a priori l 2 error estimates for finite element methods for nonlinear diffusion equations with memory this paper studies finite element approximations to the solutions of the nonlinear diffusion equations with memory an elliptic projection with memory associated with our equations is defined and then used in the derivations of optimal error estimates for semidiscrete and crank nicolson finite element approximations

direct 4 d pet list mode parametric reconstruction with a novel em algorithm the production of images of kinetic parameters is often the ultimate goal of positron emission tomography pet imaging the indirect method of pet parametric imaging also called the frame based method fm is performed by fitting the time activity curve tac for each voxel with an appropriate compartment model after image reconstruction the indirect method is simple and easily implemented however it usually leads to some loss of accuracy or precision due to the use of two separate steps this paper presents a direct 4 d method for producing 3 d images of kinetic parameters from list mode pet data in this application the tac for each voxel is described by a one tissue compartment model 1t extending previous em algorithms a new spatiotemporal complete data space was introduced to optimize the maximum likelihood function this leads to a straightforward closed form parametric image update equation this method was implemented by extending the current list mode platform molar to produce a parametric algorithm pmolar 1t using an ordered subset approach qualitative and quantitative evaluations were performed using 2 d x t and 4 d x y z t simulated list mode data based on brain receptor tracers and also with a human brain study comparisons with the indirect method showed that the proposed direct method can lead to accurate estimation of the parametric image values with reduced variance especially at low count levels in the 2 d test the direct method showed similar bias to the frame based method but with variance reduction of 23 60 in the 4 d test bias values of both methods were no more than 4 and the direct method had lower variability coefficient of variation reduction of 0 64 compared to the frame based method at the normal count level the direct method had a larger reduction in variability 27 81 and lower bias 1 5 for 4 d and 1 19 for fm at low count levels the results in the human brain study are similar with pmolar 1t showing lower noise than fm

maximum norm stability and error estimates in galerkin methods for parabolic equations in one space variable maximum norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable using the continuous in time galerkin method based on piecewise polynomial approximating functions on a quasi uniform mesh

an iterative scheme for a class of quasi variational inequalities an iterative scheme is given to obtain the approximate solution of a class of quasi variational inequalities it is shown that the approximate solution obtained by the iterative scheme converges strongly in the hilbert space to the exact solution as a special case we obtain the corresponding iterative scheme for variational inequalities

discontinuous galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity we consider a finite element in space and quadrature in time discretization of a compressible linear quasistatic viscoelasticity problem the spatial discretization uses a discontinous galerkin finite element method based on polynomials of degree r termed dg r and the time discretization uses a trapezoidal rectangle rule approximation to the volterra history integral both semi and fully discrete a priori error estimates are derived without recourse to gronwall s inequality and therefore the error bounds do not show exponential growth in time moreover the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution e g when using the standard k polynomial basis on simplicies or tensor product polynomials k on quadrilaterals when this is not the case e g using k on quadri laterals the convergence rate is suboptimal in r but remains optimal in h we also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the dg r approximation

time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

perturbed proximal point algorithms for general quasi variational like inclusions in this paper we introduce two new concepts of subdifferential and proximal mappings of proper functionals on hilbert spaces the existence and lipschitz continuity of proximal mapping of a proper functional are proved by applying these concepts we introduce and study a class of general quasi variational like inclusions and develop some new perturbed proximal point algorithms of mann and ishikawa type for finding the approximate solutions of the general quasi variational like inclusions the convergence criteria of the sequences of approximate solutions generated by these new algorithms are also discussed our algorithms and results improve and generalize many known results in the literature

sharp l 2 error estimates and superconvergence of mixed finite element methods for non fickian flows in porous media a sharper l2 error estimate is obtained for the non fickian flow of fluid in porous media by means of a mixed ritz volterra projection instead of the mixed ritz projection used in r e ewing y lin and j wang acta math univ comenian n s 70 2001 pp 75 84 moreover local l2 superconvergence for the velocity along the gauss lines and for the pressure at the gauss points is derived for the mixed finite element method via the ritz volterra projection and global l2 superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique on the basis of the superconvergence estimates some useful a posteriori error estimators are presented for this mixed finite element method

two splitting positive definite mixed finite element methods for parabolic integro differential equations abstract two novel mixed finite element procedures are established for parabolic integro differential equations which can be split into two independent symmetric positive definite sub schemes and do not need to solve a coupled system of equations without requiring the lbb consistency condition the convergence analysis shows that both methods lead to the optimal order l 2 norm error estimate for u and optimal h div norm error estimate for a numerical example is presented to illustrate the theoretical analysis

perturbed proximal point algorithms for generalized quasivariational inclusions abstract in this paper we study a class of generalized quasivariational inclusions by using the properties of the resolvent operator associated with a maximal monotone mapping in hilbert space we have established an existence theorem of solutions for generalized quasivariational inclusions suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized quasivariational inclusions as special cases some known results in this field are also discussed

time stepping schemes for systems of volterra integro differential equations linear volterra type systems of integro differential equations sides with kernels in the form of sums of complex exponentials are considered such systems arise for example in the semi discrete finite element model of dynamic viscoelasticity and of some dispersive wave equations and in the representation of dynamic subsystems on interfaces several new families of time stepping schemes are proposed here for the solution of this class of sides the accuracy stability and efficiency properties of these schemes are demonstrated via both theoretical analysis using a scalar model problem and numerical experiments applied to first and second order sides

high performance parallel numerical methods for volterra equations with weakly singular kernels non stationary discrete time waveform relaxation methods for abel systems of volterra integral equations using fractional linear multistep formulae are introduced fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed a significant expression of the error is proved which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance the numerical experiments confirm the theoretical expectations

finite element approximation of an evolution problem modeling shear band formation abstract an error analysis is provided for the spatial approximation by the finite element method of a nonlinear time dependent problem which models the antiplane shear deformations of a thermoplastic material this problem consists of a system of two coupled partial differential equations with velocity and temperature unknowns optimal order error estimates in the l 2 norm are given for each of these

sensitivity analysis framework for general quasi variational inclusions abstract it is well known that the implicit resolvent equations are equivalent to the quasivariational inclusions we use this alternative equivalent formulation to study the sensitivity of the quasivariational inclusions without assuming the differentiability of the given data since the quasi variational inclusions include classical variational inequalities quasi mixed variational inequalities and complementarity problems as special cases results obtained in this paper continue to hold for these problems in fact our results can be considered as a significant extension of previously known results

finite element approximation of diffusion equations with convolution terms approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered such problems arise from modeling of flows in fissured media convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented the advantages of implementation of the algorithm in a multiprocessing environment are discussed

finite element methods for partial volterra integro differential equations on two dimensional unbounded spatial domains abstract the precise and effective way for solving a partial volterra integro differential equation on unbounded spatial domain is to derive artificial boundary conditions for some feasible and prescribed computational domain thus the unsolvable problem on unbounded domain is converted to the computational problem on bounded domain which is often called reduced problem a feasible numerical method will then be applied to solve the reduced problem with the artificial boundary conditions which often exhibit nonlocal properties in this paper the finite element method for the spatial space and the recently developed discontinuous galerkin time stepping method for the temporal space are considered to solve the reduced problem the convergence rate explicitly with those interesting parameters spatial grid size h temporal mesh size k the location of the artificial boundary d and the truncation number m is obtained this paper is the first theoretical analysis of the numerical methods for volterra integro differential equations on unbounded spatial domains

numerical methods for treating problems of viscoelastic isotropic solid deformation mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are discussed and their discretization using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described the range of applicability of this type of formulation is reviewed in the context of geometric and constitutive linearity nonlinearity and the limitations imposed by the availability of physical data are discussed one of the above models is a volterra integral equation of the second kind in this when the kernel is separable an established technique due to goursat 1933 can be exploited to reformulate the problem as a system of ordinary differential equations this approach will be described for the special case of a linear viscoelastic isotropic homogeneous synchronous constant poisson s ratio solid this method results in a complete decoupling of the space and time dependence in this case the problem can be solved at each time level by solving first a problem of linear elasticity and then a system of ordinary differential equations for each point in the spatial mesh at which the viscoelastic displacements are required the advantages disadvantages and limitations offered by this and various other schemes for solving problems of viscoelasticity as outlined below are discussed

the generalized set valued strongly nonlinear implicit variational inequalities abstract in this paper we introduce and study a new class of generalized set valued strongly nonlinear implicit variational inequalities in hilbert spaces we construct the algorithm for solving this kind of generalized set valued strongly nonlinear implicit variational inequality problem by using the auxiliary principle technique of glowinski lions and tremolieres prove the existence of solutions for this class of generalized set valued strongly nonlinear implicit variational inequalities and the convergence of iterative sequences generated by the algorithm

an iterative algorithm for variational inequalities abstract in this paper we introduce and study a new class of nonlinear variational inequalities this new class enables us to apply variational techniques to the solution of differential equations of both odd and even orders a projection method is used to suggest an iterative algorithm for finding the approximate solution of this class we also discuss the convergence criteria of the proposed iterative algorithm several special cases are discussed which can be obtained from the general result

accelerated image reconstruction using ordered subsets of projection data the authors define ordered subset processing for standard algorithms such as expectation maximization em for image restoration from projections ordered subsets methods group projection data into an ordered sequence of subsets or blocks an iteration of ordered subsets em is defined as a single pass through all the subsets in each subset using the current estimate to initialize application of em with that data subset this approach is similar in concept to block kaczmarz methods introduced by eggermont et al 1981 for iterative reconstruction simultaneous iterative reconstruction sirt and multiplicative algebraic reconstruction mart techniques are well known special cases ordered subsets em os em provides a restoration imposing a natural positivity condition and with close links to the em algorithm os em is applicable in both single photon spect and positron emission tomography pet in simulation studies in spect the os em algorithm provides an order of magnitude acceleration over em with restoration quality maintained

mann and ishikawa type perturbed iterative algorithms for generalized quasivariational inclusions abstract in this paper we introduce a class of generalized quasivariational inclusions and show its equivalence with a class of fixed point problems by making use of the properties of proximal maps using this equivalence we develop the mann and ishikawa type perturbed iterative algorithms for this class of generalized quasivariational inclusions further using fixed point techniques we prove the existence of solutions for the class of generalized quasivariational inclusions and discuss the convergence criteria for the perturbed algorithms our algorithms and results improve and generalize many known corresponding algorithms and results

negative norm error control for second kind convolution volterra equations summary we consider a piecewise constant finite element approximation to the convolution volterra equation problem of the second kind find u such that u f phi u in a time interval 0 t an a posteriori estimate of the error measured in the w 1 p 0 t norm is developed and used to provide a time step selection criterion for an adaptive solution algorithm numerical examples are given for problems in which phi is of a form typical in viscoelasticity theory

numerical techniques for the treatment of quasistatic viscoelastic stress problems in linear isotropic solids abstract for quasistatic stress problems two alternative constitutive relationships expressing the stress in a linear isotropic viscoelastic solid body as a linear functional of the strain are available in conjunction with the equations of equilibrium these form the mathematical models for the stress problems these models are first discretized in the space domain using a finite element method and semi discrete error estimates are presented corresponding to each constitutive relationship through the use respectively of quadrature rules and finite difference replacements each semi discrete scheme is fully discretized into the time domain so that two practical algorithms suitable for the numerical stress analysis of linear viscoelastic solids are produced the semi discrete estimates are then also extended into the time domain to give spatially h 1 error estimates for each algorithm the numerical schemes are predicated on exact analytical solutions for a simple model problem and finally on design data for a real polymeric material

on the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel ii nonsmooth initial data abstract backward euler methods are considered for a parabolic integrodifferential equation with a memory term containing a weakly singular kernel the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in error analyses the stability and convergence properties of the time discretizations are derived and are applied to the semidiscrete equations built by the galerkin finite element methods in the space variables

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

discrete schemes for treating hereditary problems of viscoelasticity and applications abstract mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are presented and their discretisation using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described theoretical error estimates in appropriate sobolev space settings are given both as they arise as a result of using a gronwall inequality and also from employing a more sensitive comparison theorem which for the quasistatic problem and under physically reasonable assumptions on the relaxation function yields much sharper constants in the estimates the range of applicability of the mathematical models and hence the numerical schemes and error estimates are discussed in the context of various materials primarily polymeric materials and extensions of the techniques to the modelling of manufacturing processes such as thermoforming are presented

long time numerical solution of a parabolic equation with memory long time stability and convergence properties of two time discreti zation methods for an integro differential equation of parabolic type are studied the methods are based on the standard backward euler and second order back ward differencing methods the memory term is approximated by a quadrature rule with emphasis on such rules with reduced computational memory require ments discretization of the spatial partial differential operators by the finite element method is also considered

generalized quasi variational like inclusions with nonconvex functionals in this paper two new concepts of h subdifferential and h proximal mappings of a proper functional is introduced in hilbert spaces the existence and lipschitz continuity of h proximal mapping of a proper functional are proved by applying these concepts we introduce and study a class of generalized quasi variational like inclusions with a nonconvex functional a new iterative algorithms for finding the approximate solutions and the convergence criteria of the iterative sequences generated by the algorithm are also given these algorithm and existence result generalize many known results in the literature

discontinuous galerkin method with a posteriori l_p 0 t_i error estimate for second kind volterra problems we consider a discontinuous galerkin finite element method applied in time to a model volterra equation of the second kind a residual based computable l p 0 t i galerkin error estimate is derived for 1 le p le infty this estimate does not explicitly contain the time step and therefore the time step control must be based on a heuristic criterion the estimate can then be used to demonstrate the integrity or otherwise of the finite element solution after performing some numerical experiments we conclude that this approach is at least competetive with classical discretizations since it is computationally simple to implement but has the added advantage of reliable error feedback

non fickian convection diffusion models in porous media in this paper we propose a numerical scheme to approximate the solution of a non fickian coupled model that describes e g miscible transport in porous media the model is defined by a system of a quasilinear elliptic equation which governs the fluid pressure and a quasilinear integro differential equation which models the convection diffusion transport process the numerical scheme is based on a conforming piecewise linear finite element method for the discretization in space the fully discrete approximations is obtained with an implicit explicit method estimates for the continuous in time and the fully discrete methods are derived showing that the numerical approximation for the concentrations and the pressure are second order convergent in a discrete l 2 norm and in a discrete h 1 norm respectively

a parallel method for the numerical solution of integro differential equation with positive memory abstract an efficient parallel numerical method is proposed for an integro differential equation with positive memory instead of solving the equation in classical time marching methods which require massive storage of solutions of previous time steps in order to advance to a next time step the fourier laplace transformation in time is applied to obtain a set of complex valued elliptic problems parameterized by points on a contour in the complex plane using the independence of an elliptic problem corresponding to one contour point is independent of those elliptic problems corresponding to other contour points all elliptic problems can be solved in parallel essentially without data communications then the time domain solution can be obtained by the fourier laplace inversion formula an error analysis and the numerical implementation of this parallel method is presented

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

finite element methods for parabolic and hyperbolic partial integro differential equations on formule plusieurs methodes de galerkin a temps discret pour des equations integrodifferentielles paraboliques

the mann and ishikawa iterative approximation of solutions to variational inclusions with accretive type mappings abstract the purpose of this paper is to introduce and study the existence of solutions and convergence of mann and ishikawa iterative processes for a class of variational inclusions with accretive type mappings in banach spaces the results presented in this paper extend and improve the corresponding results of 1 9

a new expanded mixed method for parabolic integro differential equations a new expanded mixed scheme is studied and analyzed for linear parabolic integro differential equations the proposed method s gradient belongs to the simple square integrable space replacing the classical h div space the new expanded mixed projection is introduced the existence and uniqueness of solution for semi discrete scheme are proved and the fully discrete error estimates based on both backward euler scheme are obtained moreover the optimal a priori error estimates in l 2 and h 1 norm for the scalar unknown u and the error results in l 2 norm for its gradient and its flux the coefficients times the negative gradient are derived finally some numerical results are calculated to verify our theoretical analysis

generalized multivalued quasi variational inequalities in this paper we introduce and study a new class of variational inequalities known as generalized multivalued quasi variational inequalities this class is the most general and unifying one using essentially the projection technique we establish the equivalence between this class and the implicit wiener hopf equations this equivalence is used to suggest a number of iterative algorithms for solving the generalized multivalued quasi variational inequalities the convergence analysis of these algorithms is also studied the results proved in this paper represent a significant improvement and refinement of the previously known results in this area

error estimates for semidiscrete finite element methods for parabolic integro differential equations application des methodes d elements finis aux equations integro differentielles de type parabolique avec un noyau integral forme par un operateur aux derivees partielles d ordre 2

applications and numerical analysis of partial differential volterra equations a brief survey this article contains a concise survey of the numerical analysis of volterra equations and leads up to some recent results on a posteriori error estimation for finite element approximations

numerical solution of linear quasistatic hereditary viscoelasticity problems we give a space time galerkin finite element discretization of the linear quasistatic compressible viscoelasticity problem as described by an elliptic partial differential equation with a volterra memory term the discretization consists of a continuous piecewise linear approximation in space with a discontinuous piecewise constant or linear approximation in time we derive an a priori maximum energy error estimate by exploiting galerkin orthogonality and the data stability of a related discrete backward problem illustrative numerical experiments are also included as also is a brief description of our first results on a posteriori error estimation this allows for adaptive control of the space mesh but not of the time step

some aspects of variational inequalities abstract in this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence uniqueness computational properties various generalizations sensitivity analysis and their applications we also propose some open problems with sufficient information and references so that someone may attempt solution s in his her area of special interest we also include some new results which we have recently obtained

direct parametric image reconstruction in reduced parameter space for rapid multi tracer pet imaging the separation of multiple pet tracers within an overlapping scan based on intrinsic differences of tracer pharmacokinetics is challenging due to limited signal to noise ratio snr of pet measurements and high complexity of fitting models in this study we developed a direct parametric image reconstruction dpir method for estimating kinetic parameters and recovering single tracer information from rapid multi tracer pet measurements this is achieved by integrating a multi tracer model in a reduced parameter space rps into dynamic image reconstruction this new rps model is reformulated from an existing multi tracer model and contains fewer parameters for kinetic fitting ordered subsets expectation maximization osem was employed to approximate log likelihood function with respect to kinetic parameters to incorporate the multi tracer model an iterative weighted nonlinear least square wnls method was employed the proposed multi tracer dpir mt dpir algorithm was evaluated on dual tracer pet simulations 18 rm f fdg and 11 rm c met as well as on preclinical pet measurements 18 rm f flt and 18 rm f fdg the performance of the proposed algorithm was compared to the indirect parameter estimation method with the original dual tracer model the respective contributions of the rps technique and the dpir method to the performance of the new algorithm were analyzed in detail for the preclinical evaluation the tracer separation results were compared with single 18 rm f fdg scans of the same subjects measured two days before the dual tracer scan the results of the simulation and preclinical studies demonstrate that the proposed mt dpir method can improve the separation of multiple tracers for pet image quantification and kinetic parameter estimations

optimal long time l_p 0 t stability and semidiscrete error estimates for the volterra formulation of the linear quasistatic viscoelasticity problem the purpose of this article is to show how the solution of the linear quasistatic compressible viscoelasticity problem written in volterra form with fading memory may be sharply bounded in terms of the data if certain physically reasonable assumptions are satisfied the bounds are derived by making precise assumptions on the memory term which then make it possible to avoid the gronwall inequality and use instead a comparison theorem which is more sensitive to the physics of the problem once the data stability estimates are established we apply the technique also to deriving a priori error bounds for semidiscrete finite element approximations our bounds are derived for viscoelastic solids and fluids under the small strain assumption in terms of the eigenvalues of a certain matrix derived from the stress relaxation tensor for isotropic materials we can be explicit about the form of these bounds while for the general case we give a formula for their computation

maximum norm stability and maximal l p lp regularity of fems for parabolic equations with lipschitz continuous coefficients in this paper we study the semi discrete galerkin finite element method for parabolic equations with lipschitz continuous coefficients we prove the maximum norm stability of the semigroup generated by the corresponding elliptic finite element operator and prove the space time stability of the parabolic projection onto the finite element space in l infty q t l qt and l p 0 t l q omega lp 0 t lq 1

ritz volterra projections to finite element spaces and applications to integrodifferential and related equations the object of this paper is to investigate the convergence of finite element approximations to solutions of parabolic and hyperbolic integrodifferential equations and also of equations of sobolev and viscoelasticity type the concept of ritz volterra projection will be seen to unify much of the analysis for the different types of problems optimal order error estimates are obtained in l p for 2 leqq p infty and almost optimal order pointwise results given

strongly nonlinear generalized equations we study a class of generalized equations which includes as a special case various classes of quasi variational inequalities introduced by noor isac siddiqi and ansari such problems occur in mathematical programming and in various other branches of pure and applied mathematics we also propose two iterative schemes for the numerical solution of the problems considered

long time behavior of the two grid finite element method for fully discrete semilinear evolution equations with positive memory abstract based on two grid discretizations two fully discrete finite element algorithms for semilinear parabolic integro differential equations with positive memory are proposed with the backward euler scheme for the temporal discretization the basic idea of the space two grid finite element algorithms is to approximate the semilinear equations on a coarse space grid and to solve the linearized equations on a finer space grid at each time step to further decreases the amount of computational work a space time two grid algorithm based on a coarse space grid with large time stepsize t and a finer space grid with small time stepsize t for the evolutional equations is proposed in this paper the sharp long time stability and error estimates for the standard finite element method the space two grid finite element method and the space time two grid finite element method are derived it is showed that the two grid algorithms long time stability and error estimates are similar to those of the direct resolution of the semilinear problem on a fine grid

merit functions and error bounds for generalized variational inequalities we consider the generalized variational inequality and construct certain merit functions associated with this problem in particular those merit functions are everywhere nonnegative and their zero sets are precisely solutions of the variational inequality we further use those functions to obtain error bounds i e upper estimates for the distance to solutions of the problem

on the numerical approximation of an evolution problem in nonlinear viscoelasticity abstract an error analysis is furnished for the spatial approximation via the finite element method of certain nonlinear time dependent problems emanating in the theory of viscoelastic solids numerical examples are given first for a model equation and then for numerical equilibrium solutions in cases of a nonconvex double well potential and a convex potential the latter derived from the nonconvex one via relaxation by an equal area construction

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

a remark on split least squares mixed element procedures for pseudo parabolic equations abstract in this paper we introduce two novel split least squares mixed element procedures for pseudo parabolic equations by selecting the least squares functional properly each procedure can be split into two independent symmetric positive definite sub procedures one of sub procedures is for the primitive unknown variable u which is the same as the standard galerkin finite element procedure and the other is for the introduced flux variable optimal order error estimates are developed a numerical example is given to show the efficiency of the introduced schemes

finite element methods of the two nonlinear integro differential equations abstract transformation of dependent variables such as the kirchhoff transformation is a classical tool for solving nonlinear partial differential equations in larsson et al this approach was used in connection with the finite element method zlamal used it in galerkin approximation in space the euler method in time and applied it to the solution of nonlinear heat equations here we give justification of the methods for the two nonlinear integro heat equations in the case that the discretization is carried through by piecewise linear polynomials in space and by the implicit euler method in time with quadrature rules which require the storage of relatively few values of the solution the estimate of the discretization error in the maximum norm is introduced the results depend on superconvergence in the gradient for a related elliptic problem which is shown to hold only for very special meshes

general strongly nonlinear variational inequalities abstract in this paper we develop iterative algorithms for finding approximate solutions for new classes of variational and quasivariational inequalities which include as a special case some known results in this field it is shown that the solutions of the iterative schemes converge to the exact solutions

error estimates with sharp constants for a fading memory volterra problem in linear solid viscoelasticity the problem characterizing nonageing linear isothermal quasi static isotropic compressible solid viscoelasticity in the time interval 0 t is described this is essentially a volterra equation of the second kind arrived at by adding smooth fading memory to the elliptic linear elasticity equations we analyze the errors resulting from replacing the relaxation functions with practical approximations in a semidiscrete finite element approximation and in a fully discrete scheme derived by replacing the hereditary integral with the trapezoidal rule for numerical integration the error estimates are sharp in the sense that if certain bounds on the data are independent of t then so also are the constants involved in them this is a consequence of bypassing the usual gronwall lemmas with arguments that are more sensitive to the fading memory of the physical problem

kinetic modelling using basis functions derived from two tissue compartmental models with a plasma input function general principle and application to 18f fluorodeoxyglucose positron emission tomography abstract a kinetic modelling method for the determination of influx constant k i is given that utilises basis functions derived from plasma input two tissue compartmental models bafpic two forms of the basis functions are given bafpic i with k 4 0 no product loss and bafpic r with k 4 non zero simulations were performed using literature rate constant values for 18 f fluorodeoxyglucose fdg in both normal and abnormal brain pathology both homogeneous and heterogeneous tissues were simulated and this data was also used as input for other methods commonly used to determine k i non linear least squares compartmental modelling nlls autoradiographic method and patlak gjedde graphical analysis pga the four methods were also compared for real fdg positron emission tomography pet data for both k 4 0 and k 4 non zero simulated data bafpic had the best bias properties of the four methods the autoradiographic method was always the best for variability but bafpic i had lower variability than pga and nlls for non zero k 4 data the variance of bafpic r was inferior to pga but still significantly superior to nlls k i maps calculated from real data substantiate the simulation results with bafpic i having lower noise than pga voxel k i values from bafpic i correlated well with those from pga r 2 0 989 bafpic is easy to implement and combines low bias with good noise properties for voxel wise determination of k i for fdg bafpic is suitable for determining k i for other tracers well characterised by a serial two tissue compartment model and has the advantage of also producing values for individual kinetic constants and blood volume