an economical finite element approximation of generalized newtonian flows abstract we consider an economical bilinear rectangular mixed finite element scheme on regular mesh for generalized newtonian flows where the viscosity obeys a carreau type law for a pseudo plastic the key issue in the scheme is that the two components of the velocity and the pressure are defined on different meshes optimal error bounds for both the velocity and pressure are obtained by proving a discrete babuska brezzi inf sup condition on the regular quadrangulation finally we perform some numerical experiments including an example in a unit square with exact solutions a backward facing step and a four to one abrupt contraction generalized newtonian flows numerical experiments confirm our error bounds

convergence of em image reconstruction algorithms with gibbs smoothing p j green has defined an osl one step late algorithm that retains the e step of the em algorithm for image reconstruction in emission tomography but provides an approximate solution to the m step further modifications of the osl algorithm guarantee convergence to the unique maximum of the log posterior function convergence is proved under a specific set of sufficient conditions several of these conditions concern the potential function of the gibb s prior and a number of candidate potential functions are identified generalization of the osl algorithm to transmission tomography is also considered

hyperbolic parabolic singular perturbation for nondegenerate kirchhoff equations with critical weak dissipation we consider the hyperbolic parabolic singular perturbation problem for a nondegenerate quasilinear equation of kirchhoff type with weak dissipation this means that the dissipative term is multiplied by a coefficient b t which tends to 0 as t the case where b t 1 t p with p 1 but they remain true in the critical case p 1

hyperbolic parabolic singular perturbation for quasilinear equations of kirchhoff type abstract we consider a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtain parameter dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation for this purpose we show time decay estimates for hyperbolic parabolic singular perturbation problem for linear equations with a time dependent coefficient

finite element error analysis of a quasi newtonian flow obeying the carreau or power law we consider the finite element approximation of a quasi newtonian flow where the viscosity obeys the carreau or power law for sufficiently regular solutions we prove energy type error bounds for the velocity and pressure these bounds improve on existing results in the literature

review and complements on mixed hybrid finite element methods for fluid flows mixed and hybrid finite element methods for the resolution of a wide range of linear and nonlinear boundary value problems linear elasticity stokes problem navier stokes equations boussinesq equations etc have known a great development in the last few years these methods allow simultaneous computation of the original variable and its gradient both of them being equally accurate moreover they have local conservation properties conservation of the mass and the momentum as in the finite volume methods the purpose of this paper is to give a review on some mixed finite elements developed recently for the resolution of stokes and navier stokes equations and the linear elasticity problem further developments for a quasi newtonian flow obeying the power law are presented

a row action alternative to the em algorithm for maximizing likelihood in emission tomography the maximum likelihood ml approach to estimating the radioactive distribution in the body cross section has become very popular among researchers in emission computed tomography ect since it has been shown to provide very good images compared to those produced with the conventional filtered backprojection fbp algorithm the expectation maximization em algorithm is an often used iterative approach for maximizing the poisson likelihood in ect because of its attractive theoretical and practical properties its major disadvantage is that due to its slow rate of convergence a large amount of computation is often required to achieve an acceptable image here the authors present a row action maximum likelihood algorithm ramla as an alternative to the em algorithm for maximizing the poisson likelihood in ect the authors deduce the convergence properties of this algorithm and demonstrate by way of computer simulations that the early iterates of ramla increase the poisson likelihood in ect at an order of magnitude faster that the standard em algorithm specifically the authors show that from the point of view of measuring total radionuclide uptake in simulated brain phantoms iterations 1 2 3 and 4 of ramla perform at least as well as iterations 45 60 70 and 80 respectively of em moreover the authors show that iterations 1 2 3 and 4 of ramla achieve comparable likelihood values as iterations 45 60 70 and 80 respectively of em the authors also present a modified version of a recent fast ordered subsets em os em algorithm and show that ramla is a special case of this modified os em furthermore the authors show that their modification converges to a ml solution whereas the standard os em does not

analysis of an adaptive uzawa finite element method for the nonlinear stokes problem we design and study an adaptive algorithm for the numerical solution of the stationary nonlinear stokes problem the algorithm can be interpreted as a disturbed steepest descent method which generalizes uzawa s method to the nonlinear case the outer iteration for the pressure is a descent method with fixed step size the inner iteration for the velocity consists of an approximate solution of a nonlinear laplace equation which is realized with adaptive linear finite elements the descent direction is motivated by the quasi norm which naturally arises as distance between velocities we establish the convergence of the algorithm within the framework of descent direction methods

inf sup conditions for twofold saddle point problems necessary and sufficient conditions for existence and uniqueness of solutions are developed for twofold saddle point problems which arise in mixed formulations of problems in continuum mechanics this work extends the classical saddle point theory to accommodate nonlinear constitutive relations and the twofold saddle structure application to problems in incompressible fluid mechanics employing symmetric tensor finite elements for the stress approximation is presented

mildly degenerate kirchhoff equations with weak dissipation global existence and time decay abstract we consider the hyperbolic parabolic singular perturbation problem for a degenerate quasilinear kirchhoff equation with weak dissipation this means that the coefficient of the dissipative term tends to zero when t we prove that the hyperbolic problem has a unique global solution for suitable values of the parameters we also prove that the solution decays to zero as t with the same rate of the solution of the limit problem of parabolic type

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

a posteriori error estimators for nonconforming approximation of some quasi newtonian flows this paper deals with a posteriori error estimators for nonconforming approximations of quasi newtonian flows we consider the crouzeix raviart piecewise linear approximations of scalar elliptic problems and define an error estimator when omega is a simply connected domain the error is dominated by the estimator this estimator can be generalized to higher order elements we define a posteriori error estimators for the fortin soulie piecewise quadratic approximations of quasi newtonian flows and prove that the error is dominated by the estimator this estimator can be computed locally in terms of the approximate solution and is therefore suitable for adaptive refinement

hyperbolic parabolic singular perturbation for mildly degenerate kirchhoff equations time decay estimates we consider degenerate kirchhoff equations with a small parameter e in front of the second order time derivative it is well known that these equations admit global solutions when e is small enough and that these solutions decay as t with the same rate of solutions of the limit problem of parabolic type in this paper we prove decay error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem these estimates show in the same time that the difference tends to zero both as e 0 and as t concerning the decay rates it turns out that the difference decays faster than the two terms separately as t proofs involve a nonlinear step where we separate fourier components with respect to the lowest frequency followed by a linear step where we exploit weighted versions of classical energies

global solutions for dissipative kirchhoff strings with m r rp p 1 abstract we investigate the evolution problem u u m a 1 2 u 2 h au 0 u 0 u 0 u 0 u 1 where h is a hilbert space a is a self adjoint non negative operator on h with domain d a 0 is a parameter and m r r p with p u 0 u 1 d a i 2 d a i 1 2 satisfy a suitable smallness assumption and the non degeneracy condition m a 1 2 u 0 2 h 0 where p 2 i and i 2 i 1 moreover we prove for this solution decay with a polynomial rate as t these results apply to degenerate hyperbolic pdes with non local non linearities

an improved penalty method for power law stokes problems for the numerical approximation of fluid flow phenomena it is often highly desirable to decouple the equations defining conservation of momentum and conservation of mass by using a penalty function method the current penalty function methods for power law stokes fluids converge at a sublinear rate with respect to the penalty parameter in this article we show theoretically and numerically that a linear penalty function approximation to a power law stokes problem yields a higher order accuracy over the known nonlinear penalty method theoretically finite element approximation of the linear penalty function method is shown to satisfy an improved order of approximation with respect to the penalty parameter the numerical experiments presented in the paper support the theoretical results and satisfy a linear order of approximation

finite element approximation of the p laplacian in this paper we consider the continuous piecewise linear finite el ement approximation of the following problem given p 1 oo and g find u such that v vu 2vu f iniicr2 u g on a the finite element approximation is defined over i2 a union of regular tri angles yielding a polygonal approximation to q for sufficiently regular so lutions u achievable for a subclass of data g and i2 we prove optimal error bounds for this approximation in the norm wlq q 1 q p for p 2 under the additional assumption that qh c 2 numerical results demonstrating these bounds are also presented

a theoretical study of some maximum likelihood algorithms for emission and transmission tomography this paper has the dual purpose of introducing some new algorithms for emission and transmission tomography and proving mathematically that these algorithms and related antecedent algorithms converge like the em algorithms for positron single photon and transmission tomography the algorithms provide maximum likelihood estimates of pixel concentration or linear attenuation parameters one particular innovation we discuss is a computationally practical scheme for modifying the em algorithms to include a bayesian prior the bayesian versions of the em algorithms are shown to have superior convergence properties in a vicinity of the maximum we anticipate that some of the other algorithms will also converge faster than the em algorithms

global solutions for dissipative kirchhoff strings with non lipschitz nonlinear term we investigate the evolution problem u u m a 1 2 u 2 au 0 u 0 u0 u 0 u1 where h is a hilbert space a is a self adjoint nonnegative operator on h with domain d a 0i s a parameter and m r is a nonnegative function such that m 0 0a ndm is nonnecessarily lipschitz continuous in a neighborhood of 0 we prove that this problem has a unique global solution for positive times provided that the initial data u0 u 1 d a d a 1 2 satisfy a suitable smallness assumption and the nondegeneracy condition m a 1 2 u0 2 0 moreover we study the decay of the solution as t these results apply to degenerate hyperbolic pdes with nonlocal nonlinearities

a hybrid high order method for a class of quasi newtonian stokes equations on general meshes abstract in this paper we introduce a hybrid high order hho discrete scheme for numerically solving a class of incompressible quasi newtonian stokes equations in r 2 the presented hho method depends on hybrid discrete velocity unknowns at cells and edges and pressure unknowns at cells benefiting from the hybridization of unknowns the computation cost can be reduced by the technique of static condensation and the solvability of the static condensation algebra system is proved furthermore we study the hho scheme by polynomials of arbitrary degrees k k 1 on the general meshes and geometries the unique solvability of the discrete scheme is proved additionally the optimal a priori error estimates for the velocity gradient and pressure approximations are obtained finally we provide several numerical results to verify the good performance of the proposed hho scheme and confirm the optimal approximation properties on a variety of meshes and geometries

developments with maximum likelihood x ray computed tomography an approach to the maximum likelihood estimation of attenuation coefficients in transmission tomography is presented as an extension of earlier theoretical work by k lange and r carson j comput assist tomography vol 8 p 306 16 1984 the reconstruction algorithm is based on the expectation maximization em algorithm several simplifying approximations are introduced which make the maximization step of the algorithm available computer simulations are presented using noise free and poisson randomized projections the images obtained with the em type method are compared to those reconstructed with the em method of lange and carson and with filtered backprojection preliminary results show that there are potential advantages in using the maximum likelihood approaches in situations where a high contrast object such as bone is embedded in low contrast soft tissue

monotonic algorithms for transmission tomography we present a framework for designing fast and monotonic algorithms for transmission tomography penalized likelihood image reconstruction the new algorithms are based on paraboloidal surrogate functions for the log likelihood due to the form of the log likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity unlike previous methods the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events such as scatter and random coincidences the gradient and the curvature of the likelihood terms are evaluated only once per iteration since the problem is simplified at each iteration the cpu time is less than that of current algorithms which directly minimize the objective yet the convergence rate is comparable the simplicity monotonicity and speed of the new algorithms are quite attractive the convergence rates of the algorithms are demonstrated using real and simulated pet transmission scans

analysis of some finite elements for the stokes problem we study some finite elements which are used in the approximation of the stokes problem so as to obtain error estimates of optimal order resume nous etudions deux elements finis utilises pour l approximation du probleme de stokes et obtenons des estimations d erreur d ordre optimal

finite element error estimates for non linear elliptic equations of monotone type in this paper we shall consider the application of the finite element method to a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient and the derivation of error estimates for the finite element approximations such problems arise in many practical situations for example in shock free airfoil design seepage through coarse grained porous media and in some glaciological problems by making use of certain properties of the nonlinear coefficients we shall demonstrate that the variational formulations associated with these boundary value problems are well posed we shall also prove that the abstract operators accompanying such problems satisfy certain continuity and monotonicity inequalities with the aid of these inequalities and some standard results from approximation theory we show how one may derive error estimates for the finite element approximations in the energy norm

accelerated penalized weighted least squares and maximum likelihood algorithms for reconstructing transmission images from pet transmission data we present penalized weighted least squares pwls and penalized maximum likelihood pml methods for reconstructing transmission images from positron emission tomography transmission data first we view the problem of minimizing the weighted least squares wls and maximum likelihood objective functions as a sequence of nonnegative least squares minimization problems this viewpoint follows from using certain quadratic functions as surrogate functions for the wls and maximum likelihood objective functions second we construct surrogate functions for a class of penalty functions that yield closed form expressions for the iterates of the pwls and pml algorithms due to the slow convergence of the pwls and pml algorithms accelerated versions of them are developed that are theoretically guaranteed to monotonically decrease their respective objective functions in experiments using real phantom data the pml images produced the most accurate attenuation correction factors on the other hand the pwls images produced images with the highest levels of contrast for low count data

a data assimilation approach for non newtonian blood flow simulations in 3d geometries abstract blood flow simulations can play an important role in medical training and diagnostic predictions associated to several pathologies of the cardiovascular system the main challenge at the present stage is to obtain reliable numerical simulations in the particular districts of the cardiovascular system that we are interested in here we propose a data assimilation procedure in the form of a non linear optimal control problem of dirichlet type to reconstruct the blood flow profile from known data available in certain parts of the computational domain this method will allow us to obtain the boundary conditions not fully determined by the physics of the model in order to recover more accurate simulations to solve the control problem we propose a discretize then optimize do approach based on a stabilized finite element method numerical simulations on 3d geometries are performed to validate this procedure in particular we consider some idealized geometries of interest and real geometries such as a saccular aneurysm and a bypass we assume blood as an homogeneous fluid with non newtonian inelastic shear thinning behavior the results show that even in the presence of noisy data accuracy can be improved using the optimal control approach

analyse numerique des ecoulements quasi newtoniens dont la viscosite obeit a la loi puissance ou la loi de carreau we prove abstract error estimates for the approximation of the velocity and the pressure by a mixed fem of quasi newtonian flows whose viscosity obeys the power law or the carreau law these estimates are optimal in some cases they can be applied to most finite elements used for the solution of stokes s problem on prouve des estimations d erreur abstraites pour l approximation de la vitesse et la pression par une mef mixtes d ecoulements quasi newtoniens dont la viscosite obeit a la loi puissance ou la loi de carreau ces estimations sont optimales dans certains cas elles peuvent etre appliquees a la plupart des elements finis utilises pour la resolution du probleme de stokes

finite element approximation of some degenerate monotone quasilinear elliptic systems in this paper we examine the continuous piecewise linear finite element approximation of the following system given bf f equiv f j and bf g equiv g j find bf u equiv u j j 1 to r with r 1 or 2 such that nabla cdot k z nabla bf u z nabla bf u z bf f z quad z in omega subset r 2 quad bf u partial omega bf g partial omega where nabla bf u equiv partial u j partial z i 1 leq i leq 2 1 leq j leq r and k is a given matrix on omega times r 2 times r we characterize a class of matrices k for which we prove error bounds for this discretization for sufficiently regular solutions bf u achievable at least for some model problems our bounds improve on existing results in the literature it is shown that for a notable subclass of k for which only suboptimal error bounds have been previously derived the piecewise linear finite element approximation of this problem will converge

fast gradient based methods for bayesian reconstruction of transmission and emission pet images the authors describe conjugate gradient algorithms for reconstruction of transmission and emission pet images the reconstructions are based on a bayesian formulation where the data are modeled as a collection of independent poisson random variables and the image is modeled using a markov random field a conjugate gradient algorithm is used to compute a maximum a posteriori map estimate of the image by maximizing over the posterior density to ensure nonnegativity of the solution a penalty function is used to convert the problem to one of unconstrained optimization preconditioners are used to enhance convergence rates these methods generally achieve effective convergence in 15 25 iterations reconstructions are presented of an sup 18 fdg whole body scan from data collected using a siemens cti ecat931 whole body system these results indicate significant improvements in emission image quality using the bayesian approach in comparison to filtered backprojection particularly when reprojections of the map transmission image are used in place of the standard attenuation correction factors

accelerated iterative reconstruction for positron emission tomography based on the em algorithm for maximum likelihood estimation the em method that was originally developed for maximum likelihood estimation in the context of mathematical statistics may be applied to a stochastic model of positron emission tomography pet the result is an iterative algorithm for image reconstruction that is finding increasing use in pet due to its attractive theoretical and practical properties its major disadvantage is the large amount of computation that is often required due to the algorithm s slow rate of convergence this paper presents an accelerated form of the em algorithm for pet in which the changes to the image as calculated by the standard algorithm are multiplied at each iteration by an overrelaxation parameter the accelerated algorithm retains two of the important practical properties of the standard algorithm namely the selfnormalization and nonnegativity of the reconstructed images experimental results are presented using measured data obtained from a hexagonal detector system for pet the likelihood function and the norm of the data residual were monitored during the iterative process according to both of these measures the images reconstructed at iterations 7 and 11 of the accelerated algorithm are similar to those at iterations 15 and 30 of the standard algorithm for two different sets of data important theoretical properties remain to be investigated namely the convergence of the accelerated algorithm and its performance as a maximum likelihood estimator

quasi norm error bounds for the finite element approximation of a non newtonian flow we consider the finite element approximation of a non newtonian flow where the viscosity obeys a general law including the carreau or power law for sufficiently regular solutions we prove energy type error bounds for the velocity and pressure these bounds improve on existing results in the literature a key step in the analysis is to prove abstract error bounds initially in a quasi norm which naturally arises in degenerate problems of this type