large deviations for random walks on galton watson trees averaging and uncertainty in the study of large deviations for random walks in random environment a key distinction has emerged between quenched asymptotics conditional on the environment and annealed asymptotics obtained from averaging over environments in this paper we consider a simple random walk x n on a galton watson tree t i e on the family tree arising from a supercritical branching process denote by x n the distance between the node x n and the root of t our main result is the almost sure equality of the large deviation rate function for x n n under the quenched measure conditional upon t and the rate function for the same ratio under the annealed measure averaging on t according to the galton watson distribution this equality hinges on a concentration of measure phenomenon for the momentum of the walk the momentum at level n for a specific tree t is the average over random walk paths of the forward drift at the hitting point of that level this concentration or certainty is a consequence of the uncertainty in the location of the hitting point we also obtain similar results when x n is a biased walk on a galton watson tree even though in that case there is no known formula for the asymptotic speed our arguments rely at several points on a ubiquity lemma for galton watson trees due to grimmett and kesten 1984

the bymoment method for two dimensional electromagnetic scattering the bymoment method is presented for analyzing two dimensional electromagnetic wave scattering in an unbounded region the method introduces a conforming surface to geometrically decouple the interior region containing the scatterer from the exterior region extending to infinity the solution in the interior is generated using the standard finite element solution of an interior dirichlet boundary value problem the interior solution is then coupled to the exterior using green s theorem in the unbounded exterior region the validity of the method and the associated computer program is demonstrated by comparing the results to those of other methods for scattering by various cylindrical geometries

numerical dispersion in dg fetd method using brick and tetrahedral edge elements a numerical dispersion analysis nda of the discontinuous galerkin finite element time domain dg fetd method is presented vector basis functions under the nda are with brick and tetrahedral elements this nda shows that there exist both normal and spurious modes in the dg fetd method a dg fetd modeling of transverse electromagnetic tem wave propagation in a parallel waveguide is applied to verify the nda for zeroth order vector bases the effect of wave propagation direction and electrical size of elements on numerical dispersion is investigated it is shown from the nda for higher order interpolatory vector bases that the phase error of normal modes can be significantly reduced by using higher order bases

numerical dispersion of higher order nodal elements in the finite element method the discretization inherent in the finite element method results in numerical dispersion this dispersion is investigated for a time harmonic plane wave propagating through an infinite two dimensional finite element mesh composed of uniform quadrilateral and triangular elements the effects on the dispersion due to the propagation direction of the wave the order of the elements the node density and the mesh geometry are studied results are given which can serve as a guide in selecting the appropriate element order node density and mesh geometry when applying the finite element method

maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction abstract for second order linear equations and inequalities which are degenerate elliptic but which possess a uniformly elliptic direction we formulate and prove weak maximum principles which are compatible with a solvability theory in suitably weighted versions of l 2 based sobolev spaces the operators are not necessarily in divergence form have terms of lower order and have low regularity assumptions on the coefficients the needed weighted sobolev spaces are in general anisotropic spaces defined by a non negative continuous matrix weight as preparation we prove a poincare inequality with respect to such matrix weights and analyze the elementary properties of the weighted spaces comparisons to known results and examples of operators which are elliptic away from a hyperplane of arbitrary codimension are given finally in the important special case of operators whose principal part is of grushin type we apply these results to obtain some spectral theory results such as the existence of a principal eigenvalue

quenched annealed and functional large deviations for one dimensional random walk in random environment suppose that the integers are assigned random variables i taking values in the unit interval which serve as an environment this environment defines a random walk x n called a rwre which when at i moves one step to the right with probability i and one step to the left with probability 1 i when the i sequence is i i d greven and den hollander 1994 proved a large deviation principle for x n n conditional upon the environment with deterministic rate function we consider in this paper large deviations both conditioned on the environment quenched and averaged on the environment annealed for the rwre in the ergodic environment case the annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy we also give a detailed qualitative description of the resulting rate functions our techniques differ from those of greven and den hollander and allow us to present also a trajectorial quenched large deviation principle

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

a hybrid finite element method for 3 d scattering using nodal and edge elements a hybrid finite element method for three dimensional scattering is presented and numerical examples shown this approach which couples finite element discretization with the method of moments is particularly well suited for monostatic radar cross section calculations the method is based on a scalar and vector potential formulation of maxwell s equations the use of nodal elements and a highly efficient body of revolution implementation of the method of moments combined nodal and edge elements are employed to accurately model fields around corners and edges a curvature based sampling criterion is derived and shown to ensure accurate answers for highly curved scatterers numerical results and cray computer timings are compared with published results for an edge element code using radiation boundary conditions

a partitioning technique for the finite element solution of electromagnetic scattering from electrically large dielectric cylinders a finite element partitioning scheme has been developed to reduce the computational costs of modeling electrically large geometries in the partitioning scheme the cylinder is divided into many sections the finite element method is applied to each section independent of the other sections and then the solutions in each section are coupled through the use of the tangential field continuity conditions between adjacent sections since the coupling matrix is significantly smaller than the original finite element matrix it is expected that both the cpu time and memory costs can be significantly reduced the partitioning scheme is coupled to the bymoment method to account for the boundary truncation numerical results are presented to demonstrate the efficiency and accuracy of the method

reducing the phase error for finite difference methods without increasing the order the phase error in finite difference fd methods is related to the spatial resolution and thus limits the maximum grid size for a desired accuracy greater accuracy is typically achieved by defining finer resolutions or implementing higher order methods both these techniques require more memory and longer computation times in this paper new modified methods are presented which are optimized to problems of electromagnetics simple methods are presented that reduce numerical phase error without additional processing time or memory requirements furthermore these methods are applied to both the helmholtz equation in the frequency domain and the finite difference time domain fdtd method both analytical and numerical results are presented to demonstrate the accuracy of these new methods

random walks capacity and percolation on trees a collection of several different probabilistic processes involving trees is shown to have an unexpected unity this makes possible a fruitful interplay of these probabilistic processes the processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well our work has five specific aims first an exact correspondence between random walks and percolation on trees is proved extending and sharpening previous work of the author this is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes second we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation this capacitary formulation extends and sharpens work of fan on random interval coverings third we show how this formulation also applies to generalize work of evans on random labelling of trees fourth the correspondence between random walks and percolation is used to decide whether certain random walks on random trees are transient or recurrent a s in particular we resolve a conjecture of griffeath on the necessity of the nash williams criterion fifth for this last purpose we establish several new basic results on branching processes in varying environments

errors due to spatial discretization and numerical precision in the finite element method the effects of the spatial discretization and the numerical precision on a plane wave propagating through a finite element mesh are investigated in this work the spatial discretization results in dispersion in the amplitude and the phase of the wave and in a non uniform rate of convergence within an element the finite precision in the calculations used in a finite element code results in degraded accuracy these errors are investigated as a function of the node density the order of the elements and the precision of the calculations used in the finite element code the errors for first through eighth order elements are investigated both analytically and numerically

accurate solutions of maxwell s equations around pec corners and highly curved surfaces using nodal finite elements a method is presented for computing accurate solutions of maxwell s equations in the presence of perfect electrical conductors pecs with sharp corners and highly curved surfaces using conventional nodal finite elements and a scalar vector s v potential formulation this technique approximates the pec with an impedance boundary condition ibc where the impedance is small critically it couples both potentials through this boundary condition rather than setting the scalar potential to zero this permits cancellation of the tangential components of the vector potential resulting in an accurate normal electric field the cause for the inaccuracies that nodal methods experience in the presence of sharp pec corners or highly curved pec surfaces is elucidated it is then shown how the inclusion of the scalar potential cures these deficiencies permitting accurate solutions spectral analysis of the resulting finite element matrices are shown validating the boundary conditions used examples are presented comparing a benchmark solution conventional pec and ibc boundary conditions and the new s v potential ibc on a pec wedge and pec ellipse in both cases the new s v ibc produces superior results

distance from submanifolds with boundary and applications to poincare inequalities and to elliptic and parabolic problems abstract we investigate regularity of the distance function from submanifolds with boundary of r n by exploiting these regularity results we show a poincare inequality with matrix weights that degenerate along a submanifold with boundary furthermore we prove uniqueness of solutions to elliptic and parabolic equations which degenerate on submanifolds with boundary

a law of large numbers for random walks in random environment we derive a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of kalikow the approach is based on the existence of a renewal structure under an assumption of transience in the direction l this extends to a multidimensional context previous work of kesten our results also enable proving the convergence of the law of the environment viewed from the particle toward a limiting distribution

distance to ck hypersurfaces abstract it is shown by elementary means that a ck hypersurface m of positive reach in r n 1 has the property that the signed distance function to it is ck k 1 this extends and complements work of federer gilbarg and trudinger and serrin

large deviations for a random walk in random environment let px x z be an i i d collection of 0 1 valued random variables given let x n n 0 be the markov chain on z defined by x 0 0 and x n 1 x n 1 resp x n n 1 with probability px n resp 1 px n it is shown that x n n satisfies a large deviation principle with a continuous rate function that is lim n 1 n log p x n n n i a s for n 1 1 first we derive a representation of the rate function i in terms of a variational problem second we solve the latter explicitly in terms of random continued fractions this leads to a classification and qualitative description of the shape of i in the recurrent case i is nonanalytic at 0 in the transient case i is nonanalytic at c 0 c for some c 0 with linear pieces in between

nested multigrid vector and scalar potential finite element method for fast computation of two dimensional electromagnetic scattering nested multigrid techniques are combined with the ungauged vector and scalar potential formulation of the finite element method to accelerate the convergence of the numerical solution of two dimensional electromagnetic scattering problems the finite element modeling is performed on nested meshes of the same computational domain the conjugate gradient method is used to solve the resultant finite element matrix for the finest mesh while the nested multigrid vector and scalar potential algorithm acts as the preconditioner of the iterative solver numerical experiments are used to demonstrate the superior numerical convergence and efficient memory usage of the proposed algorithm

some response characteristics of finite element tidal models abstract small scale error can arise in finite element tidal models from different aspects of the spatial discretization one source is incompatibility in smoothness of interpolation of the pressure and velocity fields noise of this type can be eliminated completely by using stokes helmholtz potentials to represent the velocity and interpolating these potentials with the same smoothness as is given to the pressure another source is inhomogeneous triangulation in which the number of links is not the same at all nodes noise of this type can be suppressed by using a high resolution grid the resolution in question being that of the response rather than of the domain these points are illustrated by examining the response characteristics of idealized models it is found that in the models considered small scale noise can be ascribed to resonance anomalies associated with the method of spatial discretization in this analysis the time is continuous but some remarks are appended on the effect of time discretization and a formal proof is given of the time stepping stability of a general discretized form of laplace s tidal equations

an investigation of numerical dispersion in the vector finite element method using quadrilateral elements the discretization inherent in the vector finite element method results in the numerical dispersion of a propagating wave the numerical dispersion of a time harmonic plane wave propagating through an infinite two dimensional vector finite element mesh composed of uniform quadrilateral elements is investigated the effects on the numerical dispersion of the propagation direction of the wave the order of the polynomials used for the basis functions and the electrical size of the elements are investigated simple formulas are presented which are excellent approximations to the exact numerical dispersion the numerical dispersion is validated by a numerical example

large deviation principle for random walk in a quenched random environment in the low speed regime we consider a one dimensional random walk x n n n in a random environment of zero or strictly positive drifts we establish a full large deviation principle for x n n of the correct order n log n 2 in the low speed regime valid for almost every environment this completes the large deviation picture obtained earlier by greven and den hollander and gantert and zeitouni in the case of zero and positive drifts the proof uses coarse graining along with concentration of measure techniques

time domain finite element methods various time domain finite element methods for the simulation of transient electromagnetic wave phenomena are discussed detailed descriptions of test trial spaces explicit and implicit formulations nodal and edge facet element basis functions are given along with the numerical stability properties of the different methods the advantages and disadvantages of mass lumping are examined finally the various formulations are compared on the basis of their numerical dispersion performance

tail estimates for one dimensional random walk in random environment suppose that the integers are assigned i i d random variables x taking values in the unit interval which serve as an environment this environment defines a random walk x k called a rwre which when atx moves one step to the right with probability x and one step to the left with probability 1 x solomon 1975 determined the almost sure asymptotic speed rate of escape of a rwre for certain environment distributions where the drifts 2 x 1 can take both positive and negative values we show that the chance of the rwre deviating below this speed has a polynomial rate of decay and determine the exponent in this power law for environments which allow only positive and zero drifts we show that these large deviation probabilities decay like exp cn1 3 this differs sharply from the rates derived by greven and den hollander 1994 for large deviation probabilities conditioned on the environment as a by product we also provide precise tail and moment estimates for the total population size in a branching process with random environment

slowdown and neutral pockets for a random walk in random environment we consider a d dimensional random walk in random environment for which transition probabilities at each site are either neutral or present an effective drift pointing to the right we obtain large deviation estimates on the probability that the walk moves in a too slow ballistic fashion both under the annealed and quenched measures these estimates underline the key role of large neutral pockets of the medium in the occurrence of slowdowns of the walk

a study of discretization error in the finite element approximation of wave solutions a dispersion analysis is used to study the errors caused by the spatial discretization of the finite element method for the two dimensional scalar helmholtz equation it is shown that the error can be determined analytically for a uniform mesh of infinite extent numerical results are presented to show the effects of several parameters on the error these parameters are the nodal density the electrical size of the mesh the direction of propagation of the incident wave the type of element and the type of boundary condition

further development of vector generalized finite element method and its hybridization with boundary integrals recently vector generalized finite element method vgfem was introduced for the solution of the vector helmholtz equation and its applicability was validated for canonical problems vgfem uses a local helmholtz decomposition to construct basis functions in overlapping local domains of some canonical shape while using a canonical shape for local domains adds flexibility to the method one needs to provide information regarding boundaries of domains inhomogeneities the need for surface information proves to be a bottleneck in using the method for a larger class of problems this paper is targeted towards overcoming these deficiencies here we will introduce the modifications to this method that permit interfacing with arbitrarily shaped local domains to facilitate interfacing with existing meshing software integrate this method with boundary integrals and provide a framework for studying dispersion as will be apparent the hybridization of the method with boundary integrals is not a simple adaptation of existing methods onto the vgfem framework likewise dispersion analysis is nontrivial due to the overlapping nature of vgfem basis functions a range of practical problems has been analyzed within the presented framework and results are compared either against measurements or existing fem data to validate the presented methodology

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

reducing spurious oscillations in discontinuous wave propagation simulation using high order finite elements in this paper a modified explicit time integration scheme is proposed for simulating the propagation of discontinuous waves the spatial domain is discretized by the finite element method to obtain accurate results the standard finite element method requires a very fine mesh which is why the computational effort can be very time consuming the use of high order finite element methods such as the spectral element method based on lagrange polynomials through gauss lobatto legendre points or the iso geometric analysis using non uniform rational b splines can reduce the enormous computational costs significantly compared to the standard finite element method however explicit time integration schemes such as the central difference method cannot eliminate the spurious oscillation near the front wave the procedure proposed here is basically similar to the explicit method of noh and bathe but the semi discrete equation of motion is modified by introducing a damping parameter to suit the high order fem analysis of the discontinuous wave propagation the performance due to this modification is tested for one dimensional wave propagation problems using both lumped and consistent mass matrices then the idea of a combination of the consistent and row sum lumped mass matrices is evaluated the proposed method is studied also in two dimensions by considering the lamb problem of wave propagation the results are promising enough to provide a better simulation of the discontinuous wave propagation problem

on accuracy conditions for the numerical computation of waves abstract the helmholtz equation k 2 n 2 u f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems such problems can be solved numerically by first truncating the given unbounded domain imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization for example using a finite element method in practical applications the mesh size h and the wave number k are not independent but are constrained by the accuracy of the desired computation it will be shown that the number of points per wavelength measured by kh 1 is not sufficient to determine the accuracy of a given discretization for example the quantity k 3 h 2 is shown to determine the accuracy in the l 2 norm for a second order discretization method applied to several propagation models

the advantages of triangular and tetrahedral edge elements for electromagnetic modeling with the finite element method in this paper we examine the numerical dispersion properties of the tetrahedral edge element in comparison to other edge and nodal elements for all nodal elements as well as hexahedral edge elements the phase error is always either positive or negative for waves propagating at any incidence angle which means that the error accumulates as the wave propagates from element to element this effect can produce large errors for electrically large geometries on the other hand the tetrahedral edge elements can produce either a negative or positive phase error depending on the direction of propagation through the element for an unstructured mesh phase cancellation occurs since the orientation of the tetrahedral elements are arbitrary because of the cancellation the numerical dispersion error is very low for meshes composed of well shaped tetrahedral edge elements numerical results are presented to demonstrate this point

a novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering and radiation problems a novel hybridization of the finite element fe and boundary integral methods is presented for an efficient and accurate numerical analysis of electromagnetic scattering and radiation problems the proposed method derives an adaptive numerical absorbing boundary condition abc for the finite element solution based on boundary integral equations unlike the standard finite element boundary integral approach the proposed method is free of interior resonance and produces a purely sparse system matrix which can be solved very efficiently unlike the traditional finite element absorbing boundary condition approach the proposed method uses an arbitrarily shaped truncation boundary placed very close to the scatterer radiator to minimize the computational domain and more importantly the method produces a solution that converges to the true solution of the problem to demonstrate its great potential the proposed method is implemented using higher order curvilinear vector elements a mixed functional is designed to yield both electric and magnetic fields on an integration surface without numerical differentiation to be used in the calculation of the adaptive abc the required evaluation of boundary integrals is carried out using the multilevel fast multipole algorithm which greatly reduces both the memory requirement and cpu time the finite element equations are solved efficiently using the multifrontal algorithm a mathematical analysis is conducted to study the convergence of the method finally a number of numerical examples are given to illustrate its accuracy and efficiency

discretization errors in finite methods issues and possible solutions in this paper we review many of the sources of discretization error when finite methods is applied to high frequency electromagnetic problems a major source of error is the numerical dispersion error this error is probably the most serious error for electrically large geometries and we review past efforts to reduce this error we also present a new edge based finite difference method which offers several improvements to current finite difference this new method has less numerical dispersion error and less errors due to material discontinuities than current finite difference methods we compare its performance to vector finite elements and show its superior accuracy finally we propose a new subgridding scheme to reduce discretization error which maintains second order accuracy

finite element solution of maxwell s equations for hyperthermia treatment planning abstract a general galerkin formulation is given for the periodic form of maxwell s equations on a bounded inhomogeneous medium the formulation conserves energy exactly and its dispersion relation is studied solutions for test problems representative of hyperthermia applications show good agreement with analytic solutions a representative clinical example is shown

phase error control for fd td methods of second and fourth order accuracy for fd td methods used to solve maxwell s equations we determine the spatial resolution of the discretized domain in terms of the total computation time and the desired phase error it is shown that the spatial step should vary as spl delta x spl sim g e sub spl phi t sub c sup 1 s in order to maintain a prescribed phase error level e sub spl phi throughout the computation time t sub c where s 2 or 4 is the spatial order of accuracy of the scheme and g is a geometric factor significantly we show that the rule of thumb of using 10 20 points per wavelength to determine the spatial cell size for the standard scheme is not optimal our results are verified by numerical simulations in two dimensions with the yee 1966 scheme and a new fourth order accurate fd td scheme