identification of parameters in the 2 d ihcp abstract we present a numerical marching scheme based on mollification and the general cross validation methods to determine the diffusivity coefficient as well as the temperature and heat flux histories in the two dimensional inverse heat conduction problem when noisy data at the active boundary and the initial measured temperature distribution are given no information about the quality and or quantity of the noise is assumed a proof of stability and convergence of the algorithm together with several numerical examples are provided

on stable numerical differentiation based on a regularized volterra equation two different approaches for numeri cal differentiation are considered the first approach consists of solving a regularized volterra equation while the second approach is based on solving a disretized version of the regularized volterra equation numerical experiments show that these methods are efficient and compete fa vorably with the variational regularization method for stable calculating the derivatives of noisy functions

solving set optimization problems based on the concept of null set abstract we introduce the concept of null set in the hyperspace that is consisting of all nonempty subsets of a given normed space based on the concepts of null set and convex cone we can define two partial orderings according to the algebraic difference and hukuhara difference between any two elements in the hyperspace which will be used to define the solution concepts of set optimization problems on the other hand we transform set optimization problems into a conventional vector optimization problem under these settings we can apply the technique of scalarization to solve this transformed vector optimization problem finally we show that the optimal solution of scalarized problem is also the optimal solution of original set optimization problem

a wavelet shrinkage approach to tomographic image reconstruction abstract a method is proposed for reconstructing images from tomographic data with respect to a two dimensional wavelet basis the wavelet vaguelette decomposition wvd is used as a framework within which expressions for the necessary wavelet coefficients may be derived these coefficients are calculated using a version of the filtered back projection algorithm as a computational tool in a multiresolution fashion the necessary filters are defined in terms of the underlying wavelets denoising is accomplished through an adaptation of the wavelet shrinkage ws approach of donoho et al and amounts to a form of regularization combining these two steps yields the proposed wvd ws reconstruction algorithm which is compared to the traditional filtered backprojection method in a small study

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

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

automatic numerical differentiation by discrete mollification abstract a new very simple totally automated and powerful technique for numerical differentiation based on the computation of the derivative of a suitable filtered version of the noisy data by discrete mollification is presented several numerical examples of interest are also analyzed

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

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

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

four types of nonlinear scalarizations and some applications in set optimization abstract there are two types of criteria of solutions for the set valued optimization problem the vectorial criterion and set optimization criterion the first criterion consists of looking for efficient points of set valued map and is called set valued vector optimization problem on the other hand kuroiwa tanaka ha started developing a new approach to set valued optimization which is based on comparison among values of the set valued map in this paper we treat the second type criterion and call set optimization problem the aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships these scalarizing functions are generalization of tammer weidner s scalarizing functions for vectors as applications of the scalarizing functions for sets we present nonconvex separation type theorems gordan s type alternative theorems for set valued map optimality conditions for set optimization problem and takahashi s minimization theorems for set valued map

on the stable numerical evaluation of caputo fractional derivatives the computation of caputo s fractional derivatives in the presence of measured data is considered as an ill posed problem and treated by mollification techniques it is shown that with the appropriate choice of the radius of mollification the method is a regularizing algorithm and the order of convergence is derived error estimates are included together with numerical examples of interest

pointwise well posedness in set optimization with cone proper sets abstract this paper deals with the well posedness property in the setting of set optimization problems by using a notion of well posed set optimization problem due to zhang et al 2009 18 and a scalarization process we characterize this property through the well posedness in the tykhonov sense of a family of scalar optimization problems and we show that certain quasiconvex set optimization problems are well posed our approach is based just on a weak boundedness assumption called cone properness that is unavoidable to obtain a meaningful set optimization problem

set relations and optimality conditions in set valued maps abstract problems in set valued optimization can be solved via set optimization in this paper properties of set optimization are investigated conditions for existence of solutions are established directional derivatives are studied for set valued mappings necessary and sufficient conditions in the existence of solutions are showed with directional derivatives

about hahn banach extension theorems and applications to set valued optimization in this paper we revise several generalizations of the algebraic hanh banach extension theorem for set valued maps and give new versions in terms of set relations and selections we apply our results by establishing optimality conditions for a set valued constrained optimization problem considering several types of solutions finally we give necessary conditions for the existence of affine selections via affinelike maps in the framework of linear spaces

perturbation theory for anisotropic dielectric interfaces and application to subpixel smoothing of discretized numerical methods we derive a correct first order perturbation theory in electromagnetism for cases where an interface between two anisotropic dielectric materials is slightly shifted most previous perturbative methods give incorrect results for this case even to lowest order because of the complicated discontinuous boundary conditions on the electric field at such an interface our final expression is simply a surface integral over the material interface of the continuous field components from the unperturbed structure the derivation is based on a localized coordinate transformation technique which avoids both the problem of field discontinuities and the challenge of constructing an explicit coordinate transformation by taking the limit in which the coordinate perturbation is infinitesimally localized around the boundary not only is our result potentially useful in evaluating boundary perturbations e g from fabrication imperfections in highly anisotropic media such as many metamaterials but it also has a direct application in numerical electromagnetism in particular we show how it leads to a subpixel smoothing scheme to ameliorate staircasing effects in discretized simulations of anisotropic media in such a way as to greatly reduce the numerical errors compared to other proposed smoothing schemes

existence theorems for set optimization problems abstract in this paper we consider set optimization problems by using a partial ordering based on a set relation we establish several existence results of minimal elements more general than those existing in the literature as an application we present sufficient conditions for the existence of solutions of a set valued optimization problem under some semi continuity assumption

wavelet based filters for accurate computation of derivatives let f x be a smooth function whose derivative of a given order must be computed the signal f x is affected by two kinds of perturbation the perturbation caused by the presence of the machine epsilon m of the computer may be considered to be an extremely high frequency noise of very small amplitude the way to minimize its effect consists of choosing an appropriate value for the step size of the difference quotient the second perturbation caused by the presence of noise requires first the signal to be treated in some way it is the purpose of this work to construct a wavelet based band pass filter that deals with the two cited perturbations simultaneously in effect our wavelet acts like a smoothed difference quotient whose stepsize is of the same order as that of the usual difference quotient moreover the wavelet effectively removes the noise and computes the derivative with an accuracy equal to the one obtained by the corresponding difference quotient in the absence of noise

a stable fdtd algorithm for non diagonal anisotropic dielectrics a stable fdtd algorithm is developed for simulating maxwell s equations in anisotropic dielectric materials with principal axes not aligned with the grid the algorithm is stable because the finite difference operator that converts d to e is symmetric and positive semidefinite for contrast a previously developed asymmetric algorithm is shown to suffer from late time instabilities the presented algorithm has second order error for continuous dielectric materials and the error can be reduced to third order by richardson extrapolation applied to dielectrics with sharp interfaces the algorithm has first order error even when averaging the dielectric within partially filled grid cells however averaging the dielectric permits richardson extrapolation to obtain second order error

parameter estimation in the 1 d transport equation with advection abstract we introduce a stable numerical method for the identification of the diffusion coefficient in a one dimensional transport equation with advection when noisy data at the active boundary and initial measured population density are given this algorithm makes no assumption on the amount and or character of the noise in the data several numerical examples are supplied to show the consistency of the scheme

discrete stability analysis of the mollification method for numerical differentiation abstract the method of numerical differentiation by discrete mollification is presented in a fully discretized environment new and rigorous results for the numerical stability and error analysis of the algorithm in the presence of noisy data are derived to illustrate the accuracy and flexibility of the numerical procedure the application of the method to a function identification problem is also investigated

identification of source terms in 2 d ihcp abstract we introduce a stable numerical space marching scheme based on discrete mollification implemented as an automatic adaptive filter for the approximate identification of temperature temperature gradient and source terms in the two dimensional inverse heat conduction problem ihcp the stability and error analysis of the algorithm together with some numerical examples are provided

parameter identification using mollification for predator prey models in spatially heterogeneous environments a stable numerical method is introduced for the identification of population densitiesand spatially dependent diffusion coefficients in a lotka volterra predator prey model in which bias dispersal is present in this problem noisy data at the active boundary and initial measured population densities are given this algorithm makes no assumption on the amount and or character of the noise in the data

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

source term identification in 1 d ihcp we propose stable numerical solutions for the simultaneous identification of temperature temperature gradient and general source terms in the one dimensional inverse heat conduction problem ihcp the numerical solution consists of a regularization procedure based on the mollification method and a marching scheme for the solution of the stabilized problem the stability error analysis and implementation of the algorithm are presented together with a set of numerical results

on approximate solutions in set valued optimization problems in this paper we introduce several concepts of approximate solutions of set valued optimization problems with vector and set optimization we prove existence results and necessary and sufficient conditions by using limit sets

parameter estimation for a drying system in a porous medium we introduce a stable numerical method for the recovery of the temperature and moisture distributions in a liukov system with space dependent diffusion coefficients in this problem only cauchy noisy data at the active boundary is given and no information about the amount and or character of the noise in the data is assumed the error analysis for the algorithm is discussed and a numerical example is presented

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

on solutions of set valued optimization problems we consider two criteria of a solution associated with a set valued optimization problem a vector criterion and a set criterion we show how solutions of a vector type can help to find solutions of a set type and reciprocally as an application we obtain a sufficient condition for the existence of solutions of a set type via vector optimization

minimal element theorems revisited abstract starting with the brezis browder principle we give stronger versions of many variational principles and minimal element theorems which appeared in the recent literature relationships among the elements of different sets of assumptions are discussed and clarified i e assumptions to the metric structure of the underlying space and boundedness assumptions new results involving set valued maps and the increasingly popular set relations are obtained along the way

a fuzzy ordering on multi dimensional fuzzy sets induced from convex cones a fuzzy ordering for fuzzy sets on rn is presented by a fuzzy relation on rn rn which is induced by closed convex cones the suitability of the fuzzy order is discussed using the axioms a1 a7 in fuzzy sets and systems 118 2001 375 for fuzzy sets on rn which are incomparable with respect to the fuzzy order a method to evaluate the degree of satisfaction regarding the fuzzy order is presented by using a subsethood degree approximation by discrete cases is discussed for numerical calculation on the degree of the fuzzy order numerical examples are also given to illustrate our idea

λ c contingent derivatives of set valued maps abstract in this paper a new notion of differentiability of set valued maps is introduced based on ordered spaces techniques c lower and c upper contingent derivatives are defined conditions for c differentiability are given furthermore we show that these concepts are suitable for the formulation of optimality conditions for set valued optimization problems with set optimization criteria

modeling of sloped interfaces on a yee grid to represent material boundaries in the finite difference time domain or frequency domain method effective cell permittivity eeff can be introduced for each grid cell crossed by material interface in this paper we revisit the derivation of tensorial eeff for a sloped interface and describe possible interpolation schemes for coupling of different effective electric field and induction components near the interface we put the resulting non symmetric and symmetrized effective permittivity matrices to numerical tests in the frequency domain for very high contrast interfaces the symmetrized schemes perform worse than simple staircasing while non symmetrized interpolation retains the second order convergence

numerical identification of a nonlinear diffusion coefficient by discrete mollification the discrete mollification method is a convolution based filtering procedure suitable for the regularization of ill posed problems combined with explicit space marching finite difference schemes it provides stability and convergence for a variety of coefficient identification problems in linear parabolic equations in this paper we extend such a technique to identify some nonlinear diffusion coefficients depending on an unknown space dependent function in one dimensional parabolic models for the coefficient recovery process we present detailed error estimates and to illustrate the performance of the algorithms several numerical examples are included

automatic numerical solution of generalized 2 d ihcp by discrete mollification abstract a space marching scheme based on the mollification method and generalized cross validation is described to numerically recover the temperature and heat flux histories in the two dimensional generalized inverse heat conduction problem a proof of stability and convergence of this algorithm is provided together with several numerical examples of interest

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

stabilization of explicit methods for convection diffusion equations by discrete mollification the main goal of this paper is to show that discrete mollification is a simple and effective way to speed up explicit time stepping schemes for partial differential equations the second objective is to enhance the mollification method with a variety of alternatives for the treatment of boundary conditions the numerical experiments indicate that stabilization by mollification is a technique that works well for a variety of explicit schemes applied to linear and nonlinear differential equations

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

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

epidifferentiability and hypodifferentiability of pseudoconvex maps in set optimization problems abstract in this paper we study set valued optimization problems with set optimization criteria with this aim we introduce two classes of pseudoconvex maps whose elements are epidifferentiable or hypodifferentiable we establish optimality conditions in terms of the epiderivatives and hypoderivatives of associated maps of infima or suprema under pseudoconvexity conditions finally by connecting set optimization criteria with vector criterion we apply these results in order to solve a vector optimization problem with constraints given by a set valued map

some minimax theorems of set valued maps and their applications abstract in the paper we consider two types of scalarization functions of sets and investigate their properties moreover based on two set relations we propose two kinds of minimax and maximin values of set valued maps respectively and show some minimax theorems of set valued maps with respect to those minimax and maximin values by using several properties of the above two functions as an application of these results we give common saddle point theorems of vector valued functions

numerical stability of nonorthogonal fdtd methods in this paper a sufficient test for the numerical stability of generalized grid finite difference time domain fdtd schemes is presented it is shown that the projection operators of such schemes must be symmetric positive definite without this property such schemes can exhibit late time instabilities the origin and the characteristics of these late time instabilities are also uncovered based on this study nonorthogonal grid fdtd schemes nfdtd and the generalized yee gy methods are proposed that are numerically stable in the late time for quadrilateral prism elements allowing these methods to be extended to problems requiring very long time simulations the study of numerical stability that is presented is very general and can be applied to most solutions of maxwell s equations based on explicit time domain schemes

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

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

mollified hyperbolic method for coefficient identification problems abstract we introduce a stable numerical method for the identification of a transmissivity coefficient in a one dimensional parabolic equation it is a combination of the mollification method and a well known space marching implementation of the hyperbolic regularization procedure the new method succesfully restores a certain type of continuity with respect to the initial condition and the boundary data the accuracy of the algorithm is demonstrated by means of several examples where exact and perturbed data are considered

a perturbation method for numerical differentiation we discuss the numerical differentiation from the noisy data we propose a method i e adding a differential term for the differentiation problem an error estimate shows that the method allows for the approximate recovery of the derivative function some interesting numerical tests display the properties of the proposed method

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

a minty variational principle for set optimization abstract extremal problems are studied involving an objective function with values in order complete lattices of sets generated by so called set relations contrary to the popular paradigm in vector optimization the solution concept for such problems introduced by f heyde and a lohne comprises the attainment of the infimum as well as a minimality property the main result is a minty type variational inequality for set optimization problems which provides a sufficient optimality condition under lower semicontinuity assumptions and a necessary condition under appropriate generalized convexity assumptions the variational inequality is based on a new dini directional derivative for set valued functions which is defined in terms of a lattice difference quotient a residual operation in a lattice of sets replaces the inverse addition in linear spaces relationships to families of scalar problems are pointed out and used for proofs the appearance of improper scalarizations poses a major difficulty which is dealt with by extending known scalar results such as diewert s theorem to improper functions

a three dimensional fdtd algorithm in curvilinear coordinates em scattering the finite difference time domain fdtd algorithm for the solution of electromagnetic scattering problems is formulated in numerically defined generalized coordinates in three dimensions and implemented in a code with the lowest order bayliss turkel radiation boundary condition expressed in spherical coordinates it is shown that the algorithm is capable of accurately tracking the progress of a pulse of electromagnetic radiation through the curvilinear mesh generated by a body of revolution the only problems occurring in the vicinity of the rotation axis which represents a coordinate singularity a simple method to deal with this singular line is presented and discussed and its is shown that at least for the test problem this approximation is sufficient the algorithm discussed is useful for the solution of the exterior problem in the presence of conductors and dielectrics with complicated shapes and electrical compositions and for near field problems such as cavity penetration problems the far fields are obtained by replacing the scatterer with a virtual surface enclosing all sources

a unified approach to uncertain optimization abstract in this paper we consider uncertain scalar optimization problems with infinite scenario sets we apply methods from vector optimization in general spaces set valued optimization and scalarization techniques to develop a unified characterization of different concepts of robust optimization and stochastic programming these methods provide new insights on the interrelation between different concepts for handling uncertainties and naturally lead to new concepts of robustness

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

a new approach to numerical differentiation in this paper we consider the numerical differentiation of functions specified by noisy data a new approach which is based on an integral equation of the first kind with a suitable compact operator is presented and discussed since the singular system of the compact operator can be obtained easily tsvd is chosen as the needed regularization technique and we show that the method calls for a discrete sine transform so the method can be implemented easily and fast numerical examples are also given to show the efficiency of the method

strict approximate solutions in set valued optimization with applications to the approximate ekeland variational principle this work deals with strict solutions of set valued optimization problems under the set optimality criterion in this context we introduce a new approximate solution concept and we obtain several properties of these solutions when the error is fixed and also for their limit behavior when the error tends to zero then we prove a general existence result which is applied to obtain approximate ekeland variational principles

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

an efficient method for calculating smoothing splines using orthogonal transformations a procedure for calculating the mean squared residual and the trace of the influence matrix associated with a polynomial smoothing spline of degree 2m 1 using an orthogonal factorization is presented the procedure substantially overcomes the problem of ill conditioning encountered by a recently developed method which employs a cholesky factorization but still requires only orderm 2 n operations and ordermn storage

optimality conditions for set valued maps with set optimization abstract problems in set valued optimization can be solved via set optimization in this paper optimality conditions are studied for set valued maps with set optimization optimality requirements are established for continuous selections using directional derivatives necessary and sufficient conditions for the existence of solutions are shown for set valued maps under generalized convexity assumptions and with the notion of the contingent derivative

wavelet calculus and finite difference operators this paper shows that the naturally induced discrete differentiation operators induced from a wavelet galerkin finite dimensional approximation to a standard function space approximates differentiation with an error of order 0 h2d 2 where d is the degree of the wavelet system the degree of a wavelet system is defined as one less than the degree of the lowest order nonvanishing moment of the fundamental wavelet we consider in this paper compactly supported wavelets of the type introduced by daubechies in 1988 the induced differentiation operators are described in terms of connection coefficients which are intrinsically defined functional invariants of the wavelet system defined as l2 inner products of derivatives of wavelet basis functions with the basis functions themselves these connection coefficients can be explicitly computed without quadrature and they themselves have key moment vanishing properties proved in this paper which are dependent upon the degree of the wavelet system this is the basis for the proof of the principal results concerning the degree of approximation of the differentiation operator by the wavelet galerkin discrete differentiation operator

numerical solution of generalized ihcp by discrete mollification a numerical space marching algorithm based on discrete mollification and automatic iterative filtering by generalized cross validation is applied to the solution of a generalized one dimensional inverse heat conduction problem no information about the noise is assumed with data temperature and heat flux functions measured at a discrete set of points on the boundary x 1 0 t 1 the temperature and heat flux solution functions are approximately recovered in the unit square of the x t plane including its boundaries error bounds and numerical examples are provided

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

simultaneous space diffusivity and source term reconstruction in 2d ihcp a numerical marching scheme is presented for the simultaneous recovery of the diffusivity coefficient spatial source term temperature and heat flux distributions in the two dimensional inverse heat conduction problem when noisy data at the active boundary and the initial measured temperature distribution are given no information about the quality and or quantity of the noise is assumed a proof of stability and convergence of the algorithm are provided together with several numerical examples of interest

an inverse source problem in a semilinear time fractional diffusion equation we study an inverse source problem for a semilinear time fractional diffusion equation of second order in a bounded domain in r d the missing solely time dependent source is recovered from an additional integral measurement the existence uniqueness and regularity of a weak solution is addressed we design a numerical algorithm based on rothe s method derive a priori estimates and prove convergence of iterates towards the exact solution theoretical results are supported by a numerical experiment

further characterizations of possibility theoretical indices in fuzzy optimization abstract in this paper we investigate six types of fuzzy relations based on zadeh s possibility theory such relations were originally proposed by dubois and prade as comparison indices for fuzzy numbers and were extended by inuiguchi ichihashi and kume to general cases the fuzzy relations we consider are defined for fuzzy sets in a vector space preordered by a convex cone we reveal the connection of the fuzzy relations with set relations order like crisp relations known in the area of set optimization specifically we prove some equalities and equivalences involving the fuzzy relations and set relations under three different assumptions we finally consider a fuzzy optimization problem in a general form and define its solution concepts applying the above equivalences to the problem we obtain its relationship to certain set optimization problems

for numerical differentiation dimensionality can be a blessing finite difference methods such as the mid point rule have been applied successfully to the numerical solution of ordinary and partial differential equations if such formulas are applied to observational data in order to determine derivatives the results can be disastrous the reason for this is that measurement errors and even rounding errors in computer approximations are strongly amplified in the differentiation process especially if small step sizes are chosen and higher derivatives are required a number of authors have examined the use of various forms of averaging which allows the stable computation of low order derivatives from observational data the size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size h in this paper it is initially shown how first and higher order single variate numerical differentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding differentiation of one dimensional data the result is then extended to the multivariate differentiation of higher dimensional data the nature of the trade off between convergence and stability is explicitly characterized and the complexity of various implementations is examined

a 3 d tensor fdtd formulation for treatment of sloped interfaces in electrically inhomogeneous media a general integral finite difference time domain fdtd formulation on cubical grids for the modeling of electrically inhomogeneous media of arbitrary shape is derived the simple cubical structure is maintained and no modifications of the integral paths are necessary instead a three dimensional tensor relationship between the average electric flux density and electric field is determined beforehand and used during the simulation to account for discontinuities in the neighborhood of sloped interfaces the accuracy of the method is confirmed by comparisons with analytical solutions

galerkin method with trigonometric basis on stable numerical differentiation abstract this paper considers the p p 1 2 3 order numerical differentiation on function y in 0 2 they are transformed into corresponding fredholm integral equation of the first kind computational schemes with analytic solution formulas are designed using galerkin method on trigonometric basis convergence and divergence are all analysed in corollaries 5 1 5 2 and a priori error estimate is uniformly obtained in theorem 6 1 7 1 7 2 therefore the algorithm achieves the optimal convergence rate o 2 2 1 1 2 or 1 with periodic sobolev source condition of order 2 p besides we indicate a noise independent a priori parameter choice when the function y possesses the form of k 0 p 1 a k t k k 1 n 1 b k cos k t k 1 n 2 c k sin k t b n 1 c n 2 0 in particular in numerical differentiations for functions above good filtering effect error approaches 0 is displayed with corresponding parameter choice in addition several numerical examples are given to show that even derivatives with discontinuity can be recovered well

surface fitting and numerical gradient computations by discrete mollification abstract we review the mollification procedure for automatic fitting of surfaces defined from discrete noisy data functions in r 2 as a further application the stable numerical computation of gradient fields from discrete noisy data is also investigated the main features of the algorithm are 1 1 information about the noise is needed 2 2 the mollification parameters are chosen automatically by means of the generalized cross validation gcv procedure a complete error analysis of the method is provided together with several numerical examples of interest

discrete mollification and automatic numerical differentiation abstract an automatic method for numerical differentiation based on discrete mollification and the principle of generalized cross validation is presented with data measured at a discrete set of points of a given interval the method allows for the approximate recovery of the derivative function on the entire domain no information about the noise is assumed error estimates are included together with several numerical examples of interest