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This study deals with the numerical implementation of a formula in the enclosure method as applied to a prototype inverse initial boundary value problem for thermal imaging in a one-space dimension. A precise error estimate of the formula is given and the effect on the discretization of the used integral of the measured data in the formula is studied. The formula requires a large frequency to converge; however, the number of time interval divisions grows exponentially as the frequency increases. Therefore, for a given number of divisions, we fixed the trusted frequency region of convergence with some given error bound. The trusted frequency region is computed theoretically using the theorems provided in this paper and is numerically implemented for various cases. [ABSTRACT FROM AUTHOR]

This paper investigates photoacoustic tomography with two spatially varying acoustic parameters, the compressibility and the density. We consider the reconstruction of the absorption density parameter (imaging parameter of Photoacoustics) with complete and partial measurement data.We investigate and analyze three different numerical methods for solving the imaging problem and compare the results. [ABSTRACT FROM AUTHOR]

Finite difference (FD) solutions of wave equations have been proven useful in exploration seismology. To yield reliable and interpretable results, the numerically induced error should be minimized over a range of frequencies and angles of propagation. Grid dispersion is one of the key numerical problems and there exist some methods to solve this problem in the literature. Traditionally, the spatial FD operator coefficients are only determined in the spatial domain; however, the wave equation is solved in the temporal and spatial domain simultaneously. Recently, some methods based on the joint temporal-spatial domains have been proposed to address this problem. Variable length coefficients methods are proposed in the literature to improve efficiency while preserving accuracy by using longer operators in the low velocity regions and shorter operators in the high velocity regions. To cope with the ill-conditioning of the linear system induced by long stencil FD operators, we study in this paper a regularizing simplified least-squares model to minimize the phase velocity error in the joint temporal-spatial domain with a variable length of coefficients. Different from our previous study, we determine FD coefficients on the regular grid instead of on the staggered grid. Though the regular grid FD methods are less precise, however, with a little increase of the operator length, the precision can be improved. Stability of the numerical solutions is enhanced by the regularization. Numerical simulations made on one-dimensional to three-dimensional examples show that our scheme needs shorter operators and preserves accuracy compared with the previous methods. [ABSTRACT FROM AUTHOR]

The aim of this paper is to reconstruct local volatility from market prices of binary options. In the case of the space-dependent volatility, we obtain the stable linearization and the available integral equation to identify local volatility from observable data of binary options. We achieve the reconstruction of local volatility by numerical simulation. [ABSTRACT FROM AUTHOR]

The paper deals with the nonlinear inverse source problem of identifying an unknown timedependent point source occurring in a two-dimensional evolution advection-dispersion-reaction equation with spatially varying velocity field and dispersion tensor. The originality of this study consists in establishing a constructive identifiability theorem that leads to develop an identification method using only significant boundary observations and operating other than the classic optimization approach. To this end, we derive two dispersion-current functions that have the main property to be of orthogonal gradients which yield identifiability of the elements defining the involved unknown source from some boundary observations related to the associated state. Provided the velocity field fulfills the so-called no-slipping condition, the required boundary observations are reduced to only recording the state on the outflow boundary and its flux on the inflow boundary of the monitored domain. We establish an identification method that uses those boundary records (1) to localize the sought source position as the unique solution of a nonlinear system defined by the two dispersion-current functions, (2) to give lower and upper bounds of the total amount loaded by the unknown time-dependent source intensity function, (3) to transform the identification of this latest into solving a deconvolution problem. Some numerical experiments on a variant of the surface water BOD pollution model are presented. [ABSTRACT FROM AUTHOR]

In this paper we consider the inverse problem of simultaneously reconstructing the interface where the jump of the conductivity occurs and the Robin parameter for a transmission problem with piecewise constant conductivity and Robin-type transmission conditions on the interface. We propose a reconstruction method based on a shape optimization approach and compare the results obtained using two different types of shape functionals. The reformulation of the shape optimization problem as a suitable saddle point problem allows us to obtain the optimality conditions by using differentiability properties of the min-sup combined with a function space parameterization technique. The reconstruction is then performed by means of an iterative algorithm based on a conjugate shape gradient method combined with a level set approach. To conclude we give and discuss several numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modified inhomogeneous Helmholtz equation Δ u( x, y) u( x, y) f( x, y), x , 0 y 1, with inhomogeneous Cauchy data being given at y 0. The problem is known to be ill-posed, as the solution (if exists) does not depend continuously on the given data. We propose a regularization method to obtain a stable approximate solution of the problem and get some error estimates. Finally, a numerical example shows the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper an inverse problem is considered for a non-coercive partial differential equation, issued from a mass conservation cavitation model for a slightly compressible fluid. The cavitation phenomenon and compressibility take place and are described by the Elrod model. The existence of an optimal solution is proven. Optimality conditions are derived and some numerical results are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse heat conduction problem (IHCP). First, the existence, uniqueness and unstability solution of the inverse problem will be studied. Due to the ill-posedness of the inverse problem, we propose a meshless numerical approach based on basis function to solve this problem in the presence of noisy data. The Tikhonov regularization method with generalized discrepancy principle is applied to obtain a stable numerical approximation to the solution. The effectiveness of the algorithm is illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

A series of recent numerical experiments for parameter estimation inverse problems in epidemiology [, ] have indicated that applicability of the discrepancy principle (DP) does not depend on the structure of a particular regularizing operator. This observation confirmed the original theoretical analysis by Bakushinsky [, ] on the construction of fairly general stabilizing algorithms in Banach and Hilbert spaces. In [, ], a unified approach to the implementation of the DP for linear ill-posed problems, the Abstract Discrepancy Principle, has been proposed and justified. The current paper investigates the convergence rates of the ADP. Special cases of sectorial and self-adjoint operators are studied. [ABSTRACT FROM AUTHOR]

In this paper we present a method for the regularized solution of nonlinear inverse problems, based on Ivanov regularization (also called method of quasi solutions or constrained least squares regularization). This leads to the minimization of a nonconvex cost function under a norm constraint, where nonconvexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (nonconvex) quadratic Taylor expansions of the cost function. This leads to repeated solution of quadratic trust region subproblems with possibly indefinite Hessian. Thus, the key step of the method consists in application of an efficient method for solving such quadratic subproblems, developed by Rendl and Wolkowicz []. We here present a convergence analysis of the overall method as well as numerical experiments. [ABSTRACT FROM AUTHOR]

According to Pareto's theory of consumer demand a rational representative consumer should choose their consumption bundle as the solution of a mathematical programming problem of maximization of the utility function under their budget constraint. The inverse problem of demand analysis is to recover the utility function from the demand functions. The answer to the question of solvability of this problem is based on the revealed preference theory. If the problem is unsolvable, one should apply regularization procedure by introducing irrationality indices. When recovering the utility function one puts a priori requirements on it. In this paper, we suggest including positively-homogeneity property into these requirements. We compare the setups with and without this requirement both theoretically and empirically and provide evidence in favor of requiring this property when computing economic indices for consumer representing consumption behavior of large number of various households for large time intervals. [ABSTRACT FROM AUTHOR]

Normalized differences of several adjacent observations, referred to as pseudo-measurement errors in this paper, are used in so-called difference-based estimation methods as building blocks for the variance estimate of measurement errors. Numerical results demonstrate that pseudo-measurement errors can be used to serve the role of measurement errors. Based on this information, we propose the use of pseudo-measurement errors to determine an appropriate statistical model and then to subsequently investigate whether there is a mathematical model misspecification or error. We also propose to use the information provided by pseudo-measurement errors to quantify uncertainty in parameter estimation by bootstrapping methods. A number of numerical examples are given to illustrate the effectiveness of these proposed methods. [ABSTRACT FROM AUTHOR]

In this paper a boundary value problem is considered generated by the Sturm-Liouville equation and non-separated boundary conditions, one of which contains a spectral parameter. We give a uniqueness theorem, develop an algorithm for solving the inverse problem of reconstruction of boundary value problems with spectral data. We use the spectra of two boundary value problems and some sequence of signs as a spectral data. [ABSTRACT FROM AUTHOR]

We concerned with the asymptotic analysis for the dynamical probe method which is a reconstruction scheme to identify an anomaly inside a heat conductor from the Neumann-to-Dirichlet map. In this paper an inclusion was considered as an anomaly and we succeeded giving an improved asymptotic behavior of the indicator function defined in terms of the Neumann-to-Dirichlet map to identify not only the location of the inclusion but also some of its physical properties simultaneously. The two major improvements made for analyzing the asymptotic behavior of the indicator function are as follows. Firstly, we can know the distance to the boundary of unknown inclusion as we probe it from its outside. This improvement can avoid overshooting the boundary points as much as possible if we probe it from outside the inclusion numerically. Secondly, we can know the value of heat conductivity of inclusion as we probe close to the inclusion even without touching it. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm to bound parameters estimated from very small data sets and use this method to suggest optimal times for data collection. The method is deterministic in that we assume that one has error bars on the data. The basic idea is based on bounded error parameter identification methods and we use this framework to assign a measure of quality to the estimated parameter. Exploiting the dynamics of the underlying mathematical model allows us to describe the quality of the estimated parameter in a way that, due to the small number of data points, is not appropriate for traditional statistical techniques. The algorithm in this paper differs from traditional bounded error parameter identification techniques in that one computes the membership set by solving an inverse problem rather than a forward interval problem. [ABSTRACT FROM AUTHOR]

In the recent paper [Colombo, Physica D 236: 81–89, 2007] the author investigates an inverse problem arising in the theory of combustion. The problem studied is: determine the temperature u and the convolution memory kernel k in the evolution equation given suitable initial-boundary conditions and the following additional restriction on u: ∫Ω φ( x) u( t, x) dx = g( t), where φ and g are given functions. The main results are a local in time existence theorem and a global in time uniqueness result. In this paper we complete the study considering the linearized version of the model. We prove that, if F( u( t, x), ∇ u( t, x)) is sublinear, then the inverse problem has a unique global in time solution. [ABSTRACT FROM AUTHOR]

In this paper we investigate the numerical complexity to solve nonlinear ill-posed problems when the operator equations F( x) = yδ are solved by the iteratively regularized Gauss–Newton method (IRGNM) with inner CG-iteration. Additionally we consider a preconditioned version of the IRGNM and compare the complexity of the standard IRGNM and its preconditioned version. In the case of exponentially ill-posed problems we show the superiority of the preconditioned IRGNM, that is we prove that the preconditioning techniques presented in this paper yield a significant reduction of the total complexity. [ABSTRACT FROM AUTHOR]

In this paper, a simple and convenient new regularization method which is called modified quasi-boundary value method for solving nonlinear backward heat equation is given. Some new quite sharp error estimates between the approximate solution are provided and generalize the results in our paper [Trong, Quan, Khanh, and Tuan, Zeitschrift Analysis und ihre Anwendungen 26: 231–245, 2007, Trong and Tuan, Electron. J. Diff. Eqns. 4: 1–10, 2006, Trong and Tuan, Electron. J. Diff. Eqns. 84: 1–12, 2008]. The approximation solution is calculated by the contraction principle. A numerical experiment is given. [ABSTRACT FROM AUTHOR]

In the paper [Eppler and Harbrecht, Control & Cybernetics 34: 203–225, 2005], the authors investigated the identification of an obstacle or void of perfectly conducting material in a two-dimensional domain by measurements of voltage and currents at the boundary. In particular, the reformulation of the given nonlinear identification problem was considered as a shape optimization problem using the Kohn and Vogelius criterion. The compactness of the complete shape Hessian at the optimal inclusion was proven, verifying strictly the ill-posedness of the identification problem. The aim of the paper is to present a similar analysis for the related least square tracking formulations. It turns out that the two-norm-discrepancy is of the same principal nature as for the Kohn and Vogelius objective. As a byproduct, the necessary first order optimality condition are shown to be satisfied if and only if the data are perfectly matching. Finally, we comment on possible consequences of the two-norm-discrepancy for the regularization issue. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a method for recovering implied volatility from European option price. Under some assumptions, an integral equation can be obtained from Dupire equation, and we show the local uniqueness and stability of implied volatility. In the end of this paper, we give some numerical examples to show that the new treatment is effective. [ABSTRACT FROM AUTHOR]

In this paper the method of obtaining unimprovable (in certain sense) estimates of solutions of some integral inequalities with the operators of Volterra type is stated. The basis of this method is the theory of monotone operators in partially ordered Banach spaces. This theory allows us to reduce obtaining these estimates to solving corresponding equations. The paper consists of two parts. The first part is devoted to unimprovable estimates of solutions for linear multidimensional inequalities. In the second part the author states nonlinear inequalities which arise while researching multilinear Volterra equations of the first kind connected with modelling nonlinear dynamic systems of black body type by Volterra polynomials. [ABSTRACT FROM AUTHOR]

In this paper we study an inverse problem for a parabolic partial differential equation. The parabolic partial differential equation considered is the Fokker Planck equation associated to a system of stochastic differential equations and the inverse problem studied consists in finding from suitable data the values of the parameters that appear in the coefficients of this Fokker Planck equation. The data used in the reconstruction of the parameters are observations made at discrete times of the stochastic process solution of the system of stochastic differential equations. That is, the data of the inverse problem are a “sample” taken at discrete times of some of the components of the random vector solution of the stochastic differential equations and not, as usual, observations made on the solution of the parabolic equation. The choice of the system of stochastic differential equations and of the data used in the inverse problem are motivated by applications in mathematical finance. The stochastic differential equations presented can be used to model the dynamics of the log-returns of the index of some classes of hedge funds, such as, for example, the so called “long short equity” hedge funds and of some auxiliary variables. The solution of the inverse problem proposed is obtained through the solution of a filtering and of an estimation problem. The solution of these last two problems is based on the knowledge of the joint probability density function of the state variables of the model conditioned to the observations made and to the initial condition. This joint probability density function is solution of an initial value problem for the Kushner equation that in the circumstances considered here can be written as a sequence of initial value problems for the Fokker Planck equation associated to the system of stochastic differential equations with appropriate initial conditions. An integral representation formula for this probability density function is derived and used to develop a numerical procedure to solve the estimation problem using the maximum likelihood method. The Kushner equation provides the relation between the data and the Fokker Planck equation used to solve the inverse problem considered. The computational method proposed has been tested on synthetic data and the results obtained are presented. Some auxiliary material useful to understand this paper including some animations and some numerical experiments can be found in the website . A more general reference to the work in mathematical finance of the authors and of their coauthors is the website . [ABSTRACT FROM AUTHOR]

In this paper we study a time harmonic inverse acoustic scattering problem involving an obstacle that when hit by an incoming acoustic wave tries to appear in a location in space different from its actual location eventually with a shape and an acoustic boundary impedance different from its actual ones. We refer to this problem as inverse ghost obstacle problem and to the scatterers of this type as a special class of smart obstacles. The term "ghost" comes from the fact that the smart obstacle, when hit by the incoming wave, tries to appear as a ghost obstacle, that is as a virtual obstacle in a location in space where no real obstacle is present. We assume that the intersection between the obstacle and the ghost obstacle is empty. The obstacle tries to produce this virtual image of itself circulating on its boundary a suitable pressure current. A pressure current is a quantity whose physical dimension is pressure divided by time. The following time harmonic inverse ghost obstacle scattering problem is considered: from the knowledge of several far fields generated by the smart obstacle when hit by known time harmonic waves, of its acoustic boundary impedance and of the acoustic boundary impedance of the ghost obstacle find the location of the smart obstacle, that is find a point in the interior of the smart obstacle and eventually find the shape of the smart obstacle. The method proposed to solve this problem is a generalization of the well known Herglotz function method. We present some numerical experiments based on synthetic data that are used to test the numerical method introduced here. Some material that helps the understanding of this paper and some animations relative to the numerical experiments can be found in the website: http://www.econ.univpm.it/recchioni/w15. [ABSTRACT FROM AUTHOR]

In this paper we apply the new regularization method a daptive grid regularization, which is well-suited for ill-posed problems with discontinuous solutions and which was introduced in an earlier paper [5], to 2D linear integral equations of the first kind. After describing the method we present some numerical examples showing that this method is a powerful tool to identify discontinuities in ill-posed problems. [ABSTRACT FROM AUTHOR]

The paper is devoted to the analysis of linear ill-posed operator equations Ax = y with solution x0 in a Hilbert space setting. In an introductory part, we recall assertions on convergence rates based on general source conditions for wide classes of linear regularization methods. The source conditions are formulated by using index functions. Error estimates for the regularization methods are developed by exploiting the concept of Mathé and Pereverzev that assumes the qualification of such a method to be an index function. In the main part of the paper we show that convergence rates can also be obtained based on distance functions d( R) depending on radius R > 0 and expressing for x0 the violation of a benchmark source condition. This paper is focused on the moderate source condition x0 = A∗ v. The case of distance functions with power type decay rates d( R) = ( R–η/(1–η)) as R → ∞ for exponents 0 < η < 1 is especially discussed. For the integration operator in L2(0, 1) aimed at finding the antiderivative of a square-integrable function the distance function can be verified in a rather explicit way by using the Lagrange multiplier method and by solving the occurring Fredholm integral equations of the second kind. The developed theory is illustrated by an example, where the optimal decay order of d( R) → 0 for some specific solution x0 can be derived directly from explicit solutions of associated integral equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem of estimating simultaneously the five parameters of a jump diffusion process based on return observations of a price trajectory. We show that there occur some ill-posedness phenomena in the parameter estimation problem, because the forward operator fails to be injective and small perturbations in the data may lead to large changes in the solution. We illustrate the instability effect by a numerical case study. To obtain stable approximate solutions of the estimation problem, we use a multi-parameter regularization approach, where a least-squares fitting of empirical densities is superposed by a quadratic penalty term of fitted semi-invariants with weights. A little number of required weights is controlled by a discrepancy principle. For the realization of this control, we propose and justify a fixed point iteration, where an exponent can be chosen arbitrarily positive. A numerical case study completing the paper shows that the approach provides satisfactory results and that the amount of computation can be reduced by an appropriate choice of the free exponent. [ABSTRACT FROM AUTHOR]

Linear ill-posed problems written as the operator equation Az = ū on sets of functions z convex on multi-dimensional sets Ω are considered in the paper. A regularizing algorithm zη = R(Ah, uδ, η), where |Ah - A| ≤ h, |uδ - ū| ≤ δ, η = (h,δ), obtained in the previous papers for line segments is generalized such that the approximate solution zη tends to the exact one z uniformly on some subsets of the domain Ω. The algorithms to estimate an error of finite dimensional approximation and to find a lower zl and an upper zu functions that bound all approximation solutions are provided. [ABSTRACT FROM AUTHOR]

This paper gives a numerical method for reconstructing the original function and its derivatives from discrete input data. It is well known that this problem is ill-posed in the sense of Hadamard. The solution for the first order derivative has been proposed by [10] and [17], using the Tikhonov regularization technique. In this paper, under an assumption that the original function has a square integrable k-th order derivative, we propose a reconstruction method for the j-th order derivative where 0 ≤ j ≤ k - 1. A convergence rate estimate is obtained by taking a new choice of the Tikhonov parameter. Numerical example is given to verify the effectiveness and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned to cope with a conditional stability for an inverse problem of deciding source terms in a 1-D advection-dispersion equation. The inverse problem here is based on a mathematical model derived from a real case in a geological region in Shandong Province, China. With aids of an integral identity and analysis for a normal Sturm-Liouville problem, a conditional stability for the inverse problem is constructed. The methodology used in this paper seems to be simple and effective as compared with those in known literatures. [ABSTRACT FROM AUTHOR]

Browne and Sleemens's paper is concerning the determination of potential uniquely from the nodes for regular sturm liouville problem. In this paper, we show that the potential function in the Sturm—Liouville problem having singularity at the point zero can be determined from the position of the nodes for the eigenfunctions. [ABSTRACT FROM AUTHOR]

In this paper some problem formulations of dynamic object parameters recovery described by non-autonomous system of ordinary differential equations with multipoint unshared edge conditions are investigated. Depending on the number of additional conditions the problem is reduced to an algebraic equations system or to a problem of quadratic programming. With this purpose the paper offers a new scheme of the edge conditions transfer method called by conditions shift. The method permits to get rid from differential links and multipoint unshared initially-edge conditions. The advantage of the proposed approach is concluded by capabilities of reduction of a parametric identification problem to essential simple problems of the solution of an algebraic system or quadratic programming. [ABSTRACT FROM AUTHOR]

We consider a general non-conservative Schrödinger equation defined on an open bounded domain Ω in Rn, with C²-boundary ..., subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ0 and Γ1 are the unobserved (or uncontrolled) and observed (or controlled) parts of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level (H¹(Ω)-level) terms, which accordingly may be viewed as unbounded perturbations. The first goal of the paper is to provide Carleman-type inequalities at the H¹-level, which do not contain lower-order terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C²-solutions; and next, it turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H²-solutions, and ultimately for H¹-solutions, The passage from H² to H¹-solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L2-normal traces in the case of Dirichlet B.C., but by contrast do not produce H¹-traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H1-solutions satisfying over-determined boundary conditions, and--above all--(ii) continuous observability (or stabilization) inequalities with an explicit constant. The more demanding purely Neumann boundary conditions requires the same geometrical conditions... [ABSTRACT FROM AUTHOR]

We consider a general non-conservative Schrödinger equation defined on an open bounded domain ω in Rn, with C²-boundary Γ = ∂ Ω = Γ0 ∪ Γ 1, Γ0 ∩ Γ 1 = 0, subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ0 and Γ1 are the unobserved (or uncontrolled) and observed (or controlled) part of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level (H¹ (&Omega)-level) terms, which according may be viewed as unbounded perturbations. The first goal of the paper is to provided Carleman-type inequalities at the H¹-level, which do not contain lower-order terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C²-solutions; and next, in turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H²-solutions, and ultimately for H¹-solutions. The passage from H²- to H¹-solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L2-normal traces in the case of Dirichlet B.C., but by contrast do not produce H¹-traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H¹-solutions satisfying over-determined boundary conditions, and--above all--(ii) continuous observability (or stabilization) inequalities with an explicit... [ABSTRACT FROM AUTHOR]

In this paper the problem of finding the error of an approximate solution for ill-posed problems on some compact sets is discussed. Sets of bounded monotonic or convex functions and sets of functions with a given Lipschitz constant are considered. Thee functions are given on a segment. Two approaches are used to construct sets of approximate solutions. In the first approach we assume that a function for the right-hand side of an operator equation is given on the whole segment, in the second approach we use only grid values of this function. Applying these approaches, we obtain quadratic or linear programming problems and find functions bounding above and below these approximate solutions. In the paper we use two algorithms to find errors that determine the sets of approximate solutions for the cases when operators of the problems are positive or integral. The inverse problem for the heat conduction equation is considered as a model example. [ABSTRACT FROM AUTHOR]

In this paper, the inverse scattering problem for the nonstationary Dirac system on the half-plane is considered. The uniqueness criterion for the inverse scattering problem in terms of boundary condition is described and the restoration of the potential from the scattering operator is proved. [ABSTRACT FROM AUTHOR]

We consider a parabolic problem with degeneracy in the interior of the spatial domain and we focus on the well-posedness of the problem and on inverse source problems. The novelties of the present paper are two. First, the degeneracy point is in the interior of the spatial domain. Second, we consider Neumann boundary conditions so that no previous result can be adapted to this situation. [ABSTRACT FROM AUTHOR]

Material properties such as thermal conductivity, magnetic permeability, electric permittivity, modulus of elasticity, Poisson's ratio, thermal expansion coefficient, etc. can vary spatially throughout a given solid object as it is the case in functionally graded materials. Finding this spatial variation is an inverse problem that requires boundary values of the field quantity such as temperature, magnetic field potential or electric field potential and its derivatives normal to the boundaries. In this paper, we solve the direct problem of predicting the spatial distribution of the field variable based on its measured boundary values and on the assumed spatial distribution of the diffusion coefficient using radial basis functions, the finite volume method and the finite element method,whose accuracies are verified against analytical solutions. Minimization of the sum of normalized least-squares differences between the calculated and measured values of the field quantity at the boundaries then leads to the correct parameters in the analytic model for the spatial distribution of the spatially varying material property. [ABSTRACT FROM AUTHOR]

In l1-regularization, which is an important tool in signal and image processing, one usually is concerned with signals and images having a sparse representation in some suitable basis, e.g., in awavelet basis. Many results on convergence and convergence rates of sparse approximate solutions to linear ill-posed problems are known, but rate results for the l1-regularization in case of lacking sparsity had not been published until 2013. In the last two years, however, two articles appeared providing sufficient conditions for convergence rates in case of non-sparse but almost sparse solutions. In the present paper, we suggest a third suffi- cient condition, which unifies the existing two and, by the way, also incorporates the well-known restricted isometry property. [ABSTRACT FROM AUTHOR]

In this paper, we aim to calibration pricing models from market prices. We investigate the problem of calibrating the parameters using the trust region method from given price data. We start with the Hull-White model and formulate the problem by obtaining the first kind integral equation, and then consider the parameter recovery problem of the Black-Scholes model. We apply trust region algorithm for numerical retrieval problems. Numerical simulations are given to illustrate the feasibility of our proposed method. [ABSTRACT FROM AUTHOR]

This paper deals with numerical solution and parameters inversion for a one-dimensional non-symmetric two-sided fractional advection-dispersion equation (FADE) with zero Neumann boundary condition in a finite domain. A fully discretized finite difference scheme is set forth based on Grünwald-Letnikov's definition of the fractional derivative, and its stability and convergence are proved by estimating the spectral radius of the coefficient matrix of the scheme. Furthermore, an inverse problem of simultaneously determining the fractional order and the dispersion coefficients is investigated, and numerical inversions are carried out by using the optimal perturbation regularization algorithm. The inversion results show that the fractional order and the dispersion coefficients in the FADE can be determined successfully by the final observations. [ABSTRACT FROM AUTHOR]

The inverse scattering problem considered in this paper is to reconstruct multiple sound-soft obstacles from one incident wave and its far field pattern. Based on the ideas of the Kirsch-Kress method and the hybrid Newton method, three variant Newton-type methods are developed for reconstructing the shape of multiple obstacles. The proposed hybrid Newton-type methods I and II can choose auxiliary curves adaptively, and do not require them to be contained in the unknown multiple obstacles. The proposed hybrid Newton-type method III is simpler than the hybrid Newton method developed by Kress in terms of computational complexity since it adopts quasi-Newton iterations in numerical reconstructions. Results of numerical examples show that the proposed methods, especially the one with both adaptively choosing auxiliary curves and quasi-Newton iterations, are more efficient and stable for reconstructing multiple obstacles. [ABSTRACT FROM AUTHOR]

In this paper, an inverse Robin problem for an elliptic equation is investigated. The Robin coefficient is assumed to be a small perturbation from a reference coefficient, hence a sparse structure is a priori known. An Lq-regularized output least square formulation is considered with q ∈ (0,1). The existence of the minimizer is proved. The convergence of the finite element discretization with respect to mesh size is shown. A modified Newton algorithm is applied to solve the smoothing first order optimality system. Two-dimensional and three-dimensional numerical examples demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In the matrix completion problem, most methods to solve the nuclear norm model are relaxing it to the nuclear norm regularized least squares problem. In this paper, we propose a new unconstraint model for matrix completion problem based on nuclear norm and indicator function and design a proximal point algorithm (PPA-IF) to solve it. Then the convergence of our algorithm is established strictly. Finally, we report numerical results for solving noiseless and noisy matrix completion problems and image reconstruction. [ABSTRACT FROM AUTHOR]

The thickness, thermal conductivity and porosity of textile material are three key factors which determine the heat-moisture comfort level of the human body to a large extent based on the heat and moisture transfer process in the human body-clothing-environment system. This paper puts forward an Inverse Problem of Textile Thickness-Heat conductivity-Porosity Determination (IPT(THP)D) based on the steady-state model of heat and moisture transfer and the heat-moisture comfort indexes. Adopting the idea of the weighted least-squares method, we formulate IPT(THP)D into a function minimization problem. We employ the Particle Swarm Optimization (PSO) method to stochastically search the optimal solution of the objective function. We put the optimal solution into the corresponding direct problem to verify the effectiveness of the proposed numerical algorithms and the validity of the IPT(THP)D. [ABSTRACT FROM AUTHOR]

In this paper, the adaptive multigrid conjugate gradient method is applied to the inversion of the space-dependent permeability function in a nonlinear diffusion equation, which plays an important role in promoting the permeability estimation within multiphase porous media flow. The forward problem is discretized using finite-difference methods and the estimation is formulated as a least-square minimization problem with a regularization term. Firstly, the conjugate gradient method for optimization is used to solve this parameter identification problem. Secondly, to reduce the amount of calculation, the idea of multigrid inversion and the necessary condition of 'the optimal solution should be the fixed point of multigrid inversion' is considered. Then, by combining the multigrid multiscale method with the conjugate gradient method, a new inversion method called the adaptive multigrid conjugate gradient method is designed. The results of numerical simulations indicate that the proposed method can effectively reduce the required computation burden and have good ability in anti-noise. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the inverse problem of a Schrödinger equation defined on an open, bounded, connected set Ω of a complete, n-dimensional Riemannian manifold M with metric g. It is subject to a non-homogenous Dirichlet boundary term. We aim at determining the potential coefficient by means of a Neumann boundary measurement on a portion Γ1 of the boundary Γ of Ω. Under sharp conditions on the complementary part Γ0 = Γ∖Γ1, and under weak regularity requirements on the data, we establish two canonical results of the inverse problem: (i) global uniqueness and (ii) global Lipschitz stability. The lower bound inequality corresponding to the upper bound inequality contained in (ii) is also given. Our proofs rely on four main ingredients: (a) sharp Carleman estimate at the H1-level for Schrödinger equations on Riemannian manifolds [Control Methods in PDE-Dynamical Systems (Snowbird 2005), Contemp. Math. 426, American Mathematical Society, Providence (2007), 339-404], (b) related continuous observability inequality at the H1-level [Control Methods in PDE-Dynamical Systems (Snowbird 2005), Contemp. Math. 426, American Mathematical Society, Providence (2007), 339-404], (c) a continuous observability inequality at the L2-level [J. Inverse Ill-Posed Probl. 12 (2004), 43-123], [Functional Analysis and Evolution Equations, Birkhäuser, Basel (2008), 613-636], (d) optimal regularity theory for Schrödinger equations with Dirichlet boundary data ([Differential Integral Equations 5 (1992), 521-535], [Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Encyclopedia Math. Appl. 2, Cambridge University Press, Cambridge, 2000] as well as the new Theorem 3.6). [ABSTRACT FROM AUTHOR]

It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth, i.e., the approximate solution may lack many details that the desired exact solution might possess. Two different approaches, both referred to as fractional Tikhonov methods have been introduced to remedy this shortcoming. This paper investigates the convergence properties of these methods by reviewing results published previously by various authors. We show that both methods are order optimal when the regularization parameter is chosen according to the discrepancy principle. The theory developed suggests situations in which the fractional methods yield approximate solutions of higher quality than Tikhonov regularization in standard form. Computed examples that illustrate the behavior of the methods are presented. [ABSTRACT FROM AUTHOR]

In this paper, a Volterra integral equation of the first kind in convolutions on the semiaxis when the integral operator kernel and the right-hand side of the equation have a bounded support is considered. An inverse problem of reconstructing the solution to the equation and the integral operator kernel from values of the right-hand side is formulated. Necessary and sufficient conditions for the inverse problem solvability are obtained. A uniqueness and stability theorem is proved. Explicit formulas for reconstruction of the solution and kernel are obtained. [ABSTRACT FROM AUTHOR]

In this paper, multilevel Jacobi and Gauss-Seidel type iteration methods with compression technique are developed for solving ill-posed integral equations by making use of the multiscale structure of the matrix representation of the integral operator. The methods are based on the combination of Tikhonov regularization and multiscale Galerkin methods, and lead to fast solutions of discrete regularization methods for the equations. Choice for an a posteriori regularization parameter is proposed. An optimal convergence order for the method with the choices of parameters is established. Numerical experiments are given to illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]