determining conductivity by boundary measurements ii interior results abstract in a recent paper the authors showed that an unknown real analytic conductivity gamma may be determined from static boundary measurements in this document they extend this analysis by demonstrating that a similar result holds for piecewise real analytic conductivities in addition for the special case of a layered structure it is shown that a three times continuously differentiable conductivity is identifiable by boundary measurements originator supplied keywords include inversion convergence algorithms estimates coefficients

analysis of mixed finite element methods on locally refined grids we consider the mixed finite element method for locally refined triangulations a local projection operator hat pi h is defined to satisfy the commutation property that is required in the general theory of mixed methods our results can be applied to every known space of arbitrary order over rectangles or triangles optimal order error estimates and superconvergence for the flux along the gauss lines are established

effects of measurement precision and finite numbers of electrodes on linear impedance imaging algorithms impedance imaging systems reconstruct approximations to the conductivity inside a body from electrical measurements made on the boundary the linearized problem was studied by calderon who showed that if the conductivity differs little from a constant then it can be reconstructed approximately in his work calderon assumed that arbitrary current densities can be applied anywhere on the body s surface and the resulting voltages perfectly measured in practice however imaging systems apply currents through a finite number of electrodes on the surface and measure the resulting voltages with a limited precision in this paper linear algorithms are described for approximating the electrical conductivity in a disk from finitely many limited precision measurements the effects of finite precision and electrode number on a two dimensional version of calderon s method are studied it is shown that as the measurement errors go to zero and the electrodes proliferate in a space filling manner the finite electro

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

superconvergence and extrapolation for mixed finite element methods on rectangular domains asymptotic expansions for the rt raviart thomas mixed finite element approximation by the lowest order rectangular element associated with a second order elliptic equation on a rectangular domain are derived superconvergence for the vector field along the gauss lines is obtained as a result of the expansion a procedure of postprocessed extrapolation is presented for the scalar field as well as procedures of pure richardson extrapolation for both the vector and the scalar fields

mixed finite element methods on nonmatching multiblock grids we consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids a mortar finite element space is introduced on the nonmatching interfaces we approximate in this mortar space the trace of the solution and we impose weakly a continuity of flux condition a standard mixed finite element method is used within the blocks optimal order convergence is shown for both the solution and its flux moreover at certain discrete points superconvergence is obtained for the solution and also for the flux in special cases computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory

a uniqueness theorem for an inverse boundary value problem in electrical prospection we show that a near constant conductivity of a two dimensional body can be uniquely determined by steady state direct current measurements at the boundary mathematically we show that the coefficient in the operator is uniquely determined by its dirichlet integrals

an approach for preference ranking of alternatives abstract an approach for determining the ranking of a set of alternatives is developed it is based on concepts of conditional probability the approach evaluates the set of alternatives and arrives at a preference ranking of either the entire set or a subset of available alternatives results of preliminary tests involving generated problems are favorable

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

on the construction of a potential from cauchy data abstract we investigate the problem of recovering a potential q y in the differential equation u q y u 0 x y 0 1 0 1 u 0 y u 1 y u x 0 0 u x 1 f x u y x 1 g x the method of separation of variables reduces the recovery of q y to a nonstandard inverse sturm liouville problem an asymptotic formula is developed that suggests that under appropriate conditions on the cauchy pair f g q y is uniquely determined up to the mean moreover the recovery of q y is comparable to finding a function from its polynomial moments a reconstruction scheme is suggested and numerical examples are considered

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

bayesian decision process for cost efficient dynamic ranking via crowdsourcing rank aggregation based on pairwise comparisons over a set of items has a wide range of applications although considerable research has been devoted to the development of rank aggregation algorithms one basic question is how to efficiently collect a large amount of high quality pairwise comparisons for the ranking purpose because of the advent of many crowdsourcing services a crowd of workers are often hired to conduct pairwise comparisons with a small monetary reward for each pair they compare since different workers have different levels of reliability and different pairs have different levels of ambiguity it is desirable to wisely allocate the limited budget for comparisons among the pairs of items and workers so that the global ranking can be accurately inferred from the comparison results to this end we model the active sampling problem in crowdsourced ranking as a bayesian markov decision process which dynamically selects item pairs and workers to improve the ranking accuracy under a budget constraint we further develop a computationally efficient sampling policy based on knowledge gradient as well as a moment matching technique for posterior approximation experimental evaluations on both synthetic and real data show that the proposed policy achieves high ranking accuracy with a lower labeling cost

identification of small inhomogeneities of extreme conductivity by boundary measurements a theorem on continuous dependence we consider an electrostatic problem for a conductor consisting of finitely many small inhomogeneities of extreme conductivity embedded in a spatially varying reference medium firstly we establish an asymptotic formula for the voltage potential in terms of the reference voltage potential the location of the inhomogeneities and their geometry secondly we use this representation formula to prove a lipschitz continuous dependence estimate for the corresponding inverse problem this estimate bounds the difference in the location and in certain geometric properties of two sets of inhomogeneities by the difference in the boundary voltage potentials corresponding to a fixed current distribution

matrix estimation by universal singular value thresholding consider the problem of estimating the entries of a large matrix when the observed entries are noisy versions of a small random fraction of the original entries this problem has received widespread attention in recent times especially after the pioneering works of emmanuel candes and collaborators this paper introduces a simple estimation procedure called universal singular value thresholding usvt that works for any matrix that has a little bit of structure surprisingly this simple estimator achieves the minimax error rate up to a constant factor the method is applied to solve problems related to low rank matrix estimation blockmodels distance matrix completion latent space models positive definite matrix completion graphon estimation and generalized bradley terry models for pairwise comparison

feeling the bern adaptive estimators for bernoulli probabilities of pairwise comparisons we study methods for aggregating pairwise comparison data among a collection of n items with the goal of estimating the outcome probabilities for future comparisons working within a flexible model that only imposes a form of strong stochastic transitivity we introduce an adaptivity index which compares the risk of our estimator to that of an oracle over appropriate sub models where the oracle knows the specific sub model in the ground truth in addition to measuring the usual worst case risk of an estimator this adaptivity index also captures the extent to which the estimator adapts to instance specific difficulty relative to an oracle estimator first we propose a three step estimator termed count randomize least squares and show that it has adaptivity index upper bounded by sqrt n up to logarithmic factors we then show that conditional on the planted clique hypothesis no computationally efficient estimator can achieve an adaptivity index smaller than sqrt n second we show that a regularized least squares estimator can achieve a poly logarithmic adaptivity index thereby demonstrating a sqrt n gap between optimal and computationally achievable adaptivity finally we prove that the standard least squares estimator which is known to be optimally adaptive in several closely related problems fails to adapt in the context of estimating pairwise probabilities

a backprojection algorithm for electrical impedance imaging the authors study a two dimensional reconstruction algorithm due to d c barber and b h brown applied to a linearized electrostatic inverse problem first the authors demonstrate how this algorithm fits within the framework of inverses of generalized radon transforms studied by g beylkin second an iterative improvement of the barber brown algorithm is constructed based on the conjugate residual method several numerical results obtained with this iterative algorithm are presented

estimation from pairwise comparisons sharp minimax bounds with topology dependence data in the form of pairwise comparisons arises in many domains including preference elicitation sporting competitions and peer grading among others we consider parametric ordinal models for such pairwise comparison data involving a latent vector w e rd that represents the qualities of the d items being compared this class of models includes the two most widely used parametric models the bradley terry luce btl and the thurstone models working within a standard minimax framework we provide tight upper and lower bounds on the optimal error in estimating the quality score vector w under this class of models the bounds depend on the topology of the comparison graph induced by the subset of pairs being compared via the spectrum of the laplacian of the comparison graph thus in settings where the subset of pairs may be chosen our results provide principled guidelines for making this choice finally we compare these error rates to those under cardinal measurement models and show that the error rates in the ordinal and cardinal settings have identical scalings apart from constant pre factors

on the accuracy of the finite volume element method for diffusion equations on composite grids this paper develops the finite volume element fve method which is similar to the so called control volume finite element method but tailored to composite grid applications fve is described for the general two dimensional diffusion equation in divergence form o h 3 2 discretization error estimates are then developed for the case of triangular elements these results are improved to o h 2 using a special modification involving rectilinear elements at the grid interface

a non mortar mixed finite element method for elliptic problems on non matching multiblock grids abstract we consider the approximation of second order elliptic equations on domains that can be described as a union of sub domains or blocks we assume that a grid is defined on each block independently so that the resulting grid over the entire domain need not be conforming i e match across the block boundaries several techniques have been developed to approximate elliptic equations on multiblock grids that utilize a mortar finite element space defined on the block boundary interface itself we define a mixed finite element method that does not use such a mortar space the method has an advantage in the case where adaptive local refinement techniques will be used in that there is no mortar grid to refine as is typical of mixed methods our method is locally conservative element by element it is also globally conservative across the block boundaries theoretical results show that the approximate solution converges at the optimal rate to the true solution we present computational results to illustrate and confirm the theory

rank centrality ranking from pairwise comparisons the question of aggregating pairwise comparisons to obtain a global ranking over a collection of objects has been of interest for a very long time be it ranking of online gamers e g msr s trueskill system and chess players aggregating social opinions or deciding which product to sell based on transactions in most settings in addition to obtaining a ranking finding scores for each object e g player s rating is of interest for understanding the intensity of the preferences in this paper we propose rank centrality an iterative rank aggregation algorithm for discovering scores for objects or items from pairwise comparisons the algorithm has a natural random walk interpretation over the graph of objects with an edge present between a pair of objects if they are compared the score which we call rank centrality of an object turns out to be its stationary probability under this random walk to study the efficacy of the algorithm we consider the popular bradley terry luce btl model equ

asymptotic expansions and l error estimates for mixed finite element methods for second order elliptic problems asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and richardson extrapolation can be applied to increase the accuracy of the approximations a new procedure which is called the error corrected method is presented as a further application of the asymptotic error expansion for the first order bdm approximation of the scalar field the key point in deriving the asymptotic expansions for the error is an establishment ofl 1 error estimates for mixed finite element approximations for the regularized green s functions as another application of thel 1 error estimates for the regularized green s functions we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems

superconvergence of new mixed finite element spaces in this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in etna vol 37 pp 189 201 2010 among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids this space has the smallest unknowns however the scalar variable is only suboptimal in general thus we have employed a post processing technique for the scalar variable as a byproduct we have obtained a superconvergence on a rectangular grid the superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory but that of a scalar variable requires a new technique especially for k 1 numerical experiments are provided to support the theory

superconvergence of the velocity along the gauss lines in mixed finite element methods superconvergence for the velocity along the gauss lines is derived for the mixed finite element method for the second order elliptic equation in rectangular domains these estimates are based on the rectangular finite elements of raviart thomas type