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This paper presents a modified quasi-reversibility method for computing the exponentially unstable solution of a nonlocal terminal-boundary value parabolic problem with noisy data. Based on data measurements, we perturb the problem by the so-called filter regularized operator to design an approximate problem. Different from recently developed approaches that consist in the conventional spectral methods, we analyze this new approximation in a variational framework, where the finite element method can be applied. To see the whole skeleton of this method, our main results lie in the analysis of a semilinear case and we discuss some generalizations where this analysis can be adapted. As is omnipresent in many physical processes, there are likely myriad models derived from this simpler case, such as source localization problems for brain tumors and heat conduction problems with nonlinear sinks in nuclear science. With respect to each noise level, we benefit from the Faedo--Galerkin method to study the weak solvability of the approximate problem. Relying on the energy-like analysis, we provide detailed convergence rates in L2-H1 of the proposed method when the true solution is sufficiently smooth. Depending on the dimensions of the domain, we obtain an error estimate in Lr for some r > 2. Proof of the backward uniqueness for the quasi-linear system is also depicted in this work. To prove the regularity assumptions acceptable, several physical applications are discussed. [ABSTRACT FROM AUTHOR]

Dynamic loads widely exist in numerous applications of mechanical engineering structures, and their identification signifies an important issue for studying mechanical consequence of engineering structures and characterizing their dynamic characteristics. This research aims to reconstruct the dynamic loads in the deterministic structure of thin-walled cylindrical shell and airfoil structure. A new fast convergent iteration regularization method is developed and proposed for identifying dynamic loads. The stability, effectiveness, and convergence of this method are proved according to the regularization theory. The optimum asymptotic convergence order of the regularized solution is also provided according to the Morozov's discrepancy principle. In addition, two engineering examples are investigated to validate the effectiveness of the identification method proposed. The results demonstrate the applicability of the proposed method in mechanical engineering. [ABSTRACT FROM AUTHOR]

We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method. [ABSTRACT FROM AUTHOR]

Highlights • New regularization approach : We proposed a new approach for linear discrete ill-posed problems based on adding an artificial perturbation matrix with a bounded norm to the model matrix A. The objective of this artificial perturbation is to improve the singular value structure of A. This perturbation affects the fidelity of the model y = Ax 0 + z , and as a result, the equality relation becomes invalid. We show that using such modification provides a solution with better numerical stability. • New regularization parameter selection method : We develop a new regularization parameter selection approach that selects the regularizer in a way that minimizes the mean-squared error (MSE) between x 0 and its estimate x ^ , i.e., E ∥ x ^ − x 0 ∥ 2 2 . • Generality : A key feature of the approach is that it does not impose any prior assumptions on x 0. The vector x 0 can be deterministic or stochastic, and in the later case we do not assume any prior statistical knowledge. In addition, knowledge of the noise variance σ z 2 is not required. This makes the proposed approach applicable to a large number of linear discrete ill-posed problems. Abstract Estimating the values of unknown parameters in ill-posed problems from corrupted measured data presents formidable challenges in ill-posed problems. In such problems, many of the fundamental estimation methods fail to provide meaningful stabilized solutions. In this work, we propose a new regularization approach combined with a new regularization-parameter selection method for linear least-squares discrete ill-posed problems called constrained perturbation regularization approach (COPRA). The proposed COPRA is based on perturbing the singular-value structure of the linear model matrix to enhance the stability of the problem solution. Unlike many regularization methods that seek to minimize the estimated data error, the proposed approach is developed to minimize the mean-squared error of the estimator, which is the objective in many estimation scenarios. The performance of the proposed approach is demonstrated by applying it to a large set of real-world discrete ill-posed problems. Simulation results show that the proposed approach outperforms a set of benchmark regularization methods in most cases. In addition, the approach enjoys the shortest runtime and offers the highest level of robustness of all the tested benchmark regularization methods. [ABSTRACT FROM AUTHOR]

We propose and analyze a family of successive projection methods whose step direction is the same as the Landweber method for solving nonlinear ill-posed problems that satisfy the Tangential Cone Condition (TCC). This family encompasses the Landweber method, the minimal error method, and the steepest descent method; thus, providing an unified framework for the analysis of these methods. Moreover, we define new methods in this family, which are convergent for the constant of the TCC in a range twice as large as the one required for the Landweber and other gradient type methods. The TCC is widely used in the analysis of iterative methods for solving nonlinear ill-posed problems. The key idea in this work is to use the TCC in order to construct special convex sets possessing a separation property, and to successively project onto these sets. Numerical experiments are presented for a nonlinear two-dimensional elliptic parameter identification problem, validating the efficiency of our method. [ABSTRACT FROM AUTHOR]

Image enhancement with forward-and-backward (FAB) diffusion lacks a sound theory and is numerically very challenging due to its diffusivities that are negative within a certain gradient range. In our paper, we address both problems. First we establish a comprehensive theory for space-discrete and time-continuous FAB diffusion processes. It requires approximating the gradient magnitude with a non-standard discretisation. Then, we show that this theory carries over to the fully discrete case, when an explicit time discretisation with a fairly restrictive step-size limit is applied. To come up with more efficient algorithms, we propose three accelerated schemes: (i) an explicit scheme with global time step size adaptation that is also well suited for parallel implementations on GPUs, (ii) a randomised two-pixel scheme that offers optimal adaptivity of the time step size, (iii) a deterministic two-pixel scheme which benefits from less restrictive consistency bounds. Our experiments demonstrate that these algorithms allow speed-ups by up to three orders of magnitude without compromising stability or introducing visual artefacts. [ABSTRACT FROM AUTHOR]

Linear systems of equations with a matrix whose singular values decay to zero with increasing index number without a significant gap are commonly referred to as linear discrete ill-posed problems. We are interested in solving large systems of this kind when the right-hand side has k > 1 column vectors. The systems may be regarded as one system of equations with a block vector (with k columns) as the right-hand side and then solved by a block iterative method, or as k linear systems of equations (one for each right-hand side vector) that can be solved independently. Thus, the solution is a block vector with k columns. In many applications, including the restoration of color images, the right-hand side represents measurements that are contaminated by errors. Block iterative methods compute all columns of the solution block vector simultaneously. We will illustrate the performance of standard block GMRES methods and global GMRES methods, which also are block methods, and show that they may determine computed solutions of lower quality than when each column of the solution block vector is computed independently by a "standard" iterative method. We introduce a new local block GMRES method that can overcome the problems associated with block GMRES methods applied to linear discrete ill-posed problems. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider a variant of projected Tikhonov regularization method for solving Fredholm integral equations of the first kind. We give a theoretical analysis of this method in the Hilbert space L 2 ( a , b ) $L^{2}(a,b)$ setting and establish some convergence rates under certain regularity assumption on the exact solution and the kernel k ( ⋅ , ⋅ ) $k(\cdot,\cdot)$ . Some numerical results are also presented.

Subject of Study.The paper deals with further development of the method of computational experiments for solving ill-posed problems, e.g., the inverse spectroscopy problem. This method produces an effective (nonoverstated) estimate for solution error of the first-kind equation. Method of Research. An equation is solved by the Tikhonov regularization method. We have obtained nonoverstated estimate for solution error and a new principle for choosing the regularization parameter on the basis of the truncating singular number spectrum of an operator. It is proposed to estimate the truncation magnitude by results of solving model (training, learning) examples close to an initial example (problem). This method takes into account an additional information about the solution. Main Results. We have derived a new, more accurate estimate for regularized solution error using the truncation parameter g. Ways for determining g according to the results of solving model examples are proposed. The method of modeling or training is applied to solving the inverse spectroscopy problem (restoration of a fine spectrum structure by solving integral equation on the basis of an experimental spectrum and the spread function of a spectral device). The method makes it possible to resolve close lines and select weak lines. Practical Relevance. The proposed method can be used to restore smoothed and noisy spectra, in other words, to enhance the resolution of spectral devices by mathematical and computer processing of experimental spectra.

Recently filter-based regularization methods have been well investigated for ill-posed problems when the forward operators are compact. There are many ill-posed problems connected with pseudo-differential operators. But there is no uniform method for this kind of problems. The work on generalization of filter-based regularization methods to pseudo-differential operator is necessary. In this paper, we present a regularization framework for solving the mildly ill-posed problems involved pseudo-differential operators. A general regularization method for this kind of problems is given. The order-optimal error estimates are derived under the usual source conditions. As an example, a new fractional Tikhonov regularization method could be cast into the general framework. Numerical experiments are conducted for showing the validity of the new fractional Tikhonov method. [ABSTRACT FROM AUTHOR]

In this paper, based on the cubic B-spline finite element (CBSFE) and the radial basis functions (RBFs) methods, the inverse problems of finding the nonlinear source term for system of reaction-diffusion equations are studied. The approach of the proposed methods are to approximate unknown coefficients by a polynomial function whose coefficients are determined from the solution of minimization problem based on the overspecified data. In fact, this work considers a comparative study between the cubic B-spline finite element method and radial basis functions method. The stability and convergence analysis for these problems are investigated and some examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

In recent years, the Cauchy problem for inhomogeneous Helmholtz equation (CPHE) is often considered as the wave number k is a constant number. At present, we investigate a three dimensional CPHE with inhomogeneous Cauchy conditions given at z = 0 while wave number k is perturbed. The problem is well-known to be ill-posed in the sense of Hadamard. Therefore, we regularize the problem by applying the truncation method and possess error estimate between the exact solution and the regularized solution. A numerical experiment is given for the purpose of illustrating our method. [ABSTRACT FROM AUTHOR]

Range Restricted GMRES (RRGMRES) can be considered as regularizing iterative method and its iterates as regularized solutions. In this paper, we use two preconditioned versions of this method for solving ill-posed inverse problems. Our proposed preconditioner matrix is appropriate to use directly for linear systems of the form Ax=b with square ill-conditioned coefficient matrix. We show numerically that the Preconditioned RRGMRES method has more stable solution compared to RRGMRES. [ABSTRACT FROM AUTHOR]

Considering a general linear ill-posed equation, we explore the duality arising from the requirement that the discrepancy should take a given value based on the estimation of the noise level, as is notably the case when using the Morozov principle. We show that, under reasonable assumptions, the dual function is smooth, and that its maximization points out the appropriate value of Tikhonov’s regularization parameter. The numerical relevance of our approach is established by means of an illustrative example from nonparametric instrumental regression, a standard problem in statistics. [ABSTRACT FROM AUTHOR]

This paper proposes highly accurate one-step backward-forward algorithms for solving multi-dimensional backward heat conduction problems (BHCPs). The BHCP is renowned for being ill-posed because the solutions are generally unstable and highly dependent on the given data. In this paper, the present algorithm combines algebraic equations with a high-order Lie-group scheme to construct one-step algorithms called the backward fictitious integrate method (BFTIM) and the forward fictitious integrate method (FFTIM). First, the original parabolic equation is transformed into a new parabolic equation of an evolution type by introducing a fictitious time variable. Then, the numerical integration of the discretized algebraic equations must satisfy the constraints of the cone structure, Lie-group and Lie algebra at each fictitious time step. Finally, the algorithms with the minimum fictitious time steps along the manifold of the Lie-group scheme approach the true solution with one step when given an initial guess. In addition, this paper provides a strategy to determine the initial guess, which is the reciprocal relationship of the initial condition (IC) and the final condition (FC). More importantly, the IC and FC can be recovered by the BFTIM and FFTIM according to the relation between the IC and FC, even under large noisy measurement data. Five numerical examples of the BHCP are tested and numerical results demonstrate that the present schemes are more effective and stable. In general, the numerical implementations of the BFTIM and FFTIM are simple and have one-step convergence speeds. [ABSTRACT FROM AUTHOR]

The well-known approach to solve the ill-posed problem is Tikhonov regularization scheme. But, the approximate solution of Tikhonov scheme may not contain all the details of the exact solution. To circumference this problem, weighted Tikhonov regularization has been introduced. In this article, we propose two a posteriori parameter choice rules to choose the regularization parameter for weighted Tikhonov regularization and establish the optimal rate of convergence O(δα+1α+2) for the scheme based on these proposed rules. The numerical results are documented to demonstrate the significance of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we investigate the problem of identifying the conductivity in electrical impedance tomography from one boundary measurement. A variational method with total variation regularization is here proposed to tackle this problem. We discretize the PDE as well as the conductivity with piecewise linear, continuous finite elements. We prove the stability and convergence of this technique. For the numerical solution we propose a projected Armijo algorithm. Finally, a numerical experiment is presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

A possible modelling approach in the large eddy simulation (LES) of reactive flows is to deconvolve resolved scalars. Indeed, by inverting the LES filter, scalars such as mass fractions are reconstructed. This information can be used to close budget terms of filtered species balance equations, such as the filtered reaction rate. Being ill-posed in the mathematical sense, the problem is very sensitive to any numerical perturbation. The objective of the present study is to assess the ability of this kind of methodology to capture the chemical structure of premixed flames. For that purpose, three deconvolution methods are tested on a one-dimensional filtered laminar premixed flame configuration: the approximate deconvolution method based on Van Cittert iterative deconvolution, a Taylor decomposition-based method, and the regularised deconvolution method based on the minimisation of a quadratic criterion. These methods are then extended to the reconstruction of subgrid scale profiles. Two methodologies are proposed: the first one relies on subgrid scale interpolation of deconvolved profiles and the second uses parametric functions to describe small scales. Conducted tests analyse the ability of the method to capture the chemical filtered flame structure and front propagation speed. Results show that the deconvolution model should include information about small scales in order to regularise the filter inversion.a priorianda posterioritests showed that the filtered flame propagation speed and structure cannot be captured if the filter size is too large. [ABSTRACT FROM PUBLISHER]

We focus on the nonhomogeneous backward heat problem of finding the initial temperature θ = θ(x, y) = u(x, y, 0) such that ... where Ω = (0, π) × (0, π). In the problem, the source ƒ = ƒ(x, y, t) and the final data h = h(x, y) are determined through random noise data gij(t) and dij satisfying the regression models ... where (Xi, Yj) are grid points of Ω The problem is severely ill-posed. To regularize the instable solution of the problem, we use the trigonometric least squares method in nonparametric regression associated with the projection method. In addition, convergence rate is also investigated numerically. [ABSTRACT FROM AUTHOR]

Despite the strong focus of regularization on ill-posed problems, the general construction of such methods has not been fully explored. Moreover, many previous studies cannot be clearly adapted to handle more complex scenarios, albeit the greatly increasing concerns on the improvement of wider classes. In this note, we rigorously study a general theory for filter regularized operators in a Hilbert space for nonlinear evolution equations which have occurred naturally in different areas of science. The starting point lies in problems that are in principle ill-posed with respect to the initial/final data – these basically include the Cauchy problem for nonlinear elliptic equations and the backward-in-time nonlinear parabolic equations. We derive general filters that can be used to stabilize those problems. Essentially, we establish the corresponding well-posed problem whose solution converges to the solution of the ill-posed problem. The approximation can be confirmed by the error estimates in the Hilbert space. This work improves very much many papers in the same field of research. [ABSTRACT FROM AUTHOR]

We consider the problem of numerical analytic continuation of an analytic function f ( z ) = f ( x + i y ) on a strip domain Ω + = { z = x + i y ∈ C | x ∈ R , 0 < y < y 0 } , where the data is given approximately only on the line y = 0 . This is a severely ill-posed problem. Motivated by the advantage of iterative methods for solving ill-posed problems, we propose a new modified iterative method to solve this problem under both a priori and a posteriori parameter choice rules. Moreover, some sharp error estimates between the exact solution and its approximation are proved. Some interesting numerical examples are conducted for showing that the newly-developed method works well. [ABSTRACT FROM AUTHOR]

Two classes of methods for approximate matrix inversion with convergence orders p=3∗2 k+1 (Class 1) and p=5∗2 k−1 (Class 2), k≥1 an integer, are given based on matrix multiplication and matrix addition. These methods perform less number of matrix multiplications compared to the known hyperpower method or pth-order method for the same orders and can be used to construct approximate inverse preconditioners for solving linear systems. Convergence, error, and stability analyses of the proposed classes of methods are provided. Theoretical results are justified with numerical results obtained by using the proposed methods of orders p=7,13 from Class 1 and the methods with orders p=9,19 from Class 2 to obtain polynomial preconditioners for preconditioning the biconjugate gradient (BICG) method for solving well- and ill-posed problems. From the literature, methods with orders p=8,16 belonging to a family developed by the effective representation of the pth-order method for orders p=2 k, k is integer k≥1, and other recently given high-order convergent methods of orders p=6,7,8,12 for approximate matrix inversion are also used to construct polynomial preconditioners for preconditioning the BICG method to solve the considered problems. Numerical comparisons are given to show the applicability, stability, and computational complexity of the proposed methods by paying attention to the asymptotic convergence rates. It is shown that the BICG method converges very quickly when applied to solve the preconditioned system. Therefore, the cost of constructing these preconditioners is amortized if the preconditioner is to be reused over several systems of same coefficient matrix with different right sides. [ABSTRACT FROM AUTHOR]

We devise a fast and reliable computational framework for the elasticity imaging inverse problem of detecting cancerous tumors in the human body using an output least-squares (OLS) approach. From a mathematical standpoint, this inverse problem requires identifying a parameter in a mixed variational problem. We develop, in a continuous setting, a first-order adjoint method and two second-order adjoint methods. The continuous formulae are then used to devise a scheme for an efficient computation of the gradient and the Hessian of the OLS objective. We give detailed numerical examples. [ABSTRACT FROM PUBLISHER]

The solution of inverse problems has many applications in mathematical physics. Regularization methods can be applied to obtain the solution of ill-conditioned inverse problems by solving a family of neighboring well-posed problems. Thus, it is significant to investigate the regularization methods to increase the accuracy and efficiency of the solution of inverse problems. In this work, a new regularization filter and the related regularization method based on the singular system theory of compact operator are proposed to solve ill-posed problems. The Cauchy problem of Laplace equation of the first kind is a kind of well-known ill-posed problem. Numerical tests show that the proposed regularization method can solve the Cauchy problems more efficiently under a proper selection of regularization parameters. Numerical results also show that the proposed method is especially effective in solving ill-posed problems with big perturbations. [ABSTRACT FROM AUTHOR]

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Golub–Kahan iterative bidiagonalization represents the core algorithm in several regularization methods for solving large linear noise-polluted ill-posed problems. We consider a general noise setting and derive explicit relations between (noise contaminated) bidiagonalization vectors and the residuals of bidiagonalization-based regularization methods LSQR, LSMR, and CRAIG. For LSQR and LSMR residuals we prove that the coefficients of the linear combination of the computed bidiagonalization vectors reflect the amount of propagated noise in each of these vectors. For CRAIG the residual is only a multiple of a particular bidiagonalization vector. We show how its size indicates the regularization effect in each iteration by expressing the CRAIG solution as the exact solution to a modified compatible problem. Validity of the results for larger two-dimensional problems and influence of the loss of orthogonality is also discussed. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problems of nonlinear partial differential equations of the normal form in the class of analytic functions. We apply semi-discrete finite difference approximation which discretizes the problems only with respect to the time variable, and we give a proof for its convergence. The result implies that there are cases of convergence of finite difference schemes applied to ill-posed Cauchy problems. [ABSTRACT FROM AUTHOR]

The inequality constraint in stochastic configuration networks (SCNs) is key to their universal approximation capability. SCNs incrementally add new units to neural networks under a supervisory mechanism. However, with an increasing number of hidden nodes, the output weights which are evaluated by the least squares method, are frequently unstable. This instability phenomenon (arbitrarily small perturbations leading to large changes in the solutions) can be classified as an ill-posed problem. Instability is one of the most important manifestations of ill-posed problems, and the Tikhonov regularization method, which is widely used to address ill-posed problems, is undesirable. This study examined the supervisory mechanism of SCNs and algebraic properties of the hidden output matrix, thereby proposing a new greedy stochastic configuration network (GSCN) for ill-posed problems. First, under the premise of ensuring universal approximation capability, GSCN tries to optimize the randomly assigned input weights and biases based on the hunter–prey optimization (HPO) algorithm. The optimized hidden parameters which can reduce residual error the most are selected for the newly added hidden nodes. Second, singular value decomposition (SVD) and orthogonal–triangular (QR) decomposition with column pivoting are introduced to extract a linearly independent subset of the hidden output matrix for ill-posed problems in SCNs. Finally, one function approximation, six benchmark regressions, two classification datasets, and bearing fault diagnosis are employed to evaluate the performance of GSCN. Experimental results indicate that GSCN shows superior performance with respect to fast convergence, generalization, and stability compared with other contrast methods. This code is available at https://github.com/Gzu-public-bigdata/GSCN. [ABSTRACT FROM AUTHOR]

The article is devoted to the (singular) integral representation of the electromagnetic field ((S)IP EMF). The questions relating to the place IP EMF in electrodynamics, as well as their use for constructing a rigorous self-consistent theory of antennas, which is by far the most part is a theory of the far zone. The definition of ill-posed problem of antenna theory. Given IP EMF for the most common classes of radiating structures. The results of numerical modeling are presented.

In the fields of continuous casting and the roll stepped cooling, the heat transfer coefficient is piecewise linear. However, few papers discuss the solution of the backward heat conduction problem in this situation. Therefore, the aim of this paper is to solve the backward heat conduction problem, which has the piecewise linear heat transfer coefficient. Firstly, the ill-posed of this problem is discussed and the truncated regularized optimization scheme is introduced to solve this problem. Secondly, because the regularization parameter is the key factor for the regularization method, this paper presents an improved method for choosing the regularization parameter to reduce the iterative number and proves the fourth-order convergence of this method. Furthermore, the numerical simulation experiments show that, compared with other methods, the improved method of fourth-order convergence effectively reduces the iterative number. Finally, the truncated regularized optimization scheme is used to estimate the initial temperature, and the results of numerical simulation experiments illustrate that the inverse values match the exact values very well.

Via Carleman estimateswe prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the conormal derivative of the temperature on a relatively open subset of the lateral boundary of the space-time domain. [ABSTRACT FROM AUTHOR]

We consider the solution of large linear systems of equations that arise from the discretization of ill-posed problems. The matrix has a Kronecker product structure and the right-hand side is contaminated by measurement error. Problems of this kind arise, for instance, from the discretization of Fredholm integral equations of the first kind in two space-dimensions with a separable kernel and in image restoration problems. Regularization methods, such as Tikhonov regularization, have to be employed to reduce the propagation of the error in the right-hand side into the computed solution. We investigate the use of the global Golub–Kahan bidiagonalization method to reduce the given large problem to a small one. The small problem is solved by employing Tikhonov regularization. A regularization parameter determines the amount of regularization. The connection between global Golub–Kahan bidiagonalization and Gauss-type quadrature rules is exploited to inexpensively compute bounds that are useful for determining the regularization parameter by the discrepancy principle. [ABSTRACT FROM AUTHOR]

We study the problem of nonparametric estimation of the risk-neutral densities from options data. The underlying statistical problem is known to be ill-posed and needs to be regularized. We propose a novel regularized empirical sieve approach for the estimation of the risk-neutral densities which relies on the notion of the minimal martingale entropy measure. The proposed approach can be used to estimate the so-called pricing kernels which play an important role in assessing the risk aversion over equity returns. The asymptotic properties of the resulting estimate are analyzed and its empirical performance is illustrated. [ABSTRACT FROM AUTHOR]

This paper proposes a new methodology for moving force identification (MFI) from the responses of bridge deck. Based on the existing time domain method (TDM), the MFI problem eventually becomes solving the linear algebraic equation in the form Ax = b . The vector b is usually contaminated by an unknown error e generating from measurement error, which often called the vector e as ‘‘noise’’. With the ill-posed problems that exist in the inverse problem, the identification force would be sensitive to the noise e . The proposed truncated generalized singular value decomposition method (TGSVD) aims at obtaining an acceptable solution and making the noise to be less sensitive to perturbations with the ill-posed problems. The illustrated results show that the TGSVD has many advantages such as higher precision, better adaptability and noise immunity compared with TDM. In addition, choosing a proper regularization matrix L and a truncation parameter k are very useful to improve the identification accuracy and to solve ill-posed problems when it is used to identify the moving force on bridge. [ABSTRACT FROM AUTHOR]

The heat equation backward in time subject to the constraint , with , , , being given, is regularized by the well-posed mollified problem where An a posteriori parameter choice rule for this regularization method is suggested, which yields the error estimate Furthermore, we establish stability estimates of Hölder type for all derivatives of the solutions with respect to x and t. [ABSTRACT FROM AUTHOR]