iterative regularization for elliptic inverse problems elliptic inverse problems can be formulated using coefficient dependent energy least squares functionals resulting in a smooth convex objective functional a variational inequality emerges as a necessary and sufficient optimality condition the principle of iterative regularization when coupled with the auxiliary problem principle results in a strongly convergent scheme for the solution of elliptic inverse problems

the multi resolution method applied to the sideways heat equation abstract we consider the sideways heat equation u x x x t u t x t 0 x 1 t 0 the solution u x t on the boundary x 1 is a known function g t this is an ill posed problem since the solution if it exists does not depend continuously on the boundary i e small changes on the boundary may result in big changes in the solution in this paper we shall use the multi resolution method based on the shannon mra to obtain a well posed approximating problem and obtain an estimate for the difference between the exact solution and the solution of the approximating problem defined in v j

a quadratic convergence yielding iterative method for the implementation of lavrentiev regularization method for ill posed equations george and elmahdy 2012 considered an iterative method which converges quadratically to the unique solution is a nonlinear monotone operator defined on a real hilbert space x the convergence analysis of the method was based on a majorizing sequence in this paper we are concerned with the problem of expanding the applicability of the method considered by george and elmahdy 2012 by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular lipschitz type hypotheses considered in earlier studies such as george and elmahdy 2012 mahale and nair 2009 mathe and perverzev 2003 nair and ravishankar 2008 semenova 2010 and tautanhahn 2002 we show that the adaptive scheme considered by perverzev and schock 2005 for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate in the concluding section the method is applied to numerical solution of the inverse gravimetry problem

numerical solution to a sideways parabolic equation the inverse heat conduction problem can be considered to be a sideways parabolic equation in the quarter plane this is a model of a problem where one wants to determine the temperature on both sides of a thick wall but one side is inaccessible to measurements a numerical procedure for this severely ill posed problem is suggested which consists of two steps namely a mollification of the data and a marching difference scheme the numerical method is proved to be stable several computational results are presented and discussed copyright 2001 john wiley sons ltd

indentification of some source densities of the distribution type abstract in this paper we investigate the solvability of an ill posed two dimensional fredholm integral equation of the first kind which allows the solutions of distribution type the problem is first transformed into a well posed differential integral equation using output least squares approach with a regularization of bounded variations a globally convergent iterative method is proposed and some numerical results are presented the methodology discussed may be applied for the identification of the boundary shape of the defects of a dielectric material or the interface between different materials

the influence of the kinetic energy in equilibrium of hamiltonian systems we provide a simple and explicit example of the influence of the kinetic energy in the stability of the equilibrium of classical hamiltonian systems of the type h q p q we construct a potential energy of class ck with a critical point at 0 and two different positive defined matrices b1andb2 both independent of q and show that the equilibrium 0 0 is stable according to lyapunov for the hamiltonian h1 q while for h2 q the equilibrium is unstable moreover we give another example showing that even in the analytical situation the kinetic energy has influence in the stability in the sense that there is an analytic potential energy and two kinetic energies also analytic t1 and t2 such that the attractive basin of 0 0 is a two dimensional manifold in the system of hamiltonian t1 and a one dimensional manifold in the system of hamiltonian t2

the mollification method and the numerical solution of the inverse heat conduction problem by finite differences abstract the inverse heat conduction problem involves the calculation of surface heat flux and or temperature histories from transient measured temperatures inside solids we consider the one dimensional semi infinite linear case and present a new solution algorithm based on a data filtering interpretation of the mollification method that automatically determines the radius of mollification depending on the amount of noise in the data and finite differences a fully explicit and stable space marching scheme is developed we describe several numerical experiments of interest showing that the new procedure is accurate and stable with respect to perturbations in the data even for small dimensionless time steps

on the balancing principle for some problems of numerical analysis we discuss a choice of weight in penalization methods the motivation for the use of penalization in computational mathematics is to improve the conditioning of the numerical solution one example of such improvement is a regularization where a penalization substitutes an ill posed problem for a well posed one in modern numerical methods for pdes a penalization is used for example to enforce a continuity of an approximate solution on non matching grids a choice of penalty weight should provide a balance between error components related with convergence and stability which are usually unknown in this paper we propose and analyze a simple adaptive strategy for the choice of penalty weight which does not rely on a priori estimates of above mentioned components it is shown that under natural assumptions the accuracy provided by our adaptive strategy is worse only by a constant factor than one could achieve in the case of known stability and convergence rates finally we successfully apply our strategy for self regularization of volterra type severely ill posed problems such as the sideways heat equation and for the choice of a weight in interior penalty discontinuous approximation on non matching grids numerical experiments on a series of model problems support theoretical results

a mollified space marching finite differences algorithm for the inverse heat conduction problem with slab symmetry abstract the one dimensional inverse heat conduction problem ihcp for a slab is considered a new solution algorithm based on a data filtering interpretation of the mollification method is presented and a fully explicit space marching finite differences scheme is developed after showing the numerical stability of the algorithm the efficiency of the method is demonstrated by means of several examples the numerical results show the procedure to be very useful in the presence of noisy data even for some nonlinear cases

wavelet regularization for an inverse heat conduction problem in this paper we consider an inverse heat conduction problem which appears in some applied subjects this problem is ill posed in the sense that the solution if it exists does not depend continuously on the data the meyer wavelets are applied to formulate a regularized solution which is convergent to exact one on an acceptable interval when data error tends to zero

the interior radon transform the interior radon transform arises from a limited data problem in computerized tomography when only rays travelling through a specified region of interest are measured this problem occurs due to technical restrictions of the sampling apparatus or in an endeavour to reduce the x ray dose the corresponding operator mathcal r i is investigated as a mapping between weighted l 2 spaces the main result is a singular value decomposition svd for this operator for functions of unbounded support in mathbb r 2 the proof is based on the construction of intertwining differential operators the techniques used are unified in the sense that svds for other radon transforms with rotational symmetry can easily be derived in the same way consistency conditions are also obtained for mathcal r i for functions of bounded and unbounded support many of the results generalize to higher dimensions they are stated whenever they follow directly from the two dimensional case

determining surface temperatures from interior observations we consider the inverse heat transfer problem for the case of the one dimensional heat equation in a quarter plane and present a new solution algorithm together with error bounds of logarithmic convexity type our method is based on regularizing the initial value problem for the heat equation run sideways we describe a numerical experiment with rapidly varying temperature profiles and explain how the method can be extended to problems with temperature dependent thermal properties

numerical differentiation from a viewpoint of regularization theory in this paper we discuss the classical ill posed problem of numerical differentiation assuming that the smoothness of the function to be differentiated is unknown using recent results on adaptive regularization of general ill posed problems we propose new rules for the choice of the stepsize in the finite difference methods and for the regularization parameter choice in numerical differentiation regularized by the iterated tikhonov method these methods are shown to be effective for the differentiation of noisy functions and the order optimal convergence results for them are proved

an a posteriori parameter choice for ordinary and iterated tikhonov regularization of ill posed problems leading to optimal convergence rates on propose un choix a posteriori du parametre pour la regularisation de tikhonov ordinaire et iteree des problemes mal poses conduisant des vitesses de convergence optimales

sharp adaptation for inverse problems with random noise we consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems in particular multivariate ones we propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellipsoid of functions within a wide scale that includes ellipoids with polynomially and exponentially decreasing axes and at the same time satisfies asymptotically exact oracle inequalities within any class of linear estimates having monotone non increasing weights the construction of the estimator is based on a properly penalized blockwise stein s rule with weakly geometically increasing blocks as an application we construct sharp adaptive estimators in the problems of deconvolution and tomography

the space marching solution of the inverse heat conduction problem and the identification of the initial temperature distribution abstract it is shown that the usual data for the inverse heat conduction problem ihcp uniquely defines the initial condition for the direct problem in a one dimensional finite slab a suitable model problem is also analyzed for the semi infinite conductor case a space marching method is implemented to numerically solve both the ihcp and the identification of the initial temperature distribution several examples of interest are presented

a simple variational approach to a converse of the lagrange dirichlet theorem peter hagedorn 8r jean mawhin communicated by k kigc g ssner 1 introduction the lagrange dirichlet stability theorem states that the equilibrium posi tion of a discrete conservative mechanical sytem is stable in the sense of liapunov if the potential energy u assumes a strict local minimum at this equilibrium we know from painzev s example 22 see also 37 that this sufficient condition stated by lagranoe 13 and proved by lejevne dirich et 17 is not necessary for stability and the history of the inversion of the lagrange dirichlet theorem i e of the search for necessary conditions for stability is rich and interesting we refer to 27 for a discussion and a bibliography and to the more recent references 1 3 6 12 14 16 18 20 21 23 26 28 32 35 36 in 1971 ha6edorn proved for systems of class c 2 that the equilibrium position is unstable if the potential energy u has a strict maximum at this equilibrium 4 5 this result was extended in various directions by ta l fer o 30 bomtin kozlov 2 kozmv 10 liubushin 18 and sofer 29 a common feature of all those proofs initiated in 4 is the application of some method of the calculus of variations namely the theory of geodesics on finsler manifolds to jacobi s form of the principle of least action this leads to technical difficulties which are overcome in the above mentioned papers through a lot of effort and ingenuity the aim of this short paper is to show that the replacement of jacobi s principle of least action by a variant of another variational principle recently introduced by van groesen 33 34 in the study of normal mode vibrations of conservative nonlinear systems allows a simple and elementary treatment in the setting of a hilbert space instead of a finsler manifold of the instabili ty theorem stated in 4 with the minimal regularity assumptions of 18 29 30 in this way the problem is reduced to proving the existence of a minimum for a coercive functional defined on a weakly closed subset of a hilbert space a problem solved at the beginning of any monograph on the direct methods of the calculus of variations see e g 19 the reduction

a note on a cauchy problem for the laplace equation regularization and error estimates abstract in this paper a cauchy problem for the laplace equation is investigated based on the fundamental solution to the elliptic equation we propose to solve this problem by the truncation method which generates well posed problem then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proved error estimates for this method are provided together with a selection rule for the regularization parameter the numerical results show that our proposed numerical methods work effectively this work extends to earlier results in qian et al 2008 14 and hao et al 2009 5

on the stable analytic continuation with a condition of uniform boundedness it is shown that if h x is any continuous function defined on some interval a b 1 1 of the real axis then in general its best l2 approximant in the class of functions holomorphic and bounded by unity in the unit disk of the complex plane is a finite blaschke product an upper bound is placed on the number of factors of the latter and a method for its construction is given the paper contains a discussion of the use of these results in performing a stable analytic continuation of a set of data points under a condition of uniform boundedness as well as some numerical examples

the approximate inverse in action ii convergence and stability the approximate inverse is a scheme for constructing stable inversion formulas for operator equations originally it is defined on l2 spaces in the present article we extend the concept of approximate inverse to more general settings which allow us to investigate the discrete version of the approximate inverse which actually underlies numerical computations indeed we show convergence if the discretization parameter tends to zero further we prove stability that is we show the regularization property finally we apply the results to the filtered backprojection algorithm in 2d tomography to obtain convergence rates

the truncation method for a two dimensional nonhomogeneous backward heat problem abstract we consider the backward heat problem u t u xx u yy f x y t x y t 0 t u x y t g x y x y with the homogeneous dirichlet condition on the rectangle 0 0 where the data f and g are given approximately the problem is severely ill posed using the truncation method for fourier series we propose a simple regularized solution which not only works on a very weak condition on the exact data but also attains due to the smoothness of the exact solution explicit error estimates which include the approximation ln 1 3 2 in h2 some numerical examples are given to illuminate the effect of our method

a new regularization method for the time fractional inverse advection dispersion problem in this paper we consider the time fractional inverse advection dispersion problem tfiadp in a quarter plane the solute concentration and dispersion flux are sought from a measured concentration history at a fixed location inside the body such a problem is obtained from the classical advection dispersion equation by replacing the first order time derivative with the caputo fractional derivative of order alpha 0 alpha 1 we show that the tfiadp is severely ill posed and we further apply a new regularization method to solve it based on the solution given by the fourier method convergence estimates are presented under a priori bound assumptions for the exact solution finally one numerical example is given to show that the proposed numerical method is effective

multilevel augmentation algorithms based on fast collocation methods for solving ill posed integral equations abstract this paper continues the theme of the recent work chen et al 2008 18 in which fast collocation methods are introduced for solving ill posed fredholm integral equations of the first kind we develop in this paper multilevel augmentation algorithms which lead to fast solutions of the discrete equations resulting from fast collocation methods regularization parameter choice strategies are given for proposed methods the theoretical analysis and numerical experiments illustrate the accuracy and efficiency of the algorithm

self regularization by projection for noisy pseudodifferential equations of negative order it is well known that pseudodifferential equations of negative order considered in sobolev spaces with small smoothness indices are ill posed on the other hand it is known that efficient discretization schemes with properly chosen discretization parameters allow to obtain a regularization effect for such equations the main accomplishment of the present paper is the principle for the adaptive choice of the discretization parameters directly from noisy discrete data we argue that the combination of this principle with wavelet based matrix compression techniques leads to algorithms which are order optimal in the sense of complexity

approximate inverse method for stable analytic continuation in a strip domain numerical analytic continuation is in general an ill posed problem and some special regularization methods are needed in this paper we apply the approximate inverse method to deal with the problem in a strip domain w z x iy c x r 0 y

an integral solution for the inverse heat conduction problem after the method of weber abstract we consider the linear inverse heat conduction problem ihcp in the one dimensional semi infinite slab and present a new solution algorithm and error bounds based on the integral representation of weber s hyperbolic approximation and a suitable filtered version of the noisy data we discuss the determination of the hyperbolic parameter and describe several numerical examples of interest showing the accuracy and stability of the method

optimality of the fully discrete filtered backprojection algorithm for tomographic inversion although the filtered backprojection algorithm fba has been the standard reconstruction algorithm in 2d computerized tomography for more than 30 years its convergence behavior is not completely settled so far relying on convergence results by rieder and faridani for the semi discrete fba siam j numer anal 41 3 869 892 2003 we show optimality of the fully discrete version for reconstructing sufficiently smooth density distributions further we introduce mfba a modified version of fba and prove its optimality under weaker smoothness requirements remarkably mfba may have a larger convergence order in the angular than in the lateral variable thus allowing optimal convergence in case of angular under sampling moreover mfba can be seen as a limit of the phantom view method introduced to increase angular resolution

a note on sideways heat equation and wavelets and constant e abstract sideways heat equation and wavelets 1 written by reginska is one of the earliest important literatures about solving ill posed problem by wavelet regularization but there exists a defect in its main conclusion the aim of this paper is to modify it and to provide a complete proof from which a new interesting constant e is introduced

a finite dimensional realization of the mollifier method for compact operator equations we introduce and analyze a stable procedure for the approximation of f if where f is the least residual norm solution of the minimal norm of the ill posed equation af g with compact operator a x y between hilbert spaces and x has some smoothness assumption our method is based on a finite number of singular values of a and some finite rank operators our results are in a more general setting than the one considered by rieder and schuster 2000 and nair and lal 2003 with special reference to the mollifier method and it is also applicable under fewer smoothness assumptions on

combined mollification future temperatures procedure for solution of inverse heat conduction problem abstract the inverse heat conduction problem involves the calculation of surface heat flux and or temperature histories from transient measured temperatures inside solids this paper proposes and investigates a new combined procedure that is based on two different methods a data filtering interpretation of the mollification method and beck s future temperature method a test case is investigated of a semi infinite body exposed to a heat flux that is initially zero has a unit increase an then drops to zero the combined procedure is shown to be accurate and stable with respect to perturbations in the data even for small dimensionless time steps the future temperatures method can be significantly improved upon by the combined procedure

fourier regularization method of a sideways heat equation for determining surface heat flux we consider a non standard inverse heat conduction problem in a quarter plane which appears in some applied subjects we want to know the surface heat flux in a body from a measured temperature history at a fixed location inside the body this is an exponentially ill posed problem in the sense that the solution if it exists does not depend continuously on the data a fourier regularization method together with order optimal logarithmic stability estimates is given a numerical example shows that the theoretical results are valid

the method of simplified tikhonov regularization for dealing with the inverse time dependent heat source problem this paper investigates the inverse problem of determining a heat source using a parabolic equation where data are given at some fixed location the problem is ill posed i e the solution if it exists does not depend continuously on the data a simplified tikhonov regularization method is given and an order optimal stability estimate is obtained a numerical example shows that the regularization method is effective and stable

wavelet and error estimation of surface heat flux we consider the determination of the surface heat flux of a body from a measured temperature history at a fixed location inside the body mathematically it is formulated as a problem for the one dimensional heat equation in a quarter plane with data given along the line x 1 and the gradient of the solution is wanted for 0 x 1 this problem is severely ill posed and there are few theoretic results the results available in the literatures are mainly devoted to the case of determining surface temperature the aim of this paper is to indicate a possibility of a new approach for solving this problem based on an application of wavelet basis decomposition of measured data the error estimate with holder type is given and the convergence of regularization approximation solution at zero is also obtained which is a difficult problem left over by some authors

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

recent advances in wavelet analysis this book covers recent advances in wavelet analysis and applications in areas including wavelets on bounded intervals wavelet decomposition of special interest to statisticians wavelets approach to differential and integral equations analysis of subdivision operators and wavelets related to problems in engineering and physics

a modified tikhonov regularization for stable analytic continuation this paper is devoted to a new regularization method for solving the numerical analytic continuation of an analytic function f z f x iy on a strip domain omega z x iy in mathbb c mid x in mathbb r 0 y y 0 where the data is given approximately only on the line y 0 this problem is severely ill posed and has important practical applications the theoretical optimal error bound for the problem is proved which is independent of the selected regularization methods a modified tikhonov regularization method with asymptotic order optimal error estimates is proposed this method can be numerically implemented easily by the fast fourier transform some numerical examples are provided and a comparison with a fourier regularization method is given which show the modified tikhonov method works very well

incomplete data problems in x ray computerized tomography in the present paper we study truncated projections for the fanbeam geometry in computerized tomography first we derive consistency conditions for the divergent beam transform then we study a singular value decomposition for the case where only the interior rays in the fan are provided as for example in region of interest tomography we show that the high angular frequency components of the searched for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition

combined function specification regularization procedure for solution of inverse heat conduction problem methode combinee basee sur 2 methodes differentes la methode de specification sequentielle proposee par beck et la methode de regularisation utilisee par tikhonov

the approximate inverse in action with an application to computerized tomography the approximate inverse is a scheme to obtain stable numerical inversion formulae for linear operator equations of the first kind yet in some applications the computation of a crucial ingredient the reconstruction kernel is time consuming and instable it may even happen that the kernel does not exist for a particular semidiscrete system to cure this dilemma we propose and analyze a technique that is based on a singular value decomposition of the underlying operator the results are applied to the reconstruction problem in 2d computerized tomography where they enable the design of reconstruction filters and lead to a novel error analysis of the filtered backprojection algorithm

incomplete data problems in x ray computerized tomography i singular value decomposition of the limited angle transform etude des problemes intrinseques relatifs aux donnees incompletes dans la tomographie aux rayons x assistee par ordinateur decomposition de la transformation de radon

sideways heat equation and wavelets the sideways heat equation is considered as a model ill posed problem the aim of this paper is to indicate a possibility of a new approach for solving this problem based on an application of wavelet basis decomposition of measured data the meyer wavelets are applied to formulate a regularized solution which is convergent to the exact one when a data error tends to zero

two regularization methods and the order optimal error estimates for a sideways parabolic equation in this paper we consider an inverse heat conduction problem which appears in some applied subjects this problem is ill posed in the sense that the solution if it exists does not depend continuously on the data a tikhonov type s regularization method and a fourier regularization method are applied to formulate regularized solutions which are stably convergent to the exact ones with order optimal error estimates a numerical example shows that the computational effect of these methods are all satisfactory

wavelets and regularization of the sideways heat equation in this paper the following inverse heat conduction problem ut uxx x 0 t 0 u x 0 0 x 0 u 1 t g t t 0 u x is bounded is considered again this problem is severely ill posed its solution if it exists does not depend continuously on the data a small perturbation in the data may cause a dramatically large error in the solution for 0 x 1 in this paper a new wavelet regularization method for this problem is given moreover we can easily find the regularization parameter j such that some sharp stable estimates between the exact solution and the approximate one in hr r norm meaning is given

adaptive estimation of linear functionals in hilbert scales from indirect white noise observations we consider adaptive estimating the value of a linear functional from indirect white noise observations for a flexible approach the problem is embedded in an abstract hilbert scale we develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the hilbert scale it is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals the case of discretized indirect white noise observations is studied and the adaptive estimator in this setting is developed

a mollification framework for improperly posed problems various authors have explored the role of mollification in the solution of improperly posed problems such as numerical differentiation and the numerical solution of first kind integral equations for example murio 13 exploiting earlier suggestions of manselli and miller 11 as well as vasin 16 has shown how mollified numerical differentiators can be constructed for the differentiation of observational data but the underlying regularity requires the existence of the second derivative of the function being differentiated on the other hand hao et al 4 proposed and examined a mollification procedure where the regularity reduced to only requiring the existence of the fractional derivative being evaluated independently anderssen and de hoog 2 examined the use of multi point moving average finite difference formulas for the numerical differentiation of observational data and show how the length of such formulas must be related to the size of the discretization of the data before convergence and stability bounded amplification error can be guaranteed as the size of the discretization is decreased this represents an explicit exemplification of the situation which occurs in mollification where some appropriate measure of the extent of the mollification must be related in some suitable way to the discretization of the data before convergence and stability results can be derived however to date such investigations have not explicitly exploited the natural translation and dilation properties of fractional differentiation operators the purpose of this paper is to show how using the classical semi group theory of hille and phillips 8 a quite general theory can be constructed for the mollification of operator equations

simplified tikhonov and fourier regularization methods on a general sideways parabolic equation the inverse heat conduction problem ihcp can be considered to be a sideways parabolic equation in the quarter plane and now the results available in the literature on ihcp mainly devoted to the standard sideways heat equation numerical methods have been developed also for more general equations but in most cases the stability theory and convergence proofs have not been generalized accordingly this paper remedies this by a simplified tikhonov and a new fourier regularization methods on a general sideways parabolic equation some known results for sideways heat equation are only the special case of the conclusions in this paper the numerical example shows that the computation effect is satisfactory

parameter selection by discrete mollification and the numerical solution of the inverse heat conduction problems abstract a procedure for the numerical solution of the one dimensional inverse heat conduction problem based on the computaion of the solution associated with a suitable filtered version of the noisy data by discrete mollification is presented and a parameter choice criterion which automatically determines the radius of mollification as a function of the amount of noise in the data is introduced several numerical examples of interest are also analyzed showing the accuracy and stability properties of the method

on the influence of the kinetic energy on the stability of equilibria of natural lagrangian systems natural lagrangian systems t on r2 described by the equation frac d dt frac partial t partial dot q frac partial t partial q frac partial pi partial q are considered where t q q is a positive definite quadratic form in q and q has a critical point at 0 it is constructively proved that there exist a c potential energy and two c kinetic energies t and t such that the equilibrium q t 0 is stable for the system t and unstable for the system t equivalently it is established that for c natural systems the kinetic energy can influence the stability in the analytic category this is not true

fourth order modified method for the cauchy problem for the laplace equation we consider the cauchy problem for the laplace equation in the half plane x 0 y r where the cauchy data is given at x 0 and the solution is sought in the interval 0 x 1 the problem is ill posed the solution if it exists does not depend continuously on the data in order to solve the problem numerically it is necessary to modify the equation so that a bound on the solution is imposed we study a modification of the equation where a fourth order mixed derivative term is added error estimates for this equation are given which show that the solution of the modified equation is an approximation of the solution of the cauchy problem for the laplace equation and it is shown that when the data error tends to zero the error in the approximate solution tends to zero logarithmically numerical implementation is considered and a simple example is given

numerical approximation of a tikhonov type regularizer by a discretized frozen steepest descent method abstract we present a frozen regularized steepest descent method and its finite dimensional realization for obtaining an approximate solution for the nonlinear ill posed operator equation f x y the proposed method is a modified form of the method considered by argyros et al 2014 the balancing principle considered by pereverzev and schock 2005 is used for choosing the regularization parameter the error estimate is derived under a general source condition and is of optimal order the provided numerical example proves the efficiency of the proposed method

analysis of finite element domain embedding methods for curved domains using uniform grids we analyze the error of a finite element domain embedding method for elliptic equations on a domain omega with curved boundary the domain is embedded in a rectangle r on which uniform mesh and linear continuous elements are employed the numerical scheme is based on an extension of the differential equation from omega to r by regularization with a small parameter epsilon for neumann and robin problems or penalty with a large parameter epsilon 1 for the dirichlet problem or a mixture of these for a mixed boundary value problem for neumann and robin problems we prove that when epsilon le h the mesh size the error in the h 1 omega norm is of the optimal order mathcal o h for the dirichlet problem when epsilon le h 1 2 the error is mathcal o h 1 2 that is not optimal if the mesh is adjusted around partial omega to fit it then the optimal convergence rate mathcal o h holds for the dirichlet problem if omega is convex and epsilon le h if omega is not convex then the convergence rate can only be improved to mathcal o h 2 3 by such mesh adjustment with the parameter being epsilon h 2 3 in this latter case a parameter smaller than h 2 3 thwarts the convergence rate which is verified by a numerical result

parameter choice strategies for least squares approximation of noisy smooth functions on the sphere we consider a polynomial reconstruction of smooth functions from their noisy values at discrete nodes on the unit sphere by a variant of the regularized least squares method of an et al siam j numer anal 50 2012 pp 1513 1534 as nodes we use the points of a positive weight cubature formula that is exact for all spherical polynomials of degree up to 2m where m is the degree of the reconstructing polynomial we first obtain a reconstruction error bound in terms of the regularization parameter and the penalization parameters in the regularization operator then we discuss a priori and a posteriori strategies for choosing these parameters finally we give numerical examples illustrating the theoretical results

fast multiscale galerkin methods for solving ill posed integral equations via a coupled system under general source conditions abstract fast multiscale regularized galerkin methods for solving fredholm integral equations of the first kind are investigated we apply a multiscale galerkin method with a matrix compression strategy to discretize the integral equation of the first kind and obtain a sparse system to dealing with the ill posedness and keeping sparsity of the discrete system we use an equivalent coupled system form of the tikhonov regularization then the multilevel augmentation method is applied to solve the coupled system a priori and a posteriori regularization parameter choice strategies are proposed convergence rates of the regularized solutions are established under general source conditions numerical results are presented to demonstrate the efficiency and accuracy of the proposed methods

on the adaptive selection of the parameter in regularization of ill posed problems we study the possibility of using the structure of the regularization error for a posteriori choice of the regularization parameter as a result a rather general form of a selection criterion is proposed and its relation to the heuristical quasi optimality principle of tikhonov and glasko z vychisl mat mat fiz 4 1964 pp 564 571 and to an adaptation scheme proposed in a statistical context by lepskii theory probab appl 36 1990 pp 454 466 is discussed the advantages of the proposed criterion are illustrated by using such examples as self regularization of the trapezoidal rule for noisy abel type integral equations lavrentiev regularization for nonlinear ill posed problems and an inverse problem of the two dimensional profile reconstruction

determining surface temperature and heat flux by a wavelet dual least squares method we apply a wavelet dual least squares method to a general sideways parabolic equation for determining surface temperature and surface heat flux connecting meyer wavelet bases with a special project method dual least squares method we can obtain a regularized solution meanwhile order optimal error estimates between the approximate solution and exact solution are proved

optimal error bound and fourier regularization for identifying an unknown source in the heat equation in this paper we investigate a problem of the identification of an unknown source from one supplementary temperature measurement at a given instant of time for the transient heat equation under an a priori condition we answer the question concerning the best possible accuracy for the problem the fourier regularization method is utilized for solving the problem and its convergent rate is analyzed numerical results are presented to illustrate the accuracy and efficiency of the method

stability for p laplace type equation in a borderline case abstract we study the dirichlet problem for a p laplacian type operator in the setting of the orlicz zygmund space q log r n q 1 and 0 more precisely our aim is to establish under which assumptions on 0 existence and uniqueness of the solution are assured and to prove continuity and stability of the associated nonlinear operator

wavelet and fourier methods for solving the sideways heat equation we consider an inverse heat conduction problem the sideways heat equation which is a model of a problem where one wants to determine the temperature on both sides of a thick wall but where one side is inaccessible to measurements mathematically it is formulated as a cauchy problem for the heat equation in a quarter plane with data given along the line x 1 where the solution is wanted for 0 leq x 1 the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data we consider stabilizations based on replacing the time derivative in the heat equation by wavelet based approximations or a fourier based approximation the resulting problem is an initial value problem for an ordinary differential equation which can be solved by standard numerical methods e g a runge kutta method we discuss the numerical implementation of fourier and wavelet methods for solving the sideways heat equation theory predicts that the fourier method and a method based on meyer wavelets will give equally good results our numerical experiments indicate that also a method based on daubechies wavelets gives comparable accuracy as test problems we take model equations with constant and variable coefficients we also solve a problem from an industrial application with actual measured data

on the instability of equilibrium when the potential has a non strict local minimum instability is studied for a mechanical system with two degrees of freedom whose potential has a non strict local minimum at the origin under suitable conditions of smoothness it is shown that the corresponding equilibrium is unstable when the potential has a local minimum along a curve passing through the origin as a particular case we get instability if the potential is analytic and has a non isolated minimum at the origin these results improve those of g hamel 1 and silla 5 on the same subject

a galerkin least squares approach for photoacoustic tomography the development of fast and accurate image reconstruction algorithms is a central aspect of computed tomography in this paper we address this issue for photoacoustic computed tomography in circular geometry we investigate the galerkin least squares method for that purpose for approximating the function to be recovered we use subspaces of translation invariant spaces generated by a single function this includes many systems that have previously been employed in photoacoustic tomography such as generalized kaiser bessel basis functions or the natural pixel basis by exploiting an isometry property of the forward problem we are able to efficiently set up the galerkin equation for a wide class of generating functions and devise efficient algorithms for its solution we establish a convergence analysis and present numerical simulations that demonstrate the efficiency and accuracy of the derived algorithm

instability of a non isolated equilibrium in this paper non isolated equilibria of conservative mechanical systems are studied under mild non degeneracy conditions such equilibria are shown to be unstable in the sense of liapunov

wavelet and spectral regularization methods for a sideways parabolic equation we consider a special sideways parabolic equation which appears in some applied subjects the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data in this paper some wavelet and spectral regularization methods for this problem are given the error estimates are also established respectively

mobius deconvolution on the hyperbolic plane with application to impedance density estimation in this paper we consider a novel statistical inverse problem on the poincare or lobachevsky upper complex half plane here the riemannian structure is hyperbolic and a transitive group action comes from the space of 2 x 2 real matrices of determinant one via mobius transformations our approach is based on a deconvolution technique which relies on the helgason fourier calculus adapted to this hyperbolic space this gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random mobius transform a motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the poincare plane exactly describes the physical system that is of statistical interest

simultaneous multiresolution strategies for nonrigid image registration multiresolution strategies are commonly used in the nonrigid registration to avoid local minima in the optimization space generally a step by step hierarchical approach is adopted in which the registration starts on a level with reduced complexity downsampled images global transformations then continuing to levels with increased complexity until the finest level is reached in this paper we propose two alternative multiresolution strategies for both the data and transformation models in which different resolution levels are considered simultaneously instead of subsequently through combining the different strategies for data and transformation we systematically define 3 3 multiresolution schemes including both existing and novel methods experiments on 10 pairs of computed tomography lung data sets showed that the best performing strategy resulted in a reduction of the upper quartile of the mean target registration error from 2 to 1 5 mm compared with the conventionally hierarchical multiresolution method while achieving smoother deformations experiments with intersubject registration of 18 3d t1 weighted mri brain scans confirmed that simultaneous multiresolution strategies produce more accurate registration results median of mean overlap increased from 0 55 to 0 57 and smoother deformation fields than the traditionally hierarchical method evaluation of robustness indicated that the largest differences in accuracy between methods are observed for structures with a relatively large initial misalignment

spectrally formulated wavelet finite element for wave propagation and impact force identification in connected 1 d waveguides in this paper a novel wavelet based spectral finite element is developed for studying elastic wave propagation in 1 d connected waveguides first the partial differential wave equation is converted to simultaneous ordinary differential equations odes using daubechies wavelet approximation in time these odes are then solved using finite element fe technique by deriving the exact interpolating function in the transformed domain spectral element captures the exact mass distribution and thus the system size required is very much smaller then conventional fe the localized nature of the compactly supported daubechies wavelet allows easy imposition of initial boundary values this circumvents several disadvantages of the conventional spectral element formulation using fast fourier transforms fft particularly in the study of transient dynamics the proposed method is used to study longitudinal and flexural wave propagation in rods beams and frame structures numerical experiments are performed to show the advantages over fft based spectral element methods the efficiency of the spectral formulation for impact force identification is also demonstrated

numerical identification of interface source functions in heat transfer problems under nonlinear boundary conditions abstract an inverse approach based on a fully explicit and unconditionally stable finite difference implementation of the mollification method for the approximate identification of transient interface sources in semi infinite or finite slab mediums subjects to nonlinear boundary conditions is presented stability bounds are derived an two typical examples corresponding to newton cooling law and stefan boltzmann radiation law are illustrated

finite dimensional realization of a quadratic convergence yielding iterative regularization method for ill posed equations with monotone operators recently jidesh et al 2015 considered a quadratic convergence yielding iterative method for obtaining approximate solution to nonlinear ill posed operator equation f x y where f d f x x is a monotone operator and x is a real hilbert space in this paper we consider the finite dimensional realization of the method considered in jidesh et al 2015 numerical example justifies our theoretical results

central difference schemes in time and error estimate on a non standard inverse heat conduction problem inverse heat conduction problems ihcp are severely ill posed in the sense that the solution if it exists does not depend continuously on the data but now the results available in the literature on ihcp are mainly devoted to the standard sideways heat equation this paper remedies this by a central difference schemes in time which itself has a regularizing effect for a non standard ihcp which appears in some applied subjects an error estimate is obtained and the error estimate also gives information about how to choose the step length in the time discretization a numerical example shows that the computational effect of this method is satisfactory

singular value decompositions and inversion methods for the exterior radon transform and a spherical transform abstract the classical radon transform r maps an integrable function in rn to its integrals over all n 1 dimensional hyperplanes and the exterior radon transform is the transform r restricted to hyperplanes that do not intersect a given disc a singular value decomposition for the exterior transform is given for spaces of square integrable functions on the exterior of the disc this decomposition in orthogonal functions explicitly produces the null space and range of the exterior transform and gives a new method for inverting the transform modulo the null space a modification of this method is given that will exactly invert functions of compact support these results generalize theorems of r m perry and the author a singular value decomposition for the radon transform that integrates over spheres in rn containing the origin is also given this follows from the singular value decomposition for r and yields the null space and a new inversion method for this transform

the approximate inverse in action iii 3d doppler tomography the approximate inverse is a powerful tool for solving first kind operator equations in a stable way its abstract convergence and stability theory developed in our articles siam j numer anal 37 1909 1929 2000 and math comp 72 1399 1415 2003 is applied to the reconstruction problem of 3d vector field tomography resulting in a reconstruction algorithm of filtered backprojection type for an analytically computed reconstruction filter reconstruction kernel convergence with rates as well as the regularization property are established

expanding the applicability of a modified gauss newton method for solving nonlinear ill posed problems we expand the applicability of a modified gauss newton method recently presented in george 2013 19 for approximate solution of a nonlinear ill posed operator equation between two hilbert spaces we use a center type lipschitz condition in our convergence analysis instead of a lipschitz type condition used in earlier studies such as george 2013 2010 19 18 this way a tighter convergence analysis is obtained and under less computational cost since the more precise and easier to compute center lipschitz instead of the lipschitz constant is used in the convergence analysis numerical examples are presented to show that our results apply but earlier ones do not apply to solve equations

wavelets and regularization of the cauchy problem for the laplace equation abstract in this paper a cauchy problem for two dimensional laplace equation in the strip 0 x 1 is considered again this is a classical severely ill posed problem i e the solution if it exists does not depend continuously on the data a small perturbation in the data can cause a dramatically large error in the solution for 0 x 1 the stability of the solution is restored by using a wavelet regularization method moreover some sharp stable estimates between the exact solution and its approximation in h r r norm is also provided

central difference method of a non standard inverse heat conduction problem for determining surface heat flux from interior observations in this paper we consider a non standard inverse heat conduction problem in a quarter plane we want to know the surface heat flux from a measured temperature history at a fixed location inside the body this is a severely ill posed problem in that arbitrarily small differences in the input temperature can result in arbitrarily large differences in the surface flux a central difference schemes in time can restore the stability of heat flux solution and error estimate can be obtained a numerical experiment shows that the method works effectively

newton lavrentiev regularization for ill posed operator equations in hilbert scales in this paper we consider the two step method for approximately solving the ill posed operator equation f x f where f d f x x is a nonlinear monotone operator defined on a real hilbert space x in the setting of hilbert scales we derive the error estimates by selecting the regularization parameter a according to the adaptive method considered by pereverzev and schock in 2005 when the available data is f d with f f email protected x is of optimal order