a multi level method for transmission eigenvalues of anisotropic media in this paper we propose a multi level finite element method for the transmission eigenvalue problem of anisotropic media the problem is non standard and non self adjoint with important applications in inverse scattering theory we employ a suitable finite element method to discretize the problem the resulting generalized matrix eigenvalue problem is large sparse and non hermitian to compute the smallest real transmission eigenvalue which is usually an interior eigenvalue we devise a multi level method using arnoldi iteration at the coarsest mesh the eigenvalue is obtained using arnoldi iteration with an adaptive searching technique this value is used as the initial guess for arnoldi iteration at the next mesh level this procedure is then repeated until the finest mesh level numerical examples are presented to show the viability of the method

sensitivity analysis of solutions to optimization problems in hilbert spaces with applications to optimal control and estimation abstract this research provides sufficient conditions for lipschitz continuity and directional differentiability of the solutions and the lagrange multipliers of constraint nonlinear programming problems in hilbert spaces the abstract results are applied to problems in optimal control and parameter estimation

identification of parabolic distributed parameter systems by regularization with differential operators abstract the theory of identification of variable coefficients in parabolic distributed parameter systems by regularization is extended to the case in which the stabilizing functional is the norm of a differential operator stability and convergence of the method are proved a detailed computational evaluation of the approach is presented via the estimation of a spatially varying conductivity in the two dimensional heat equation

a robust numerical algorithm for computing maxwell s transmission eigenvalue problems we study a robust and efficient eigensolver for computing a few smallest positive eigenvalues of the three dimensional maxwell s transmission eigenvalue problem the discretized governing equations by the nedelec edge element result in a large scale quadratic eigenvalue problem qep for which the spectrum contains many zero eigenvalues and the coefficient matrices consist of patterns in the matrix form xy 1 z both of which prevent existing eigenvalue solvers from being efficient to remedy these difficulties we rewrite the qep as a particular nonlinear eigenvalue problem and develop a secant type iteration together with an indefinite locally optimal block preconditioned conjugate gradient lobpcg method to sequentially compute the desired positive eigenvalues furthermore we propose a novel method to solve the linear systems in each iteration of lobpcg intensive numerical experiments show that our proposed method is robust although the desired real eigenvalues are surrounded by complex eigenv

finite element methods for maxwell s transmission eigenvalues the transmission eigenvalue problem plays a critical role in the theory of qualitative methods for inhomogeneous media in inverse scattering theory efficient computational tools for transmission eigenvalues are needed to motivate improvements to theory and more importantly are parts of inverse algorithms for estimating material properties in this paper we propose two finite element methods to compute a few lowest maxwell s transmission eigenvalues which are of interest in applications since the discrete matrix eigenvalue problem is large sparse and in particular non hermitian due to the fact that the problem is neither elliptic nor self adjoint we devise an adaptive method which combines the arnoldi iteration and estimation of transmission eigenvalues exact transmission eigenvalues for balls are derived and used as a benchmark numerical examples are provided to show the viability of the proposed methods and to test the accuracy of recently derived inequalities for transmission eigenvalues

an approximation theory for the identification of nonlinear distributed parameter systems an abstract approximation framework for the identification of nonlinear distributed parameter systems is developed inverse problems for nonlinear systems governed by strongly maximal monotone operators satisfying a mild continuous dependence condition with respect to the unknown parameters to be identified are treated convergence of galerkin approximations and the corresponding solutions of finite dimensional approximating identification problems to a solution of the original infinite dimensional identification problem is demonstrated using the theory of nonlinear evolution systems and a nonlinear analogue of the trotter kato approximation result for semigroups of bounded linear operators the nonlinear theory developed here is shown to subsume an existing linear theory as a special case it is also shown to be applicable to a broad class of nonlinear elliptic operators and the corresponding nonlinear parabolic partial differential equations to which they lead an application of the theory to a quas

a two grid discretization scheme of non conforming finite elements for transmission eigenvalues abstract in this paper for the helmholtz transmission eigenvalue problem we propose a two grid discretization scheme of non conforming finite elements with this scheme the solution of the transmission eigenvalue problem on a fine grid h is reduced to the solution of the primal and dual eigenvalue problem on a much coarser grid h and the solutions of two linear algebraic systems with the same positive definite hermitian and block diagonal coefficient matrix on the fine grid h we prove the resulting solution still maintains an asymptotically optimal accuracy and we report some numerical examples in two dimension and three dimension on the modified zienkiewicz element to validate the efficiency of our approach for solving transmission eigenvalues

a quasi newton method for estimating the parameter in a nonlinear hyperbolic system a quasi newton method qnm in infinite dimensional spaces for estimating parameters involved in distributed parameter systems is presented in this paper next the global convergence of a sequence generated by the algorithm qnm is also proved we apply the algorithm qnm to an identification problem of a distributed parameter system governed by a nonlinear hyperbolic partial differential equation

a new multigrid finite element method for the transmission eigenvalue problems numerical methods for the transmission eigenvalue problems are hot topics in recent years based on the work of lin and xie 2015 we build a multigrid method to solve the problems with our method we only need to solve a series of primal and dual eigenvalue problems on a coarse mesh and the associated boundary value problems on the finer and finer meshes theoretical analysis and numerical results show that our method is simple and easy to implement and is efficient for computing real and complex transmission eigenvalues

a type of adaptive c0 non conforming finite element method for the helmholtz transmission eigenvalue problem abstract the transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering in this paper for the helmholtz transmission eigenvalue problem in r d d 2 3 we discuss the a posteriori error estimates and adaptive algorithm of a type of c 0 non conforming finite elements including the morley zienkiewicz element and the modified zienkiewicz element we give the a posteriori error estimators for approximate eigenpair and prove their reliability and efficiency for eigenfunctions we also analyze their reliability for eigenvalues numerical results not only in r 2 but also in r 3 indicate that the a posteriori error estimators are sharp and the adaptive algorithm is efficient for computing real and complex transmission eigenvalues as expected

reduced sqp methods for parameter identification problems parameter estimation problems are formulated as constrained regularized optimization problems reduced sqp methods with bfgs update are analyzed to solve these infinite dimensional optimization problems rate of convergence results are given and numerical feasibility of the resulting algorithms is demonstrated

non conforming finite element methods for transmission eigenvalue problem abstract the transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory in this paper for the helmholtz transmission eigenvalue problem we give a weak formulation which is a nonselfadjoint linear eigenvalue problem based on the weak formulation we first discuss the non conforming finite element approximation and prove the error estimates of the discrete eigenvalues obtained by the adini element the morley zienkiewicz element the modified zienkiewicz element et al and we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem

a c0ipg method and its error estimates for the helmholtz transmission eigenvalue problem abstract the interior penalty methods using c 0 lagrange elements c 0 ipg developed in the last decade for the fourth order problems are an interesting topic in academia at present in this paper we apply the methods to the helmholtz transmission eigenvalue problem which is quadratic and non self adjoint and has important applications in the inverse scattering theory we propose a discrete scheme present a complete error analysis and first prove the c 0 ipg method can capture smooth eigenpairs efficiently using the argument in this paper we can prove that the c 0 ipg methods for the fourth order eigenvalue problems in the existing literature can also capture smooth eigenpairs efficiently numerical examples confirm our theoretical analysis

a mixed fem for the quad curl eigenvalue problem the quad curl problem arises in the study of the electromagnetic interior transmission problem and magnetohydrodynamics in this paper we study the quad curl eigenvalue problem and propose a mixed method using edge elements assuming stringent regularity of the solution of the quad curl source problem we prove the convergence and show that the divergence free condition can be bypassed

a generalized modified bessel function and a higher level analogue of the theta transformation formula abstract a new generalization of the modified bessel function of the second kind k z x is studied elegant series and integral representations a differential difference equation and asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess the motivation behind introducing this generalization is to have a function which gives a new pair of functions reciprocal in the koshliakov kernel cos z m 2 z 4 x sin z j 2 z 4 x and which subsumes the self reciprocal pair involving k z x its application towards finding modular type transformations of the form f z w f z i w where 1 is given as an example we obtain a beautiful generalization of a famous formula of ramanujan and guinand equivalent to the functional equation of a non holomorphic eisenstein series on s l 2 z this generalization can be considered as a higher level analogue of the general theta transformation formula we then use it to evaluate an integral involving the riemann function and consisting of a sum of products of two confluent hypergeometric functions

the imaging of small perturbations in an anisotropic media abstract in this paper we employ asymptotic analysis to determine information about small volume defects in a known anisotropic scattering medium from far field scattering data the location of the defects is reconstructed via the music algorithm from the range of the multi static response matrix derived from the asymptotic expansion of the far field pattern in the presence of small defects since the same data determines the transmission eigenvalues corresponding to the perturbed media we investigate how the presence of the defects changes the transmission eigenvalues and use this information to recover the strength of the small defects we provide convergence results on transmission eigenvalues as the size of the defects tends to zero as well as derive the first correction term in the asymptotic expansion of the simple transmission eigenvalues numerical examples are presented to show the viability of our imaging method

receding horizon boundary control of nonlinear conservation laws with shock avoidance this paper deals with the boundary control for scalar non linear hyperbolic systems of conservation laws one of the issues arising from these systems is the occurrence of singularities called shocks in the domain we will show that to avoid the shock a set of constraints on both the control and the state at the boundary has to be fulfilled then a proof of the exponential stability of the system is established provided that there is no shock and that the state at the boundary is exponentially stable these conditions are shown to be achieved by the receding horizon optimal control approach a simulation is finally carried out with the freeway traffic model to demonstrate the effectiveness of the here proposed control

a survey of the maximum principles for optimal control problems with state constraints this paper gives a survey of the various forms of pontryagin s maximum principle for optimal control problems with state variable inequality constraints the relations between the different sets of optimality conditions arising in these forms are shown furthermore the application of these maximum principle conditions is demonstrated by solving some illustrative examples

uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacles with embedded objects abstract this paper considers the inverse problem of scattering of time harmonic acoustic and electromagnetic plane waves by a bounded inhomogeneous penetrable obstacle with embedded objects inside a new method is proposed to prove that the inhomogeneous penetrable obstacle can be uniquely determined from the far field pattern at a fixed frequency disregarding its contents our method is based on constructing a well posed interior transmission problem in a small domain associated with the helmholtz or modified helmholtz equation and the maxwell or modified maxwell equations a key role is played by the smallness of the domain which ensures that the lowest transmission eigenvalue is large so that a given wave number k is not an eigenvalue of the interior transmission problem another ingredient in our proofs is a priori estimates of solutions to the transmission scattering problems with data in l p 1 p 2 which are established in this paper by using the integral equation method a main feature of the new method is that it can deal with the acoustic and electromagnetic cases in a unified way and can be easily applied to deal with inverse scattering by unbounded rough interfaces

a coupled bem and fem for the interior transmission problem in acoustics abstract the interior transmission problem itp is a boundary value problem arising in inverse scattering theory and it has important applications in qualitative methods in this paper we propose a coupled boundary element method bem and a finite element method fem for the itp in two dimensions the coupling procedure is realized by applying the direct boundary integral equation method to define the so called dirichlet to neumann dtn mappings we show the existence of the solution to the itp for the anisotropic medium numerical results are provided to illustrate the accuracy of the coupling method

eigenvector derivatives with repeated eigenvalues in this paper an algorithm is derived for computing the derivatives of eigenvalues and eigenvectors for real symmetric matrices in the case of repeated eigenvalues where the matrices are functions of real parameters such as mass density or moment of inertia the algorithm is an extension of recent work by i u ojalvo the key step in this extended derivation is to differentiate the eigenvalue equation twice the algorithm preserves the symmetry and band structure of the matrices allowing efficient computer storage and solution techniques applications include sensitivity analysis and optimization of the normal modes of finite element modeled structures such as large space structures a cantilever beam finite element example is included

a well posed conservation law with a variable unilateral constraint this paper considers the cauchy problem for a conservation law with a variable unilateral constraint its motivation being for instance the modeling of a toll gate along a highway this problem is solved by means of nonclassical shocks and its well posedness is proved then the solutions so obtained are shown to coincide with the limits of the classical solutions to suitable conservation laws with discontinuous flux function that approximate the constrained problem

the local convergence of broyden like methods on lipschitzian problems in hilbert spaces the classical local convergence analysis for broyden s method requires a finite number of real variables holder continuity of the jacobian and a rather accurate initial approximation b 0 here these three conditions are relaxed to allow for lipschitz continuous hilbert space operators that can be reasonably well approximated by a linear operator b that is close to b 0 up to an arbitrary compact perturbation to achieve local and r linear convergence broyden s method is modified by a very mild line search condition in case of a continuously differentiable system in euclidean space we obtain r superlinear convergence without restricting b 0 or requiring the jacobian to be holder continuous

koshliakov kernel and identities involving the riemann zeta function abstract some integral identities involving the riemann zeta function and functions reciprocal in a kernel involving the bessel functions j z x y z x and k z x are studied interesting special cases of these identities are derived one of which is connected to a well known transformation due to ramanujan and guinand

iterative methods for transmission eigenvalues transmission eigenvalues have important applications in inverse scattering theory they can be used to obtain useful information of the physical properties such as the index of refraction of the scattering target despite considerable effort devoted to the existence and estimation for the transmission eigenvalues the numerical treatment is limited since the problem is nonstandard classical finite element methods result in non hermitian matrix eigenvalue problems in this paper we focus on the computation of a few lowest transmission eigenvalues which are of practical importance instead of a non hermitian problem we work on a series of generalized hermitian problems we first use a fourth order reformulation of the transmission eigenproblem to construct functions involving an associated generalized eigenvalue problem the roots of these functions are the transmission eigenvalues then we apply iterative methods to compute the transmission eigenvalues we show the convergence of the numerical schemes the effectiveness of the methods is demonstrated using various numerical examples

further analysis of the arnoldi process for eigenvalue problems this paper takes another look at the convergence analysis of the arnoldi procedure for solving non hermitian eigenvalue problems two main viewpoints are put in contrast the first exploits the eigenbasis when there is one and relies on classical min max approximation theory results the second approach relies on the schur factorization its aim is to link the convergence analysis of the arnoldi process for eigenvalue problems to that of the generalized minimal residual iterations gmres for which much is known

recursive integral method for transmission eigenvalues the transmission eigenvalue problem arose in the inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing the problem is numerically challenging because of the nonself adjointness and complicated spectrum in this talk we introduce a novel recursive contour integral method using the spectrum projection the method can compute both real and complex eigenvalues and does not require any a priori information numerical examples show that the method is effective and robust

spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem we first develop an efficient spectral galerkin method and a rigorous error analysis for the generalized eigenvalue problems associated to a transmission eigenvalue problem then we present an iterative scheme based on computation of the first transmission eigenvalue to estimate the index of refraction of an inhomogeneous medium we present ample numerical results to demonstrate the effectiveness and accuracy of our approach

a high accuracy spectral method based on min max principle for biharmonic eigenvalue problems on a spherical domain abstract in this study we develop a high accuracy spectral method based on the min max principle for biharmonic eigenvalue problems in the spherical domain by analyzing the orthogonal spherical harmonic and approximation and using the min max principle we first deduce the error estimates of approximate eigenvalues then we construct an appropriate set of orthogonal spherical functions contained in h 0 2 and establish the matrix formulations for the discrete variational form whose mass matrix and stiff matrix are all sparse so that we can solve the numerical solutions efficiently finally we provide some numerical experiments to validate that the theoretical results are correct

state and parameter estimation in 1 d hyperbolic pdes based on an adjoint method an optimal estimation method for state and distributed parameters in 1 d hyperbolic system based on adjoint method is proposed in this paper a general form of the partial differential equations governing the dynamics of system is first introduced in this equation the initial condition or state variable as well as some empirical parameters are supposed to be unknown and need to be estimated the lagrangian multiplier method is used to connect the dynamics of the system and the cost function defined as the least square error between the simulation values and the measurements the adjoint state method is applied to the objective functional in order to get the adjoint system and the gradients with respect to parameters and initial state the objective functional is minimized by broyden fletcher goldfarb shanno bfgs method due to the non linearity of both direct and adjoint system the nonlinear explicit lax wendroff scheme is used to solve them numerically the presented optimal estimation approach is validated by two illustrative examples the first one about state and parameter estimation in a traffic flow and the second one in an overland flow system

estimation of two phase petroleum reservoir properties by regularization abstract an algorithm is developed based on the theory of regularization and on spline approximation to estimate the absolute permeability in two phase petroleum reservoirs from noisy well pressure data the regularization feature of the algorithm converts the ill posed estimation problem to a well posed one the algorithm which employs the partial conjugate gradient method of nazareth as its core minimization technique automatically chooses the regularization parameter based on the non regularized estimation it is shown that regularized estimation is more stable and insensitive to the choice of initial guess as compared to non regularized conventional estimation

zeros of combinations of the riemann ξ function and the confluent hypergeometric function on bounded vertical shifts abstract in 1914 hardy proved that infinitely many non trivial zeros of the riemann zeta function lie on the critical line using the transformation formula of the jacobi theta function recently the first author obtained an integral representation involving the riemann function and the confluent hypergeometric function linked to the general theta transformation using this result we show that a series consisting of bounded vertical shifts of a product of the riemann function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line thereby generalizing a previous result due to the first and the last authors along with roy and robles the latter itself is a generalization of hardy s theorem

a c0ip method of transmission eigenvalues for elastic waves abstract this paper develops a discontinuous galerkin method to compute a few smallest elasticity transmission eigenvalues which are of practical importance in inverse elastic scattering theory for high order problems comparing with classical conforming finite element methods discontinuous galerkin methods use simpler basis functions which make the numerical implementation much easier in this paper we propose an interior penalty discontinuous galerkin method using c 0 lagrange elements c 0 ip for the elastic transmission eigenvalue problem and prove the optimal convergence numerical examples are presented to validate its effectiveness both real and complex eigenvalues can be obtained

an efficient spectral galerkin approximation based on dimension reduction scheme for transmission eigenvalues in polar geometries abstract in this paper we put forward an efficient spectral galerkin approximation in view of dimension reduction scheme for transmission eigenvalue problem in polar geometries firstly we turn the original problem into an equivalent fourth order nonlinear eigenvalue problem then the fourth order nonlinear eigenvalue problem is transformed into a coupled fourth order linear eigenvalue system by introducing an auxiliary poisson equation secondly based on polar coordinate transformation we further reduce the coupled fourth order linear eigenvalue system to a series of equivalent one dimensional eigenvalue systems thirdly we derive the essential polar condition and introduce the appropriate weighted sobolev space according to the polar condition and establish the weak form and the corresponding discrete form in addition by utilizing spectral theory of compact operators we prove the error estimates of approximation eigenvalues and eigenvectors for each one dimensional eigenvalue system finally we provide ample numerical experiments and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results