paper presents a new hierarchical basis of arbitrary order for integral equations solved with the method of moments mom the basis is derived from orthogonal legendre polynomials which are modified to impose continuity of vector quantities between neighboring elements while maintaining most of their desirable features expressions are presented for wire surface and volume elements but emphasis is given to the surface elements in this case the new hierarchical basis leads to a near orthogonal expansion of the unknown surface current and implicitly an orthogonal expansion of the surface charge in addition all higher order terms in the expansion have two vanishing moments in contrast to existing formulations these properties allow the use of very high order basis functions without introducing ill conditioning of the resulting mom matrix numerical results confirm that the condition number of the mom matrix obtained with this new basis is much lower than existing higher order interpolatory and hierarchical basis functions as a consequence of the excellent condition numbers we demonstrate that even very high order mom systems e g tenth order can be solved efficiently with an iterative solver in relatively few iterations

nonconformal discretization of electric current volume integral equation with higher order hierarchical vector basis functions a nonconformal discretization of the electric current volume integral equation jvie is presented for the electromagnetic scattering analysis of inhomogeneous dielectric objects this jvie is based on higher order geometrical and current modeling where curved tetrahedral elements and higher order hierarchical vector hohv basis functions are adopted in order to implement the nonconformal discretization the half face based hohv basis function is introduced to address the nonconformal volumetric elements compared to the conventional low order conformal nonconformal vie our nonconformal hohv based jvie has the advantage of improving the computational efficiency and accuracy compared to the previous vie discretized by other higher order functions our nonconformal jvie is based on the higher order geometric modeling and the hohv bases in order to improve the efficiency the basis expansion and recombination technique is introduced to significantly accelerate the matrix filling moreover the flexibility of basis order selection is further enhanced by the mixed order schemes numerical results are given to demonstrate its accuracy efficiency and flexibility

a diagonalized multilevel fast multipole method with spherical harmonics expansion of the k space integrals diagonalization of the fast multipole method fmm for the helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves the resulting k space integral over the ewald sphere is numerically evaluated storing the k space quadrature samples of the method of moments mom basis functions constitutes a large portion of the overall memory requirements of the resulting algorithm for solving the integral equations of scattering and radiation problems in this paper it is proposed to expand the k space representation of the basis functions by spherical harmonics in order to reduce the sampling redundancy introduced by numerical quadrature rules aggregations plane wave translations and disaggregations in the realized multilevel fast multipole method mlfmm are carried out using the k space samples of a numerical quadrature rule however the incoming plane waves on the finest mlfmm level are expanded in spherical harmonics again thus due to the orthonormality of spherical harmonics the testing integrals for the individual testing functions are simplified into series over products of spherical harmonics expansion coefficients overall the resulting mlfmm can save a considerable amount of memory without compromising accuracy and numerical speed

a modified fdtd 2 4 scheme for modeling electrically large structures with high phase accuracy a new fourth order finite difference time domain fdtd scheme has been developed that exhibits extremely low phase errors at low grid resolutions compared to the conventional fdtd scheme moreover this new scheme is capable of combining with the standard yee 1966 scheme to produce a stable hybrid algorithm the problem of wave propagation through a building is simulated using this new hybrid algorithm to demonstrate the large savings in computing resources it could afford with this new development the fdtd method can now be used to successfully model structures that are thousands of wavelengths large using the present day computer technology

double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers a novel double higher order galerkin type method of moments based on higher order geometrical modeling and higher order current modeling is proposed for surface integral equation analysis of combined metallic and dielectric antennas and scatterers of arbitrary shapes the technique employs generalized curvilinear quadrilaterals of arbitrary geometrical orders for the approximation of geometry metallic and dielectric surfaces and hierarchical divergence conforming polynomial vector basis functions of arbitrary orders for the approximation of electric and magnetic surface currents within the elements the geometrical orders and current approximation orders of the elements are entirely independent from each other and can be combined independently for the best overall performance of the method in different applications the results obtained by the higher order technique are validated against the analytical solutions and the numerical results obtained by low order moment method techniques from literature the examples show excellent accuracy flexibility and efficiency of the new technique at modeling of both current variation and curvature and demonstrate advantages of large domain models using curved quadrilaterals of high geometrical orders with basis functions of high current approximation orders over commonly used small domain models and low order techniques the reduction in the number of unknowns is by an order of magnitude when compared to low order solutions

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2