an inverse pollution problem in porous media abstract in this paper we deal with an inverse source problem of integro differential parabolic equations which comes from nonlinear pollution problems in porous media we study the existence and uniqueness of solutions for the direct problem as well as the existence of quasisolutions of the inverse source problem in an appropriate class of admissible source functions

incremental identification of transport coefficients in convection diffusion systems in this paper an incremental approach for the identification of a model for transport coefficients in convection diffusion systems on the basis of high resolution measurement data is presented the transport is represented by a convection term with known convective velocity and by a diffusion term with an unknown generally state dependent transport coefficient the identification of the transport model for this transport coefficient constitutes an ill posed nonlinear inverse problem we present a novel decomposition approach in which this inverse problem is split into a sequence of inverse subproblems in the first identification step of this incremental approach a source is estimated by solving an affine linear inverse problem by means of the conjugate gradient method in the second identification step a nonlinear inverse problem has to be solved to reconstruct a transport coefficient a newton type method using the conjugate gradient method in its inner iteration is used to solve this nonlinear inverse problem of coefficient estimation finally in the third identification step a transport model structure is proposed and identified on the basis of the model free transport coefficient reconstructed in the two previous steps the ill posedness of each inverse problem is examined by using artificially perturbed transient simulation data and appropriate regularization techniques the identification methodology is illustrated for a three dimensional convection diffusion equation which has its origin in the modeling and simulation of energy transport in a laminar wavy film flow

optimal reconstruction of material properties in complex multiphysics phenomena we develop an optimization based approach to the problem of reconstructing temperature dependent material properties in complex thermo fluid systems described by the equations for the conservation of mass momentum and energy our goal is to estimate the temperature dependence of the viscosity coefficient in the momentum equation based on some noisy temperature measurements where the temperature is governed by a separate energy equation we show that an elegant and computationally efficient solution of this inverse problem is obtained by formulating it as a pde constrained optimization problem which can be solved with a gradient based descent method a key element of the proposed approach the cost functional gradients are characterized by mathematical structure quite different than in typical problems of pde constrained optimization and are expressed in terms of integrals defined over the level sets of the temperature field advanced techniques of integration on manifolds are required to evaluate numerically such gradients and we systematically compare three different methods as a model system we consider a two dimensional unsteady flow in a lid driven cavity with heat transfer and present a number of computational tests to validate our approach and illustrate its performance

error analysis of an equation error method for the identification of the diffusion coefficient in a quasi linear parabolic differential equation we consider the inverse problem of reconstructing the diffusion coefficient in a quasi linear parabolic differential equation in divergence form from measurements of the solution at a finite number of points in the interior of the domain an equation error method is developed which transforms the inverse problem into a system of linear operator equations for the diffusion coefficient and which can be solved by the conjugate gradient method in a very efficient and stable manner a detailed error analysis relates the required number of measurements with their accuracy numerical results illustrate the performance of the method

multiplicativity factors for seminorms ii let s be a seminorm on an algebra a in this paper we study multiplicativity and quadrativity factors for s i e constants 0 and 0 for which s xy s x s y and s x2 s x 2 for all x y a we begin by investigating quadrativity factors in terms of the kernel of s we then turn to the question under what conditions does s have multiplicativity factors if it has quadrativity factors we show that if a is commutative then quadrativity factors imply multiplicativity factors we further show that in the noncommutative case there exist both proper seminorms and norms that have quadrativity factors but no multiplicativity factors

a technique to develop mesh distortion immune finite elements three sufficient conditions are proposed for an iso parametric finite element formulation to be immune to mesh distortion effects even for extremely distorted element geometries for which the jacobian geometric mapping goes negative illustrative element formulations are presented to test the sufficiency of the proposed conditions numerical results confirm that the illustrative elements are indeed capable of giving mesh distortion immune solution for extreme mesh distortions this has been possible partly because the stiffness integral is free of the jacobian term in the new formulation

rotationally invariant distortion resistant finite elements the predictive capability of conventional iso parametric finite elements deteriorates with mesh distortion in the case of geometrically non linear analysis changes in geometry causing severe distortion can result in negative jacobian mapping between the local and global systems resulting in numerical breakdown this paper presents a finite element formulation that is resistant to irregular mesh geometries and large element distortions whilst remaining invariant to rigid body motion the predictive capabilities of the family of finite elements are demonstrated using a series of geometrically non linear analyses including an elastic cantilever beam and an elasto plastic double notched specimen

energy minimization using sobolev gradients application to phase separation and ordering a common problem in physics and engineering is the calculation of the minima of energy functionals the theory of sobolev gradients provides an efficient method for seeking the critical points of such a functional we apply the method to functionals describing coarse grained ginzburg landau models commonly used in pattern formation and ordering processes

a modified method for high order numerical derivatives in this paper we propose a new regularization method for computing high order numerical derivatives from one dimensional noisy data the convergence estimate under an appropriate choice of the regularization parameter is obtained some interesting numerical tests show that the proposed method is effective and stable

an unsymmetric 4 node 8 dof plane membrane element perfectly breaking through macneal s theorem summary among numerous finite element techniques few models can perfectly without any numerical problems break through macneal s theorem any 4 node 8 dof membrane element will either lock in in plane bending or fail to pass a c0 patch test when the element s shape is an isosceles trapezoid in this paper a 4 node plane quadrilateral membrane element is developed following the unsymmetric formulation concept which means two different sets of interpolation functions for displacement fields are simultaneously used the first set employs the shape functions of the traditional 4 node bilinear isoparametric element while the second set adopts a novel composite coordinate interpolation scheme with analytical trail function method in which the cartesian coordinates x y and the second form of quadrilateral area coordinates qacm ii s t are applied together the resulting element us atfq4 exhibits amazing performance in rigorous numerical tests it is insensitive to various serious mesh distortions free of trapezoidal locking and can satisfy both the classical first order patch test and the second order patch test for pure bending furthermore because of usage of the second form of quadrilateral area coordinates qacm ii the new element provides the invariance for the coordinate rotation it seems that the behaviors of the present model are beyond the well known contradiction defined by macneal s theorem copyright 2015 john wiley sons ltd

a class of seminorms on function algebras abstract let a be a function algebra on a set t in this paper we study seminorms on a of the form sc x cx where c 0 c a is a fixed element and is the sup norm on t we begin by proving that under suitable assumptions elements c d a satisfy c d on t if and only if for some p 0 b of a these results are then used in order to study multiplicativity and quadrativity factors for sc on b i e constants 0 and 0 for which sc xy sc x sc y and sc x2 sc x 2 for all x y b finally for a family t of functions in a we define the seminorm s f x sup sf x f f and provide conditions under which s f has multiplicativity and quadrativity factors by exhibiting an element c a such that s f s c on a

a novel unsymmetric 8 node plane element immune to mesh distortion under a quadratic displacement field an 8 node quadrilateral plane finite element is developed based on a novel unsymmetric formulation which is characterized by the use of two sets of shape functions viz the compatibility enforcing shape functions and completeness enforcing shape functions the former are chosen to satisfy exactly the minimum inter as well as intra element displacement continuity requirements while the latter are chosen to satisfy all the linear and higher order completeness requirements so as to reproduce exactly a quadratic displacement field numerical results from test problems reveal that the new element is indeed capable of reproducing exactly a complete quadratic displacement field under all types of admissible mesh distortions in this respect the proposed 8 node unsymmetric element emerges to be better than the existing symmetric quad8 quad8 9 quad9 quad12 and quad16 elements and matches the performance of the quartic element quad25 for test problems involving a cubic or higher order displacement field the proposed element yields a solution accuracy that is comparable to or better than that of quad8 quad8 9 and quad9 elements furthermore the element maintains a good accuracy even with the reduced 2 2 numerical integration copyright 2003 john wiley sons ltd

error analysis of a multisingular inverse coefficient problem for the sturm liouville operator based on boundary measurement in this paper we study the inverse problem of determining the unknown leading coefficient k k x of the linear sturm liouville operator au k x u x q x u x x 0 1 from boundary measurements when u x or and u x vanishes at several called singular points of the interval 0 1 as a result the considered inverse problem has simultaneously different types moderate or severely of ill conditioned situations in different parts of the interval 0 1 the presented inverse polynomial method permits use of a priori information about singular points either to increase the order of the polynomial approximation in each subinterval or to obtain an artificial cauchy data for the unknown coefficient error estimations for the polynomial approximations are presented for well conditioned as well as for ill conditioned situations the behaviour of the inverse problem solution with respect to both types dirichlet and neumann of noisy data is analyzed the obtained results are illustrated by various numerical examples

a 20 node hexahedron element with enhanced distortion tolerance this paper presents a distortion resistant 20 node hexahedron element that employs two different sets of shape functions for the trial and test functions the formulation seeks to satisfy the continuity and completeness requirements by exploiting the intrinsic properties of these two sets of shape functions several test problems are used to assess the performance of the element under various mesh distortions the ability of the proposed as well as the classical 20 node element to maintain solution accuracy under severe mesh distortions has been studied the proposed element exhibits a very high tolerance to mesh distortions in particular for problems involving linear and quadratic displacement fields the element is capable of reproducing exact solution under all admissible geometrical distortions of the mesh for test problems involving higher order displacement fields the performance of the present element is in general better than that of the classical element copyright 2004 john wiley sons ltd

differentiation by integration with jacobi polynomials in this paper the numerical differentiation by integration method based on jacobi polynomials originally introduced by mboup et al 19 20 is revisited in the central case where the used integration window is centered such a method based on jacobi polynomials was introduced through an algebraic approach 19 20 and extends the numerical differentiation by integration method introduced by lanczos 1956 21 the method proposed here rooted in 19 20 is used to estimate the nth email protected n order derivative from noisy data of a smooth function belonging to at least c n 1 q email protected n in 19 20 where the causal and anti causal cases were investigated the mismodelling due to the truncation of the taylor expansion was investigated and improved allowing a small time delay in the derivative estimation here for the central case we show that the bias error is o h q 2 where h is the integration window length for email protected c n q 2 in the noise free case and the corresponding convergence rate is o d q 1 n 1 q where d is the noise level for a well chosen integration window length numerical examples show that this proposed method is stable and effective

model problems for the multigrid optimization of systems governed by differential equations we discuss a multigrid approach to the optimization of systems governed by differential equations such optimization problems appear in many applications and are of a different nature than systems of equations our approach uses an optimization based multigrid algorithm in which the multigrid algorithm relies explicitly on nonlinear optimization models as subproblems on coarser grids our goal is not to argue for a particular optimization based multigrid algorithm but instead to demonstrate how multigrid can be used to accelerate nonlinear programming algorithms furthermore using several model problems we give evidence both theoretical and numerical that the optimization setting is well suited to multigrid algorithms some of the model problems show that the optimization problem may be more amenable to multigrid than the governing differential equation in addition we relate the multigrid approach to more traditional optimization methods as further justification for the use of an optimization based multigrid algorithm

nonlinear wavelet image processing variational problems compression and noise removal through wavelet shrinkage this paper examines the relationship between wavelet based image processing algorithms and variational problems algorithms are derived as exact or approximate minimizers of variational problems in particular we show that wavelet shrinkage can be considered the exact minimizer of the following problem given an image f defined on a square i minimize over all g in the besov space b11 l1 i the functional f g l2 i 2 g b11 l1 i we use the theory of nonlinear wavelet image compression in l2 i to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i i d mean zero gaussian noise a new signal to noise ratio snr which we claim more accurately reflects the visual perception of noise in images arises in this derivation we present extensive computations that support the hypothesis that near optimal shrinkage parameters can be derived if one knows or can estimate only two parameters about an image f the largest for which f bq lq i 1 q 2 1 2 and the norm f bq lq i both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than donoho and johnstone s visushrink procedure an example suggests however that donoho and johnstone s 1994 1995 1996 sureshrink method which uses a different shrinkage parameter for each dyadic level achieves a lower error than our procedure

on optimal reconstruction of constitutive relations abstract in this investigation we develop and validate a computational method for reconstructing constitutive relations based on measurement data applicable to problems arising in nonequilibrium thermodynamics and continuum mechanics this parameter estimation problem is solved as pde constrained optimization using a gradient based technique in the optimize then discretize framework the principal challenge is that the control variable i e the relation characterizing the constitutive property is not a function of the independent variables in the problem but of the state dependent variable the proposed method allows one to reconstruct a smooth constitutive relation defined over a broad range of the dependent variable it relies on three main ingredients a computationally friendly expression for the cost functional gradient sobolev gradients used in lieu of discontinuous l 2 gradients and a systematic technique for shifting the identifiability region the performance of this approach is illustrated by the reconstruction of the temperature dependence of the thermal conductivity in a one dimensional model problem

alternative ways for formulation of hybrid stress elements an element stiffness matrix can be derived by the conventional potential energy principle and indirectly also by generalized variational principles such as the hu washizu principle and the hellinger reissner principle the present investigation has the objective to show an approach which is concerned with the formulation of incompatible elements for solid continuum and for plate bending problems by the hellinger reissner principle it is found that the resulting scheme is equivalent to that considered by tong 1982 for the construction of hybrid stress elements in tong s scheme the inversion of a large flexibility matrix can be avoided it is concluded that the introduction of additional internal displacement modes in mixed finite element formulations by the hellinger reissner principle and the hu washizu principle can lead to element stiffness matrices which are equivalent to the assumed stress hybrid method

plane isoparametric hybrid stress elements invariance and optimal sampling formulation and applications of the hybrid stress finite element model to plane elasticity problems are examined conditions for invariance of the element stiffness are established for two dimensional problems the results of which are easily extended to three dimensional cases next the hybrid stress functional for a 3 d continuum is manipulated into a more convenient form in which the location of optimal stress strain sampling points can be identified to illustrate these concepts 4 and 8 node plane isoparametric hybrid stress elements which are invariant and of correct rank are developed and compared with existing hybrid stress elements for a 4 node element lack of invariance is shown to lead to spurious zero energy modes under appropriate element rotation alternative 8 node elements are considered and the best invariant element is shown to be one in which the stress compatibility equations are invoked results are also presented which demonstrate the validity of the optimal sampling points the effects of reduced orders of numerical integration and the behaviour of the elements for nearly incompressible materials

skin friction and pressure the footprints of turbulence abstract the problems of exact state reconstruction and approximate state estimation based on wall information in a wall bounded incompressible unsteady flow are addressed it is shown that if in an arbitrarily small neighborhood of time t precise measurements are made of the two components of wall skin friction and the wall pressure all terms in the taylor series expansions of the unsteady flow state near the wall at time t may be determined in the linear setting this determination may be made based on skin friction measurements alone combining this fact with the analyticity of solutions of the nonlinear navier stokes equation and the unique continuation theorem for analytic functions in theory complete reconstruction of a fully developed turbulent flow in a channel at any reynolds number at time t is possible given only information about the unsteady flow available at the wall in a neighborhood of time t without knowledge of the initial conditions of the flow thus skin friction and pressure measurements on the wall in a neighborhood of time t provide a unique footprint of the entire unsteady turbulent flow state no other flow can have the same footprint indeed higher order terms are shown to uniformly improve the correlation of truncated taylor series expansions with the dns of a turbulent flow near the wall however such series extrapolations amplify measurement noise as they require differentiation in both space and time of the measurements and the radius of convergence of the taylor series expansions is less than 10 wall units the so called linear stochastic estimation technique in which the polynomials forming the basis of the series expansion are replaced by well behaved functions such as pod modes on the entire flow domain also demonstrates very poor convergence in light of these limitations on direct extrapolations from measurements in the practical setting an adjoint based algorithm is presented and numerically tested for estimating the state of an entire turbulent channel flow system based on a time history of noisy measurements at the wall this algorithm effectively uses the unsteady nonlinear navier stokes equation itself as a filter to find the flow solution that is most consistent with the available measurements

fourier truncation method for high order numerical derivatives we consider a classical ill posed problem numerical differentiation with a new method we propose fourier truncation method to compute high order numerical derivatives a holder type stability estimate is obtained a numerical implementation is described numerical examples show that the proposed method is effective and stable

numerical differentiation by a fourier extension method with super order regularization abstract based on the idea of fourier extension we develop a new method for numerical differentiation the tikhonov regularization method with a super order penalty term is presented to deal with the illposdness of the problem and the regularization parameter can be chosen by a discrepancy principle for various smooth conditions the solution process of the new method is uniform and order optimal error bounds can be obtained numerical experiments are also presented to illustrate the effectiveness of the proposed method

gradient method in hilbert besov spaces for the optimal control of parabolic free boundary problems abstract this paper presents computational analysis of the inverse stefan type free boundary problem where information on the boundary heat flux is missing and must be found along with the temperature and the free boundary we pursue optimal control framework introduced in abdulla 2013 2016 where boundary heat flux and free boundary are components of the control vector and optimality criteria consist of the minimization of the quadratic declinations from the available measurements of the temperature distribution at the final moment phase transition temperature on the free boundary and the final position of the free boundary we develop gradient descent algorithm based on frechet differentiability in hilbert besov spaces complemented with preconditioning or increase of regularity of the frechet gradient through implementation of the riesz representation theorem three model examples with various levels of complexity are considered extensive comparative analysis through implementation of preconditioning and tikhonov regularization calibration of preconditioning and regularization parameters effect of noisy data comparison of simultaneous identification of control parameters vs nested optimization is pursued

a shape free 8 node plane element unsymmetric analytical trial function method summary the unsymmetric fem is one of the effective techniques for developing finite element models immune to various mesh distortions however because of the inherent limitation of the metric shape functions the resulting element models exhibit rotational frame dependence and interpolation failure under certain conditions in this paper by introducing the analytical trial function method used in the hybrid stress function element method an effort was made to naturally eliminate these defects and improve accuracy the key point of the new strategy is that the monomial terms the trial functions in the assumed metric displacement fields are replaced by the fundamental analytical solutions of plane problems furthermore some rational conditions are imposed on the trial functions so that the assumed displacement fields possess fourth order completeness in cartesian coordinates the resulting element model denoted by us atfq8 can still work well when interpolation failure modes for original unsymmetric element occur and provide the invariance for the coordinate rotation numerical results show that the exact solutions for constant strain stress pure bending and linear bending problems can be obtained by the new element us atfq8 using arbitrary severely distorted meshes and produce more accurate results for other more complicated problems copyright 2012 john wiley sons ltd

a wavelet galerkin method for high order numerical differentiation numerical differentiation is a classical ill posed problem in this paper we propose a wavelet galerkin method for high order numerical differentiation by an appropriate choice of the regularization parameter an order optimal stability estimate of holder type is obtained some numerical examples show that the method is effective and stable

new nonconforming finite elements on arbitrary convex quadrilateral meshes in this paper we construct new nonconforming finite elements on the meshes consisting of arbitrary convex quadrilaterals especially for the quadratic and cubic cases for each case we first define a quadrilateral element that adopts edge moments as the degrees of freedom dofs and then enforce a linear constraint on this element we have for the quadratic case eight degrees of freedom per element and for the cubic case eleven dofs per element respectively the dimensions and the bases of different types for the global finite element spaces are provided we consider the approximations of two dimensional second order elliptic problems for both of these elements error estimates with optimal convergence order in both broken h 1 norm and l 2 norm are given moreover we consider the discretization of the stokes equations adopting our quadratic element to approximate each component of the velocity along with piecewise discontinuous p 1 element for the pressure this mixed scheme is stable and optimal error estimates both for the velocity and the pressure are also achieved numerical examples verify our theoretical analysis

the kuramoto sivashinsky equation a bridge between pde s and dynamical systems abstract through extensive numerical simulation we have characterized the transition to chaos of the solutions to the kuramoto sivashinsky equation the attracting solution manifolds undergo a complex bifurcation sequence including multimodal fixed points invariant tori traveling wave trains and homoclinic orbits we relate this behavior to earlier work where the kuramoto sivashinsky equation was shown to behave as a finite dimensional dynamical system of ordinary differential equations

effects of element distortions on the performance of isoparametric elements we discuss the effects of element distortions on the performance of displacement based isoparametric quadrilateral finite elements suitable example problems for both the lagrangian and the serendipity types of elements are used to show numerically the effects of element distortions the effects of angular and curved edge distortions are evaluated analytically in terms of the abilities of the elements to represent exactly various polynomial displacement fields we then solve a plane stress problem adaptively to demonstrate the effect of element distortions on the accuracy of the stress distribution the conclusion is that the lagrangian elements are not affected by angular distortions and are therefore the more reliable elements for general use

a computational framework for the regularization of adjoint analysis in multiscale pde systems this paper examines the regularization opportunities available in the adjoint analysis and optimization of multiscale pde systems regularization may be introduced into such optimization problems by modifying the form of the evolution equation and the forms of the norms and inner products used to frame the adjoint analysis typically l2 brackets are used in the definition of the cost functional the adjoint operator and the cost functional gradient if instead we adopt the more general sobolev brackets the various fields involved in the adjoint analysis may be made smoother and therefore easier to resolve numerically the present paper identifies several relationships which illustrate how the different regularization options fit together to form a general framework the regularization strategies proposed are exemplified using a id kuramoto sivashinsky forecasting problem and computational examples are provided which exhibit their utility a multiscale preconditioning algorithm is also proposed that noticeably accelerates convergence of the optimization procedure application of the proposed regularization strategies to more complex optimization problems of physical and engineering relevance is also discussed

distortion immune nine node displacement based quadrilateral thick plate finite elements that satisfy constant bending patch test we employ the linked interpolation concept to develop a higher order 9 node quadrilateral plate finite element with curved sides which passes the constant bending patch test for arbitrary node positions the linked interpolation for the plate displacements is expanded with three bubble parameters to get polynomial completeness necessary to satisfy the patch test in contrast to some other techniques the elements developed in this way retain a symmetric stiffness matrix at a marginal computational expense at the element level two new elements are generated using this concept and tested on several examples with curved sides or some kind of geometric distortion

8 and 12 node plane hybrid stress function elements immune to severely distorted mesh containing elements with concave shapes abstract by simply improving the first version of hybrid stress element method proposed by pian several 8 and 12 node plane quadrilateral elements which are immune to severely distorted mesh containing elements with concave shapes are successfully developed firstly instead of the stresses the stress function is regarded as the functional variable and introduced into the complementary energy functional then the fundamental analytical solutions in global cartesian coordinates of are taken as the trial functions for 2d finite element models and meanwhile the corresponding unknown stress function constants are introduced thus the resulting stress fields must be more reasonable because both the equilibrium and the compatibility relations can be satisfied thirdly by using the principle of minimum complementary energy these unknown stress function constants can be expressed in terms of the displacements along element boundaries which can be interpolated directly by the element nodal displacements finally the complementary energy functional can be rewritten in terms of element nodal displacement vector and thus the element stiffness matrix of such hybrid stress function hs f element is obtained this technique establishes a universal frame for developing reasonable hybrid stress elements based on the principle of minimum complementary energy and the first hybrid stress element proposed by pian is just a special case within this frame following above procedure two 8 node and two 12 node quadrilateral plane elements are constructed by employing different fundamental analytical solutions of airy stress function numerical results show that the 8 node and 12 node models can produce the exact solutions for pure bending and linear bending problems respectively even the element shape degenerates into triangle and concave quadrangle furthermore these elements do not possess any spurious zero energy mode and rotational frame dependence

a systematic construction of b bar functions for linear and non linear mixed enhanced finite elements for plane elasticity problems in a previous paper a modified hu washizu variational formulation has been used to derive an accurate four node plane strain stress finite element denoted qe2 for the mixed element qe2 two enhanced strain terms are used and the assumed stresses satisfy the equilibrium equations a priori for the linear elastic case in this paper an alternative approach is discussed the new formulation leads to the same accuracy for linear elastic problems as the qe2 element however it turns out to be more efficient in numerical simulations especially for large deformation problems using orthogonal stress and strain functions we derive b functions which avoid numerical inversion of matrices the b strain matrix is sparse and has the same structure as the strain matrix b obtained from a compatible displacement field the implementation of the derived mixed element is basically the same as the one for a compatible displacement element the only difference is that we have to compute a b strain matrix instead of the standard b matrix accordingly existing subroutines for a compatible displacement element can be easily changed to obtain the mixed enhanced finite element which yields a higher accuracy than the q4 and qm6 elements copyright 1999 john wiley sons ltd

as flat as possible how does one determine a surface which is as flat as possible such as those created by soap film surfaces what does it mean to be as flat as possible in this paper we address this question from two distinct points of view one local and one global in nature continuing with this theme we put a temporal twist on the question and ask how to evolve a surface so as to flatten it as efficiently as possible this elementary discussion provides a platform to introduce a wide range of advanced topics in partial differential equations and helps students build geometric and analytic understanding of solutions of certain elliptic and parabolic partial differential equations

adjoint based optimization of pde systems with alternative gradients in this work we investigate a technique for accelerating convergence of adjoint based optimization of pde systems based on a nonlinear change of variables in the control space this change of variables is accomplished in the differentiate then discretize approach by constructing the descent directions in a control space not equipped with the hilbert structure we show how such descent directions can be computed in general lebesgue and besov spaces and argue that in the besov space case determination of descent directions can be interpreted as nonlinear wavelet filtering of the adjoint field the freedom involved in choosing parameters characterizing the spaces in which the steepest descent directions are constructed can be leveraged to accelerate convergence of iterations our computational examples involving state estimation problems for the 1d kuramoto sivashinsky and 3d navier stokes equations indeed show significantly improved performance of the proposed method as compared to the standard approaches

numerical identification of a spatially varying diffusion coefficient we consider the inverse problem of identifying a spatially varying diffusion coefficient on the basis of an observed solution to the forward problem under appropriate conditions this inverse problem can be solved as a first order hyperbolic problem in the unknown coefficient we provide a modified upwind difference scheme for this hyperbolic problem and prove that its convergence rate is 0 h when certain conditions are met

an inverse polynomial method for the identification of the leading coefficient in the sturm liouville operator from boundary measurements an inverse polynomial method of determining the unknown leading coefficient k k x of the linear sturm liouville operator au k x u x q x u x x 0 1 is presented as an additional condition only two measured data at the boundary x 0 x 1 are used in absence of a singular point u x 0 u x 0 x 0 1 the inverse problem is classified as a well conditioned if there exists at least one singular point then the inverse problem is classified as moderately ill conditioned u x 0 0 x 0 0 1 u x 0 x x 0 u x 0 x 0 1 and severely ill conditioned u x 0 u x 0 0 x 0 0 1 u x 0 u x 0 x x 0 for each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a cauchy problem for to the first order differential equation with respect the unknown function k k x an approximate analytical solution of the each cauchy problems are derived in explicit form numerical simulations all the above cases are given for noise free and noisy data an accuracy of the presented approach is demonstrated on numerical test solutions

high order numerical derivatives for one dimensional scattered noisy data based on the smoothing spline approximation in this paper we propose a regularization method for computing high order numerical derivatives from one dimensional noisy data the convergence rates under two different choices of the regularization parameter are obtained numerical examples show that the proposed method is effective and stable

identification of a temperature dependent heat conductivity from single boundary measurements considering the identification of a temperature dependent conductivity in a quasilinear elliptic heat equation from single boundary measurements we proof uniqueness in dimensions n ge 2 taking noisy data into account we apply tikhonov regularization in order to overcome the instabilities by using a problem adapted adjoint we give convergence rates under substantially weaker and more realistic conditions than required by the general theory our theory is supported by numerical tests

on the numerical radius and its applications abstract we give a brief account of the numerical radius of a linear bounded operator on a hilbert space and some of its better known properties both finite and infinite dimensional aspects are discussed as well as applications to stability theory of finite difference approximations for hyperbolic initial value problems

hexahedral volume coordinate method hvcm and improvements on 3d wilson hexahedral element abstract a new volume coordinate method for developing 3d hexahedral elements called hexahedral volume coordinate method hvcm is systematically established in this paper i several characteristic parameters of a hexahedron are defined and the degeneration conditions under which a hexahedron degenerates into other special polyhedrons are given ii the volume coordinates l 1 l 2 l 3 l 4 l 5 l 6 of any point within a convex hexahedron are defined iii transformation relations between the volume coordinates and the cartesian or isoparametric coordinates are presented iv the differential formulas for volume coordinates in hexahedral elements are given this new coordinate system can keep not only the advantages of local natural coordinate system but also a linear relation with the cartesian coordinate system then for checking the validity of the new hvcm it is used to formulate three new incompatible eight node hexahedral elements hvcc8 hvcc8 es and hvcc8 em by a similar procedure of famous wilson s incompatible mode numerical results show that the present elements exhibit much better performance than that of conventional isoparametric elements in most distorted mesh cases especially for macneal s thin beam problem it demonstrates that the new hvcm is a powerful tool for constructing high performance hexahedral finite element models

quadrative seminorms and jordan structures on algebras abstract we call a seminorm s on an algebraaquadrative if there is a positive such that s a2 s a 2 for all a in a if s is quadrative we ask what polynomial operations onaare continuous in the topology defined by s we show that when a polynomial lies in a certain large jordan algebra it is continuous this is true for example of xyx and x2y yx2 and it will follow that for some 0 and all a b in a s a b a s a 2 s b a n d s a 2 b b a 2 s a 2 s b we further show that if the polynomial lies outside any one of a certain set of jordan ideals then for some quadrative s ona continuity does not hold such ideals exist for instance for xy xy2 and y2x xy2

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

stable norms from theory to applications and back abstract the purpose of this survey paper is to give a brief review of certain aspects of stability of norms and subnorms acting on algebras over a field f either r or c a norm n on an associative algebra a over f shall be called stable if for some positive constant n a m n a m for all a a m 1 2 3 a norm shall be called strongly stable if the above inequality holds with 1 we begin the paper by discussing several results regarding norm stability including conditions under which norms on certain algebras are stable the second part of the paper is devoted to applications where we employ the notion of norm stability to obtain criteria for the convergence of a well known family of finite difference schemes for the initial value problem associated with the parabolic system u x t t 1 j k s a jk 2 u x t x j x k 1 j s b j u x t x j cu x t where ajk bj and c are constant matrices ajk being hermitian the third and last part of the paper deals with the question of stability for subnorms acting on subsets of power associative algebras that are closed under scalar multiplication and under raising to powers a subnorm f on such a set s is a real valued function satisfying f a 0 for all 0 a s and f a f a for all a s and f

descent methods for inverse problems of fundamental importance in modelling is the problem of recovering coe cient functions in a difierential equation from appropriate measurements on the solution thisisinvestigatedbymeansofsteepestdescenttechniquesusingcertainnew banach space gradients