scattering by a hybrid asymptotic finite element method abstract we describe a hybrid asymptotic finite element method with application to the solution of the helmholtz equation for large wavenumber thus this method is suitable to compute the acoustic field radiated or scattered by acoustically large objects the method is based on patching a short wavelength asymptotic expansion of the outer field to an fem interpolation of the inner field the patching takes place on an artificial boundary continuity of the field and its normal derivative across the artificial boundary is enforced weakly in a variational setting the variational principle is described as is an element like implementation strategy two example applications are included diffraction by a truncated wedge and radiation by a source on a wave bearing surface

diffraction from simple shapes by a hybrid asymptotic finite element method the application of a hybrid asymptotic finite element method to the problem of scattering from two dimensional submerged objects is considered the hybrid method is based on patching a short wavelength asymptotic expansion of the scattered field to a finite element interpolation of the near field in patching the diffracted field shape functions with unknown amplitude are forced to agree smoothly with the solution in the near field along a curve at a prescribed distance from the diffraction points a new hybrid finite element on this artificial boundary represents the effect of the outer domain on the solution within this new boundary this allows the replacement of the original boundary value problem with an asymptotically equivalent boundary value problem the domain of which is small and efficiently discretized the method is applied to diffraction by a blunted wedge which in this context represents a degenerate prism the hybrid scattering solution shall be compared to an analytic field representat

dtn based mixed dimensional coupling using a boundary stress recovery technique abstract a new computational approach for the mixed dimensional modeling of time harmonic waves in elastic structures is proposed a two dimensional 2d structure is considered which includes a part that is assumed to behave in a one dimensional 1d way the 2d and 1d structural regions are discretized by using 2d and 1d finite element fe formulations the hybrid model if designed properly is much more efficient than the standard 2d model taken for the entire problem one important issue related to such hybrid 2d 1d models is the way the 2d 1d coupling is done and the coupling error generated here three closely related coupling methods are considered they are all based on the dirichlet to neumann dtn map associated with the 1d problem on the interface the first coupling method is the dtn method in its usual form where the dtn map is calculated numerically and where the 1d and 2d problems are solved separately the second coupling method is the one devised by carka mear and landis cml which is equivalent to the former one but in which the 2d and 1d interface solutions are solved for simultaneously in the third coupling method the dtn map is enforced on the interface iteratively direct application of these three methods results in low accuracy as a recent study shows therefore these methods are used here in conjunction with a boundary stress recovery bsr technique originally proposed by hughes which provides the same order of accuracy for the stress as for the primary variable the performance of the three methods is demonstrated and they are compared via numerical examples conclusions are drawn on their relative merit

isogeometric collocation on planar multi patch domains abstract we present an isogeometric framework based on collocation to construct a c 2 smooth approximation of the solution of the poisson s equation over planar bilinearly parameterized multi patch domains the construction of the used globally c 2 smooth discretization space for the partial differential equation is simple and works uniformly for all possible multi patch configurations the basis of the c 2 smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches edges and vertices of the multi patch domain for the selection of the collocation points which is important for the stability and convergence of the collocation problem two different choices are numerically investigated the first approach employs the tensor product greville abscissae as collocation points and shows for the multi patch case the same convergence behavior as for the one patch case 1 which is suboptimal in particular for odd spline degree the second approach generalizes the concept of superconvergent points from the one patch case cf 2 3 4 to the multi patch case again these points possess better convergence properties than greville abscissae in case of odd spline degree

phase retrieval via wirtinger flow theory and algorithms we study the problem of recovering the phase from magnitude measurements specifically we wish to reconstruct a complex valued signal boldsymbol x in mathbb c n about which we have phaseless samples of the form y r left langle boldsymbol a r boldsymbol x rangle right 2 r 1 ldots m knowledge of the phase of these samples would yield a linear system this paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm in a nutshell this algorithm starts with a careful initialization obtained by means of a spectral method and then refines this initial estimate by iteratively applying novel update rules which have low computational complexity much like in a gradient descent scheme the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements indeed the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources in theory a variation on this scheme leads to a near linear time algorithm for a physically realizable model based on coded diffraction patterns we illustrate the effectiveness of our methods with various experiments on image data underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval

phaseless near field far field transformation utilizing combinations of probe signals phaseless near field far field transformations based on nonconvex optimization suffer from local stationary points in order to avoid the corresponding suboptimal solutions the incorporation of combinations of probe signals into the cost functional is proposed and investigated the nonconvex optimization based on the wirtinger calculus relies on forward and adjoint radiation operator evaluations via the fast irregular antenna field transformation algorithm fiafta and is performed by a memory limited quasi newton method this allows the solution of large scale problems with utilization of the full flexibility of the fiafta in terms of measurement probe correction and arbitrary sample locations results for synthetic and measured near field data reveal significant improvements when the knowledge of probe signal combinations is utilized especially in applications with a large number of unknowns

on the potentialities of 3d 1d coupled models in hemodynamics simulations this work comprises a step towards the quantitative and qualitative analysis of coupled local and global hemodynamics phenomena in the arterial system the aim of this work is to present some numerical examples to put in evidence the importance of the use of 3d 1d coupled models in hemodynamics problems when carrying out simulations of rather complex situations accordingly some cases for which classical modeling cannot be applied are identified and solved the results obtained here allow us to assess some interrelations between local pointwise quantities defined at the level of the 3d model and global mean quantities defined at the level of the 1d model

outflow boundary conditions for three dimensional finite element modeling of blood flow and pressure in arteries flow and pressure waves emanate from the heart and travel through the major arteries where they are damped dispersed and reflected due to changes in vessel caliber tissue properties and branch points as a consequence solutions to the governing equations of blood flow in the large arteries are highly dependent on the outflow boundary conditions imposed to represent the vascular bed downstream of the modeled domain the most common outflow boundary conditions for three dimensional simulations of blood flow are prescribed constant pressure or traction and prescribed velocity profiles however in many simulations the flow distribution and pressure field in the modeled domain are unknown and cannot be prescribed at the outflow boundaries an alternative approach is to couple the solution at the outflow boundaries of the modeled domain with lumped parameter or one dimensional models of the downstream domain we previously described a new approach to prescribe outflow boundary conditions for simulations of blood flow based on the dirichlet to neumann and variational multiscale methods this approach termed the coupled multidomain method was successfully applied to solve the non linear one dimensional equations of blood flow with a variety of models of the downstream domain this paper describes the extension of this method to three dimensional finite element modeling of blood flow and pressure in the major arteries outflow boundary conditions are derived for any downstream domain where an explicit relationship of pressure as a function of flow rate or velocities can be obtained at the coupling interface we developed this method in the context of a stabilized semi discrete finite element method flow rate and pressure distributions are shown for different boundary conditions to illustrate the dramatic influence of alternative boundary conditions on these quantities

radiation pattern evaluation from near field intensities on planes this paper describes analyzes and implements an approach to far field determination from phaseless measurements over two planar surfaces a proper formulation of the problem is considered as a quadratic inverse one whose data is the square amplitude of the near field a solution is introduced as the global minimum of an appropriate functional next to perform such a minimization procedure a finite dimensional representation of the field radiated by sources whose plane wave spectrum becomes negligible outside a fixed angular domain is used a detailed investigation of the properties of the mapping connecting the unknowns to the data makes it possible to analyze how to escape from the local minima possibly met in the course of the minimization procedure to this end the crucial role of the availability of phaseless data over two different sarfa ces and of appropriate weights in the functional definition is emphasized and a reliable iterative procedure converging on the solution regardless of the starting point is thus obtained this property is confirmed by experimental results concerning near zone data from a shaped reflector at 9 ghz it can be readily appreciated that when only the field intensity is detected the complexity and the cost of the equipment required for near field techniques in antenna testing and diagnostics can be reduced to a very large extent

a finite element solver and energy stable coupling for 3d and 1d fluid models the paper develops a solver based on conforming finite element method fem for a 3d 1d coupled incompressible flow problem new coupling conditions are introduced to ensure a suitable bound for the cumulative energy of the model we study the stability and the accuracy of the discretization method and the performance of some state of the art linear algebraic solvers for such flow configurations motivated by the simulation of the flow over inferior vena cava ivc filter we consider the coupling of a 1d fluid model and a 3d fluid model posed in a domain with anisotropic inclusions the relevance of our approach to realistic cardiovascular simulations is demonstrated by computing blood flows over a model ivc filter

on data increasing in phase retrieval via quadratic inversion flattening manifold and local minima in this article we address the question of traps in phase retrieval via quadratic inversion first we clarify the genesis of local minima from a geometrical perspective by showing how the shape of the manifold of data and the position of the data point are related to the presence of local minima in the objective functional later we recall a mathematical condition for the lack of traps afterward we illustrate how the ratio between the dimension of data space m and the number of real unknowns n r impacts on the manifold of data and consequently on the presence of local minima in particular we show that if the dimension of data space is high enough the manifold of data is sufficiently large and no traps appear in the functional in the last part of this article with reference to a particular unknown sequence we establish the minimum value of the ratio m n r such that no traps should appear in the functional in this condition phase retrieval via quadratic inversion exhibits a global convergence behavior consequently the actual solution of the problem can be reached regardless of the initial guess which can be chosen completely at random

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

optimization based amp for phase retrieval the impact of initialization and ell_ 2 regularization we consider an ell 2 regularized non convex optimization problem for recovering signals from their noisy phaseless observations we design and study the performance of a message passing algorithm that aims to solve this optimization problem we consider the asymptotic setting m n rightarrow infty m n rightarrow delta and obtain sharp performance bounds where m is the number of measurements and n is the signal dimension we show that for complex signals the algorithm can perform accurate recovery with only m 64 pi 2 4 n approx 2 5n measurements also we provide a sharp analysis on the sensitivity of the algorithm to noise we highlight the following facts about our message passing algorithm 1 adding ell 2 regularization to the non convex loss function can be beneficial and 2 spectral initialization has a marginal impact on the performance of the algorithm the sharp analyses in this paper not only enable us to compare the performance of our method with other phase recovery schemes but also shed light on designing better iterative algorithms for other non convex optimization problems

discrete dirichlet to neumann maps for unbounded domains abstract it is shown how to construct a discrete counterpart of the dirichlet to neumann dtn map for use on an artificial boundary b introduced in exterior boundary value problems following the idea of deakin and dryden this discrete map provides an approximate non reflecting or absorbing boundary condition for use in formulating a problem in the finite computational domain bounded by b different discrete maps are constructed for use with finite difference and finite element methods in the solution of some simple problems shows that use of the discrete dirichlet to neumann ddtn map yields nearly the same accuracy as that of using the continuous dtn map

a unified variational approach for coupling 3d 1d models and its blood flow applications in this work we propose an extended variational formulation in order to handle the problem of the flow of an incompressible fluid in compliant vessels with discontinuous fields this approach makes possible to tackle in a consistent variational manner problems like the coupling between models of different dimensionality the main ideas of the work are presented for a simple heat transfer problem and then are used in the context of modelling blood flow in large vessels a computational implementation of the proposed formulation is also shown and discussed by presenting examples of practical applications

phase retrieval using alternating minimization phase retrieval problems involve solving linear equations but with missing sign or phase for complex numbers information more than four decades after it was first proposed the seminal error reduction algorithm of gerchberg and saxton and fienup is still the popular choice for solving many variants of this problem the algorithm is based on alternating minimization i e it alternates between estimating the missing phase information and the candidate solution despite its wide usage in practice no global convergence guarantees for this algorithm are known in this paper we show that a resampling variant of this approach converges geometrically to the solution of one such problem finding a vector bf x from bf y bf a where bf y vert bf a t bf x vert and vert bf z vert denotes a vector of element wise magnitudes of bf z under the assumption that bf a is gaussian empirically we demonstrate that alternating minimization performs similar to recently proposed convex techniques for this problem which are based on lifting to a convex matrix problem in sample complexity and robustness to noise however it is much more efficient and can scale to large problems analytically for a resampling version of alternating minimization we show geometric convergence to the solution and sample complexity that is off by log factors from obvious lower bounds we also establish close to optimal scaling for the case when the unknown vector is sparse our work represents the first theoretical guarantee for alternating minimization albeit with resampling for any variant of phase retrieval problems in the non convex setting

optimal modal reduction of dynamic subsystems extensions and improvements the problem of optimal reduction of a linear dynamic subsystem is revisited the subsystem may represent for example a complex but minor part of a large elastic structure the goal is to drastically reduce the number of degrees of freedom of a subsystem attached through an interface to a main system in such a way as to affect the dynamic behavior of the main system in the least possible way in a recent publication the optimal modal reduction omr algorithm was developed to this end this algorithm seeks a reduction of the subsystem that will have minimal effect in the l2 norm on the dirichlet to neumann dtn map on the interface here this algorithm is extended and improved in a number of ways first a family of alternative formulations are derived for the dtn map which lead to alternative omr algorithms one of them called omr j 2 is shown to yield better results than the original formulation second the omr algorithm which was originally developed for undamped subsystems is extended to subsystems undergoing rayleigh damping in addition an enhanced derivation of the original omr algorithm is presented and the good pointwise performance of omr is explained by relating to minimization with respect to the h1 norm the extensions mentioned above are discussed theoretically as well as demonstrated via numerical examples experiments and discussion include comparison of the omr algorithm to simple coarsening of the subsystem discretization in all cases central finite difference discretization in space and explicit time stepping are employed to solve the scalar wave equation copyright 2010 john wiley sons ltd

the dirichlet to neumann map for two dimensional crack problems a coupled analytic finite element method is presented for two dimensional crack problems two classes of problems are studied the first considers problems where non linear constitutive processes occur in a region near the crack tip and the remotely applied loading can be characterized by the linear elastic k field and perhaps the t stress in this case the finite element method is applied in a circular region around the crack tip where non linear constitutive response is occurring and stiffness contributions associated with a numerically implemented dirichlet to neumann map are imposed on the circular boundary to account for the large surrounding elastic domain and the remote applied loading the second class of problems considers entirely linear elastic domains with irregular external boundaries and or complex applied loadings here the discrete dirichlet to neumann map is used to represent a circular region surrounding the crack tip and finite elements are used for the external region in this case the mixed mode stress intensity factors and the t stress are retrieved from the map

the nitsche method applied to a class of mixed dimensional coupling problems abstract a computational approach for the mixed dimensional modeling of time harmonic waves in elastic structures is proposed a two dimensional 2d structure is considered that includes a part which is assumed to behave in a one dimensional 1d way the 2d and 1d structural regions are discretized using 2d and 1d finite element formulations the coupling of the 2d and 1d regions is performed weakly by using the nitsche method the advantage of using the nitsche method to impose boundary and interface conditions has been demonstrated by various authors here this advantage is shown in the context of mixed dimensional coupling the computational aspects of the method are discussed and it is compared to the slightly simpler penalty method both theoretically and numerically numerical examples are presented in various configurations where the 1d model is either confined laterally or laterally free and where the 2d part is either simply connected or doubly connected the performance is investigated for various wave numbers and various extents of the 1d region varying material properties and distributed loads in the 1d and 2d parts are also considered it is concluded that the nitsche method is a viable technique for mixed dimensional coupling of elliptic problems of this type

generalized mathematical homogenization from theory to practice a generalized mathematical homogenization gmh theory for coupling atomistic and continuum descriptions at finite temperature has been extended to account for many body potentials gmh gives rise to the constitutive law free coarse scale equations where coupled continuum thermo mechanical equations are directly derived from molecular dynamics equations an integrated computational approach which seamlessly integrates abaqus for coarse scale finite element method computations with a molecular dynamics code has been developed the accuracy and efficiency of the method has been studied for crystal silicon structures including failure predictions of nanowires

the sources reconstruction method for amplitude only field measurements an extension of the sources reconstruction method srm for antenna diagnostics using amplitude only field measurements is presented while previous works limit the application to canonical domain cases the combination of the srm capabilities for handling with arbitrary geometry domains jointly with phase retrieval techniques allows antenna diagnostics extension to complex geometry antenna and field acquisition domains the consideration of the radiation inverse problem with a general integral equation formulation using arbitrary geometry field and currents domains and field phaseless information supposes a challenging ill posed problem that is solved using iterative minimization techniques for non linear problems different examples are presented from simple to general antenna diagnostics cases

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

excitation retrieval of microwave linear arrays from phaseless far field data a methodology to recover the excitations of microwave linear arrays from the measurements of far field magnitude only is proposed the approach combines tools from convex optimization to solve the phase retrieval problem and a simple measurement procedure to mitigate the non uniqueness of the solution numerical simulations in various representative and realistic configurations of noise and measurement sampling are presented and discussed they show that it is possible to perfectly retrieve the array excitations from only the knowledge of far field magnitude by simply solving a convex optimization problem in addition the proposed approach that only calls for readily available routines is stable with respect to noise since the reconstruction performances degrades gracefully as the signal to noise ratio decreases