reconstruction of a time dependent source term from a single boundary measurement in maxwell s equations with nonlinear generalized ohm s law abstract hyperbolic maxwell s equation with an unknown time dependent source is investigated we consider a nonlinear generalized ohm s law in our model the source is reconstructed from a single boundary measurement over a part of the boundary we use well known rothe s method to show the existence of a solution in the case of regular solution we provide a uniqueness proof as well to support theoretical results a numerical experiment is provided

iterative across time solution of linear differential equations krylov subspace versus waveform relaxation the aim of this paper is two fold first we propose an efficient implementation of the continuous time waveform relaxation wr method based on block krylov subspaces second we compare this new wr krylov implementation against krylov subspace methods combined with the shift and invert sai technique some analysis and numerical experiments are presented since the wr krylov and sai krylov methods build up the solution simultaneously for the whole time interval and there is no time stepping involved both methods can be seen as iterative across time methods the key difference between these methods and standard time integration methods is that their accuracy is not directly related to the time step size

parallelization of the rational arnoldi algorithm rational krylov methods are applicable to a wide range of scientific computing problems and the rational arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational krylov space typically the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration we explore the option of solving several linear systems simultaneously thus constructing the rational krylov basis in parallel if this is not done carefully the basis being orthogonalized may become badly conditioned leading to numerical instabilities in the orthogonalization process we introduce the new concept of continuation pairs which gives rise to a near optimal parallelization strategy that allows to control the growth of the condition number of this nonorthogonal basis as a consequence we obtain a significantly more accurate and reliable parallel rational arnoldi algorithm the computational benefits are illustrated using several numerical examples from different application areas

on optimal convergence rate of the rational krylov subspace reduction for electromagnetic problems in unbounded domains we solve an electromagnetic frequency domain induction problem in mathbf r 3 for a frequency interval using rational krylov subspace rks approximation the rks is constructed by spanning on the solutions for a certain a priori chosen set of frequencies we reduce the problem of the optimal choice of these frequencies to the third zolotaryov problem in the complex plane having an approximate closed form solution and determine the best cauchy hadamard convergence rate the theory is illustrated with numerical examples for maxwell s equations arising in 3d magnetotelluric geophysical exploration

three dimensional modeling of ip effects in time domain electromagnetic data understanding the effects of induced polarization ip effects on time domain electromagnetic data requires the ability to simulate common survey techniques when taking chargeability into account most existing techniques preform this modeling in the frequency domain prior to transforming their results to the time domain even though this technique can allow for chargeable material to be easily incorporated its application for some problems can be computationally limiting we developed a new technique for forward modeling the time domain electromagnetic response of chargeable materials in three dimensions the frequency dependence of ohms law translates to an ordinary differential equation when considered in the time domain the system of ordinary partial differential equations was then discretized using an implicit time stepping algorithm that yielded absolute stability this approach allowed us to operate directly in the time domain and avoid frequency to time domain transformations although this approach can be applied directly to materials exhibiting debye dispersions other cole cole dispersions resulted in fractional derivatives in time to overcome this difficulty pade approximations were used to represent the frequency dependence as a rational series of integer order terms the resulting method was then simplified to generate a reduced time domain model that can be used to forward model the ip decay curves in the absence of any electromagnetic coupling we found numerical examples in which the method produced accurate results the potential application of the method was demonstrated by modeling the full time domain electromagnetic response of a gradient array ip survey and the occurrence of negative transients in airborne time domain electromagnetic data

an approximate framework for quantum transport calculation with model order reduction a new approximate computational framework is proposed for computing the non equilibrium charge density in the context of the non equilibrium green s function negf method for quantum mechanical transport problems the framework consists of a new formulation called the x formulation for single energy density calculation based on the solution of sparse linear systems and a projection based nonlinear model order reduction mor approach to address the large number of energy points required for large applied biases the advantages of the new methods are confirmed by numerical experiments

on adaptive choice of shifts in rational krylov subspace reduction of evolutionary problems we compute u t exp ta varphi using rational krylov subspace reduction for 0 leq t infty where u t varphi in mathbf r n and 0 a a in mathbf r n times n a priori optimization of the rational krylov subspaces for this problem was considered in v druskin l knizhnerman and m zaslavsky siam j sci comput 31 2009 pp 3760 3780 there was suggested an algorithm generating sequences of equidistributed shifts which are asymptotically optimal for the cases with uniform spectral distributions here we develop a recursive greedy algorithm for choice of shifts taking into account nonuniformity of the spectrum the algorithm is based on an explicit formula for the residual in the frequency domain allowing adaptive shift optimization at negligible cost the effectiveness of the developed approach is demonstrated in an example of the three dimensional diffusion problem for maxwell s equation arising in geophysical exploration we compare our approach with the one using the above mentioned equidistributed sequences of shifts numerical examples show that our algorithm is able to adapt to the spectral density of operator a for examples with near uniform spectral distributions both algorithms show the same convergence rates but the new algorithm produces superior convergence for cases with nonuniform spectra

on conjugacies between piecewise smooth circle maps abstract let f i i 1 2 be piecewise c 1 circle homeomorphisms with two break points log d f i i 1 2 are absolutely continuous on each continuity interval of d f i and d log d f i l p for some p 1 suppose the jump ratios of f 1 and f 2 at their break points do not coincide but f 1 f 2 have the same total jumps i e the product of jump ratios and identical irrational rotation number of bounded type then the map h conjugating f 1 and f 2 is a singular function that is it is continuous on s 1 but d h x 0 almost everywhere with respect to lebesgue measure

solution of the time domain inverse resistivity problem in the model reduction framework part i one dimensional problem with siso data many time domain problems in engineering applications can be described by means of parameter dependent time invariant dynamic systems we are interested in parameter estimation by fitting available transient measurements using the nonlinear least square method as the main application we consider the control source electromagnetic csem method of geophysical exploration governed by the diffusion maxwell system where the unknown parameters describe the spatial distribution of electrical resistivity we propose a novel model reduction approach for constructing an efficient approximation of the jacobian the reduction is based on subspace projection of the state variable and it allows us to split the time and space dependence of the derivative we examine several popular rational krylov subspaces and single out the h 2 optimal subspace that not only minimizes the approximation error but completely annuls its influence on the inversion result preliminary numerical experiments with a simplified one dim

renormalizations and rigidity theory for circle homeomorphisms with singularities of the break type circle homeomorphisms with singularities of the break type are considered in the case when rotation numbers have periodic continued fraction expansion we establish hyperbolicity for renormalizations and then use it in order to prove the following rigidity result namely we show that any two homeomorphisms with a single break point are smoothly conjugate to each other provided they have the same quadratic irrational rotation number and the same size of a break

a general method for fdtd modeling of wave propagation in arbitrary frequency dispersive media a general formulation is presented for finite difference time domain fdtd modeling of wave propagation in arbitrary frequency dispersive media two algorithmic approaches are outlined for incorporating dispersion into the fdtd time stepping equations the first employs a frequency dependent complex permittivity denoted form 1 and the second employs a frequency dependent complex conductivity denoted form 2 a pade representation is used in z transform space to represent the frequency dependent permittivity form 1 or conductivity form 2 this is a generalization over several previous methods employing either debye lorentz or drude models the coefficients of the pade model may be obtained through an optimization process leading directly to a finite difference representation of the dispersion relation without introducing discretization error stability criteria for the dispersive fdtd algorithms are given we show that several previously developed dispersive fdtd algorithms can be cast as special cases of our more general framework simulation results are presented for a one dimensional 1 d air muscle example considered previously in the literature and a three dimensional 3 d radiation problem in dispersive lossy soil using measured soil data

complex bounds for real maps in this paper we prove complex bounds also referred to as a priori bounds for c 3 and in particular for analytic maps of the interval any c 3 mapping of the interval has an asymptotically holomorphic extension to a neighbourhood of the interval we associate to such a map a complex box mapping which provides a kind of markov structure for the dynamics moreover we prove universal geometric bounds on the shape of the domains and on the moduli between components of the range and domain such bounds show that the first return maps to these domains are well controlled and consequently such bounds form one of the corner stones in many recent results in one dimensional dynamics for example renormalization theory rigidity density of hyperbolicity and local connectivity of julia sets

convergence analysis of an extended krylov subspace method for the approximation of operator functions in exponential integrators we analyze the convergence of an extended krylov subspace method for the approximation of operator functions that appear in exponential integrators for operators the size of the polynomial part of the extended krylov subspace is restricted according to the smoothness of the initial data this restriction for the continuous operator has a significant influence on the approximation of matrix functions evaluated for matrices stemming from space discretizations of the continuous operator we prove convergence of the method for the continuous operator and in the discrete case this leads to a convergence independent of the norm of the discretized operator uniformly over all possible grids the analysis is illustrated by numerical experiments

solution of large scale evolutionary problems using rational krylov subspaces with optimized shifts we consider the computation of u t exp ta varphi using rational krylov subspace reduction for 0 le t infty where u t varphi in mathbf r n and 0 a a in mathbf r n times n the objective of this work is the optimization of the shifts for the rational krylov subspace rks we consider this problem in the frequency domain and reduce it to a classical zolotaryov problem the latter yields an asymtotically optimal solution with real shifts we also construct an infinite sequence of shifts yielding a nested sequence of the rkss with the same optimal cauchy hadamard convergence rate the effectiveness of the developed approach is demonstrated on an example of the three dimensional diffusion problem for maxwell s equation arising in geophysical exploration

renormalization horseshoe and rigidity for circle diffeomorphisms with breaks in this paper we build the renormalization horseshoe for the circle homeomorphisms which are c2 smooth everywhere except for one point and at that point have a jump in first derivative we also show that two such homeomorphisms are c1 smoothly conjugate for a certain class of rotation numbers which include non diophantine numbers with arbitrarily high rate of growth

recurrence relations for orthogonal rational functions it is well known that members of families of polynomials that are orthogonal with respect to an inner product determined by a nonnegative measure on the real axis satisfy a three term recursion relation analogous recursion formulas are available for orthogonal laurent polynomials with a pole at the origin this paper investigates recursion relations for orthogonal rational functions with arbitrary prescribed real or complex conjugate poles the number of terms in the recursion relation is shown to be related to the structure of the orthogonal rational functions

the attractor of renormalization and rigidity of towers of critical circle maps we demonstrate the existence of a global attractor with a cantor set structure for the renormalization of critical circle mappings the set is invariant under a generalized renormalization transformation whose action on is conjugate to the two sided shift with a countable alphabet

on monotonicity of the lanczos approximation to the matrix exponential abstract we prove strictly monotonic error decrease in the euclidian norm of the krylov subspace approximation of exp a where and a are respectively a vector and a symmetric matrix in addition we show that the norm of the approximate solution grows strictly monotonically with the subspace dimension

adaptive rational krylov subspaces for large scale dynamical systems the rational krylov space is recognized as a powerful tool within model order reduction techniques for linear dynamical systems however its success has been hindered by the lack of a parameter free procedure which would effectively generate the sequence of shifts used to build the space in this paper we propose an adaptive computation of these shifts the whole procedure only requires us to inject some initial rough estimate of the spectral region of the matrix while further information is automatically generated during the process the approach is a full generalization to the nonsymmetric case of the idea first proposed in druskin et al 2010 18 and it is used for two important problems in control the approximation of the transfer function and the numerical solution of large lyapunov equations the procedure can be naturally extended to other related problems such as the solution of the sylvester equation and parametric or higher order systems several numerical experiments are proposed to assess the quality of the rational projection space over its most natural competitors

a lanczos model order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media in this paper we present a krylov subspace model order reduction technique for time and frequency domain electromagnetic wave fields in linear dispersive media starting point is a self consistent first order form of maxwell s equations and the constitutive relation this form is discretized on a standard staggered yee grid while the extension to infinity is modeled via a recently developed global complex scaling method by applying this scaling method the time or frequency domain electromagnetic wave field can be computed via a so called stability corrected wave function since this function cannot be computed directly due to the large order of the discretized maxwell system matrix krylov subspace reduced order models are constructed that approximate this wave function we show that the system matrix exhibits a particular physics based symmetry relation that allows us to efficiently construct the time and frequency domain reduced order models via a lanczos type reduction algorithm the frequency domain models allow for frequency sweeps meaning that a single model provides field approximations for all frequencies of interest and dominant field modes can easily be determined as well numerical experiments for two and three dimensional configurations illustrate the performance of the proposed reduction method

encoding via conjugate symmetries of slow oscillations for globally coupled oscillators we study properties of the dynamics underlying slow cluster oscillations in two systems of five globally coupled oscillators these slow oscillations are due to the appearance of structurally stable heteroclinic connections between cluster states in the noise free dynamics in the presence of low levels of noise they give rise to long periods of residence near cluster states interspersed with sudden transitions between them moreover these transitions may occur between cluster states of the same symmetry or between cluster states with conjugate symmetries given by some rearrangement of the oscillators we consider the system of coupled phase oscillators studied by hansel et al phys rev e 48 3470 1993 in which one can observe slow noise driven oscillations that occur between two families of two cluster periodic states in the noise free case there is a robust attracting heteroclinic cycle connecting these families the two families consist of symmetric images of two inequivalent periodic orbits that have the same symmetry for n 5 oscillators one of the periodic orbits has one unstable direction and the other has two unstable directions examining the behavior on the unstable manifold for the two unstable directions we observe that the dimensionality of the manifold can give rise to switching between conjugate symmetry orbits by applying small perturbations to the system we can easily steer it between a number of different marginally stable attractors finally we show that similar behavior occurs in a system of phase energy oscillators that are a natural extension of the phase model to two dimensional oscillators we suggest that switching between conjugate symmetries is a very efficient method of encoding information into a globally coupled system of oscillators and may therefore be a good and simple model for the neural encoding of information

lanczos method for heat conduction analysis an analysis technique that uses the lanczos vectors to construct a reyleigh ritz solution for the transient heat conduction problem is presented the lanczos vectors are derived from the heat supply vector and incorporate all the information peculiar to a given problem they are orthonormal with respect to the heat capacitance matrix and are generated with a minimum of computational effort furthermore an error bound is derived that can be used to terminate the lanczos algorithm as soon as the number of vectors required to obtain a desired degree of accuracy has been generated a summary of the algorithm is presented and the accuracy and efficiency of the method is illustrated using a numerical example

an error analysis for rational galerkin projection applied to the sylvester equation in this paper we suggest a new formula for the residual of galerkin projection onto rational krylov spaces applied to a sylvester equation and establish a relation to three different underlying extremal problems for rational functions these extremal problems enable us to compare the size of the residual for the above method with that obtained by adi in addition we deduce several new a priori error estimates for galerkin projection onto rational krylov spaces both for the sylvester and for the lyapunov equation

effects of nonresonant interaction in ensembles of phase oscillators we consider general properties of groups of interacting oscillators for which the natural frequencies are not in resonance such groups interact via nonoscillating collective variables like the amplitudes of the order parameters defined for each group we treat the phase dynamics of the groups using the ott antonsen ansatz and reduce it to a system of coupled equations for the order parameters we describe different regimes of cosynchrony in the groups for a large number of groups heteroclinic cycles corresponding to a sequential synchronous activity of groups and chaotic states where the order parameters oscillate irregularly are possible

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

renormalization horseshoe for critical circle maps we prove that the invariant horseshoe for the renormalization of critical circle maps is globally uniformly hyperbolic with one dimensional unstable direction this extends our earlier result on hyperbolicity of periodic orbits of the renormalization and completes the proof of lanford s program

solution of time convolutionary maxwell s equations using parameter dependent krylov subspace reduction we suggest a new algorithm for the solution of the time domain maxwell equations in dispersive media after spacial discretization we obtain a large system of time convolution equations then this system is projected onto a small subspace consisting of the laplace domain solutions for a preselected set of laplace parameters this approach is a generalization of the rational krylov subspace approach for the solution of non dispersive maxwell s systems we show that the projected system preserves such properties of the initial system as stability and passivity as an example we consider the 3d quasistationary induced polarization problem with the cole cole conductivity model important for geophysical oil exploration our numerical experiments show that the introduction of the induced polarization does not have significant effect on convergence

accelerated source sweep analysis using a reduced order model approach this communication is concerned with the development of a model order reduction mor approach for the acceleration of a source sweep analysis using the volume electric field integral equation efie formulation in particular we address the prohibitive computational burden associated with the repeated solution of the two dimensional electromagnetic wave scattering problem for source sweep analysis the method described within is a variant of the krylov subspace approach to mor that captures at an early stage of the iteration the essential features of the original system as such these approaches are capable of creating very accurate low order models numerical examples are provided that demonstrate the speed up achieved by utilizing these mor approaches when compared against a method of moments mom solution accelerated by use of the fast fourier transform fft

determination of a time dependent convolution kernel from a boundary measurement in nonlinear maxwell s equations a nonlinear hyperbolic maxwell equation with an unknown solely time dependent convolution kernel is studied the missing kernel is recovered from an additional normal component measurement over the whole boundary the existence of a solution to this inverse problem is shown moreover a constructive algorithm for approximations is designed and its convergence is established uniqueness is proved for a regular solution theoretical results are supported by a numerical experiment

on the convergence of krylov subspace methods for matrix mittag leffler functions in this paper we analyze the convergence of some commonly used krylov subspace methods for computing the action of matrix mittag leffler functions as is well known such functions find application in the solution of fractional differential equations we illustrate the theoretical results by some numerical experiments

generic rigidity for circle diffeomorphisms with breaks we prove that c r smooth r 2 circle diffeomorphisms with a break i e circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity are generically i e for almost all irrational rotation numbers not c 1 varepsilon rigid for any varepsilon 0 this result complements our recent proof joint with khanin geom funct anal 24 2002 2028 2014 that such maps are generically c 1 rigid it stands in remarkable contrast to the result of yoccoz ann sci ec norm sup 17 333 361 1984 that c r smooth circle diffeomorphisms are generically c r 1 varkappa rigid for any varkappa 0

solving metamaterial maxwell s equations via a vector wave integro differential equation in this paper we discuss the time domain metamaterial maxwell s equations one major contribution of this paper is that after some effort we find that the metamaterial maxwell s equations can be beautifully reduced to a vector wave integro differential equation involving just one unknown which is quite similar to that obtained from the standard maxwell s equations in vacuum then we study the existence and uniqueness of this new modeling equations and propose a fully discrete finite element method to solve this model numerical results justifying our analysis are presented this discovery shall make simulation of metamaterials much more efficient than the previous works

paraexp a parallel integrator for linear initial value problems a novel parallel algorithm for the integration of linear initial value problems is proposed this algorithm is based on the simple observation that homogeneous problems can typically be integrated much faster than inhomogeneous problems an overlapping time domain decomposition is utilized to obtain decoupled inhomogeneous and homogeneous subproblems and a near optimal krylov method is used for the fast exponential integration of the homogeneous subproblems we present an error analysis and discuss the parallel scaling of our algorithm the efficiency of this approach is demonstrated with numerical examples

strong effects of network architecture in the entrainment of coupled oscillator systems random networks of coupled phase oscillators representing an approximation for systems of coupled limit cycle oscillators are considered entrainment of such networks by periodic external forcing applied to a subset of their elements is numerically and analytically investigated for a large class of interaction functions we find that the entrainment window with a tongue shape becomes exponentially narrow for networks with higher hierarchical organization however the entrainment is significantly facilitated if the networks are directionally biased i e closer to the feedforward networks furthermore we show that the networks with high entrainment ability can be constructed by evolutionary optimization processes the neural network structure of the master clock of the circadian rhythm in mammals is discussed from the viewpoint of our results

parabolic and hyperbolic contours for computing the bromwich integral some of the most eectiv e methods for the numerical inversion of the laplace transform are based on the approximation of the bromwich contour integral the accuracy of these methods often hinges on a good choice of contour and several such contours have been proposed in the literature here we analyze two recently proposed contours namely a parabola and a hyperbola using a representative model problem we determine estimates for the optimal parameters that dene these contours an application to a fractional diusion equation is presented

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

large scale gauss newton inversion of transient controlled source electromagnetic measurement data using the model reduction framework abstracttransient data controlled source electromagnetic measurements are usually interpreted via extracting few frequencies and solving the corresponding inverse frequency domain problem coarse frequency sampling may result in loss of information and affect the quality of interpretation however refined sampling increases computational cost fitting data directly in the time domain has similar drawbacks i e its large computational cost in particular when the gauss newton gn algorithm is used for the misfit minimization that cost is mainly comprised of the multiple solutions of the forward problem and linear algebraic operations using the jacobian matrix for calculating the gn step for large scale 2 5d and 3d problems with multiple sources and receivers the corresponding cost grows enormously for inversion algorithms using conventional finite difference time domain fdtd algorithms a fast 3d forward solver based on the rational krylov subspace rks reduction algorithm using an optimal subsp

on convergence of krylov subspace approximations of time invariant self adjoint dynamical systems abstract we extend the rational krylov subspace algorithm from the computation of the action of the matrix exponential to the solution of stable dynamical systems a d dt u t b t u t 0 0 a d dt i 0 m a i d dt si i where m n a i a i r n n s 0 and u t b t r n b t 0 0 not assuming that evolution of b t is described by a low dimensional subspace of r n we show that the reduced equation is stable and derive an a priori error bound via rational approximation of the exponential on the boundary of the nonlinear numerical range of a we also describe a simple and easily computable external bound of this numerical range the obtained results are applied to the infinite order problem arising in the solution of the dispersive maxwell s system

approximation of functions of large matrices with kronecker structure we consider the numerical approximation of f mathcal a b f a b where b in mathbb r n b rn and mathcal a a is the sum of kronecker products that is mathcal a m 2 otimes i i otimes m 1 in mathbb r n times n a m2 i i m1 rn n here f is a regular function such that f mathcal a f a is well defined we derive a computational strategy that significantly lowers the memory requirements and computational efforts of the standard approximations with special emphasis on the exponential and on completely monotonic functions for which the new procedure becomes particularly advantageous our findings are illustrated by numerical experiments with typical functions used in applications

the rkfit algorithm for nonlinear rational approximation the rkfit algorithm outlined in m berljafa and s ga ttel generalized rational krylov decompositions with an application to rational approximation siam j matrix anal appl 2015 is a krylov based approach for solving nonlinear rational least squares problems this paper puts rkfit into a general framework allowing for its extension to nondiagonal rational approximants and a family of approximants sharing a common denominator furthermore we derive a strategy for the degree reduction of the approximants as well as methods for their conversion to partial fraction form for the efficient evaluation and root finding we also discuss commons and differences of rkfit and the popular vector fitting algorithm a matlab implementation of rkfit is provided and numerical experiments including the fitting of a mimo dynamical system and an optimization problem related to exponential integration demonstrate its applicability

error estimates and evaluation of matrix functions via the faber transform the need to evaluate expressions of the form f a or f a b where f is a nonlinear function a is a large sparse n times n matrix and b is an n vector arises in many applications this paper describes how the faber transform applied to the field of values of a can be used to determine improved error bounds for popular polynomial approximation methods based on the arnoldi process applications of the faber transform to rational approximation methods and in particular to the rational arnoldi process also are discussed

an iterative solution method for solving f a x b using krylov subspace information obtained for the symmetric positive definite matrix a abstract the conjugate gradients method generates successive approximations x i for the solution of the linear system ax b where a is symmetric positive definite and usually sparse it will be shown how intermediate information obtained by the conjugate gradients cg algorithm or by the closely related lanczos algorithm can be used to solve f a x b iteratively in an efficient way for suitable functions f the special case f a a 2 is discussed in particular we also consider the problem of solving ax b for different right hand sides b a variant on a well known algorithm for that problem is proposed which does not seem to suffer from the usual loss of orthogonality in the standard cg and lanczos algorithms

hybrid finite difference integral equation solver for 3d frequency domain anisotropic electromagnetic problems the modeling of the controlled source electromagnetic csem and single well and crosswell electromagnetic em configurations requires fine gridding to take into account the 3d nature of the geometries encountered in these applications that include geological structures with complicated shapes and exhibiting large variations in conductivities such as the seafloor bathymetry the land topography and targets with complex geometries and large contrasts in conductivities such problems significantly increase the computational cost of the conventional finite difference fd approaches mainly due to the large condition numbers of the corresponding linear systems to handle these problems we employ a volume integral equation ie approach to arrive at an effective preconditioning operator for our fd solver we refer to this new hybrid algorithm as the finite difference integral equation method fdie this fdie preconditioning operator is divergence free and is based on a magnetic field formulation similar to the lippman schwinger ie method this scheme allows us to use a background elimination approach to reduce the computational domain resulting in a smaller size stiffness matrix furthermore it yields a linear system whose condition number is close to that of the conventional lippman schwinger ie approach significantly reducing the condition number of the stiffness matrix of the fd solver moreover the fd framework allows us to substitute convolution operations by the inversion of banded matrices which significantly reduces the computational cost per iteration of the hybrid method compared to the standard ie approaches also well established fd homogenization and optimal gridding algorithms make the fdie more appropriate for the discretization of strongly inhomogeneous media some numerical studies are presented to illustrate the accuracy and effectiveness of the presented solver for csem single well and crosswell em applications

bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators abstract we study the bifurcation and dynamical behaviour of the system of n globally coupled identical phase oscillators introduced by hansel mato and meunier in the cases n 3 and n 4 this model has been found to exhibit robust slow switching oscillations that are caused by the presence of robust heteroclinic attractors this paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors we consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function these bifurcations preserve the permutation symmetry of the system we then investigate the implications of these bifurcations for the sensitivity to detuning i e the size of the smallest perturbations that give rise to loss of frequency locking for n 3 we find three types of heteroclinic bifurcation that are codimension one with symmetry on varying two parameters in the coupling function we find three curves giving a an s 3 transcritical homoclinic bifurcation b a saddle node heteroclinic bifurcation and c a z 3 heteroclinic bifurcation we also identify several global bifurcations with symmetry that organize the bifurcation diagram these are codimension two with symmetry for n 4 oscillators we determine many but not all codimension one bifurcations with symmetry including those that lead to a robust heteroclinic cycle a robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region furthermore we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system while for other parts of the phase plane it can coexist with stable limit cycles we finish with a discussion of bifurcations that appear for this coupling function and general n as well as for more general coupling functions

transient electromagnetic response of a three dimensional body in a layered earth the three dimensional 3 d electromagnetic scattering problem is first formulated in the frequency domain in terms of an electric field volume integral equation three dimensional responses are then fourier transformed with sine and cosine digital filters or with the decay spectrum the digital filter technique is applied to a sparsely sampled frequency sounding which is replaced by a cubic spline interpolating function prior to convolution with the digital filters typically 20 to 40 frequencies at five to eight points per decade are required for an accurate solution a calculated transient is usually in error after it has decayed more than six orders in magnitude from early to late time the decay spectrum usually requires ten frequencies for a satisfactory solution however the solution using the decay spectrum appears to be less accurate than the solution using the digital filters particularly after early times checks on the 3 d solution include reciprocity and convergence checks in the frequenc

solution of 3d time domain electromagnetic problems using optimal subspace projection abstracttime domain problems for controlled source electromagnetic exploration require accurate discretization of the solution for multiple spacial and temporal scales therefore forward simulation using conventional computational methods becomes computationally expensive even without accounting for induced polarization ip effects these effects create another complication caused by the presence of a convolution integral in the time domain maxwell system we suggested a novel fast and robust algorithm to solve the 3d time domain electromagnetic em problems that can be considered as a generalization of the spectral lanczos decomposition method the new method also allowed us to incorporate the ip effects without significant cost increase the discretized large scale maxwell system was projected onto a small subspace consisting of the laplace domain solutions the so called parameter dependent krylov subspace for an optimally chosen set of laplace parameters the projected system preserved stabilit

stability and phase error analysis of fd td in dispersive dielectrics four fd td extensions for the modeling of pulse propagation in debye or lorentz dispersive media are analyzed through studying the stability and phase error properties of the coupled difference equations corresponding to maxwell s equations and to the equations for the dispersion for good overall accuracy the author shows that all schemes should be run at their courant stability limit and that the timestep should finely resolve the medium timescales particularly for debye schemes it should be at least spl delta t 10 sup spl minus 3 spl tau while for lorentz schemes it should be spl delta t 10 sup spl minus 2 spl tau where spl tau is a typical medium relaxation time a numerical experiment with a debye medium confirms this the author has determined that two of the discretizations for debye media are totally equivalent in the lorentz medium case the author establishes that the method that uses the polarization differential equation to model dispersion is stable for all wavenumbers and that the method using the local in time constitutive relation is weakly unstable for modes with wavenumber k such that k spl delta x spl pi 2

optimal rational functions for the generalized zolotarev problem in the complex plane it has been long recognized that the determination of optimal parameters for the classical alternating direction implicit adi method leads to the zolotarev problem min r in r nn frac max r z z in e min r z z in f for disjoint compact sets e f subseteq mathbb c where rnn is the collection of rational functions of order n in the case where e and f are real intervals it was more recently pointed out that if they have different lengths it is of interest to generalize the foregoing problem to the set rmn with unequal numerator degree m and denominator degree n the object of the paper is to investigate the generalized zolotarev problem in the complex plane a method is proposed to construct the optimal rational function which is then applied to the particular example of two line segments e on the real axis f parallel to the imaginary axis numerical experiments show that the improvement on the classical solution may be amazingly great explicit expressions are also provided for special values of data

solution of nonlinear time dependent pdes through componentwise approximation of matrix functions exponential propagation iterative epi methods provide an efficient approach to the solution of large stiff systems of odes compared to standard integrators however the bulk of the computational effort in these methods is due to products of matrix functions and vectors which can become very costly at high resolution due to an increase in the number of krylov projection steps needed to maintain accuracy in this paper it is proposed to modify epi methods by using krylov subspace spectral kss methods instead of standard krylov projection methods to compute products of matrix functions and vectors numerical experiments demonstrate that this modification causes the number of krylov projection steps to become bounded independently of the grid size thus dramatically improving efficiency and scalability as a result for each test problem featured as the total number of grid points increases the growth in computation time is just below linear while other methods achieved this only on selected test problems or not at all

an adaptive scheme for a class of interpolatory model reduction methods for frequency response problems frequency sweep problems arise in many structural dynamic acoustic and structural acoustic applications in each case they incur the evaluation of a frequency response function for a typically large number of frequencies because each function evaluation requires the solution of an often large scale system of equations frequency sweep problems are computationally intensive interpolatory model order reduction is a powerful tool for reducing their cost however the performance of this tool depends on the location and number of the interpolation frequency points it also depends on the number of consecutive frequency derivatives of the response function that are matched at each frequency point so far these two choices have been made in the literature in a heuristic manner in contrast this paper proposes an automatic adaptive strategy based on monitoring the euclidean norm of the relative residual associated with the function to be evaluated over the frequency band of interest more specifically the number of interpolation points and the number of matched frequency derivatives are adaptively increased until the global euclidean norm of the relative residual is reduced below a user specified tolerance the robustness accuracy and computational efficiency of this adaptive strategy are highlighted with the solution of several frequency sweep problems associated with large scale structural dynamic acoustic and structural acoustic finite element models copyright 2012 john wiley sons ltd

slow switching in a population of delayed pulse coupled oscillators we show that peculiar collective dynamics called slow switching arises in a population of leaky integrate and fire oscillators with delayed all to all pulse couplings by considering the stability of cluster states and symmetry possessed by our model we argue that saddle connections between a pair of the two cluster states are formed under general conditions slow switching appears as a result of the system s approach to the saddle connections it is also argued that such saddle connections are easy to arise near the bifurcation point where the state of perfect synchrony loses stability we develop an asymptotic theory to reduce the model into a simpler form with which an analytical study of the cluster states becomes possible

linearization of generalized interval exchange maps a standard interval exchange map is a one to one map of the interval that is locally a translation except at nitely many singularities we dene for such maps in terms of the rauzy veech continuous fraction algorithm a diophantine arithmetical condition called restricted roth type which is almost surely satised in parameter space let t0 be a standard interval exchange map of restricted roth type and let r be an integer 2 we prove that amongst c r 3 deformations of t0 that are c r 3 tangent to t0 at the singularities those that are conjugated to t0 by a c r dieomorphism close to the identity form a c 1 submanifold of codimension g 1 2r 1 s here g is the genus and s is the number of marked points of the translation surface obtained by suspension of t0 both g and s can be computed from the combinatorics of t0

dimension reduction of large scale second order dynamical systems via a second order arnoldi method a structure preserving dimension reduction algorithm for large scale second order dynamical systems is presented it is a projection method based on a second order krylov subspace a second order arnoldi soar method is used to generate an orthonormal basis of the projection subspace the reduced system not only preserves the second order structure but also has the same order of approximation as the standard arnoldi based krylov subspace method via linearization the superior numerical properties of the soar based method are demonstrated by examples from structural dynamics and microelectromechanical systems

robust rigidity for circle diffeomorphisms with singularities we prove that under certain regularity conditions imposed on the renormalizations of two circle diffeomorphisms with singularities their c 1 smooth equivalence follows from exponential convergence of those renormalizations as an easy corollary any two analytical critical circle maps with the same order of critical points and the same irrational rotation number are c 1 smoothly conjugate