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This work elaborated an improved method to multiframe super-resolution (SR), which involves a nonlocal first-order regularization combined with a nonlocal p-Laplacian term. The nonlocal TV term excels at edge preserving, whilst the nonlocal p-Laplacian is commonly used to perfectly reconstruct image textures. Firstly, we discuss the existence and uniqueness of a solution to our new model in a well posed framework. Then, we derive a modified Primal-dual iteration to compute the super-resolved solution. Furthermore, we introduce a new bilevel optimization approach to learn two regularization parameters. The included tests validate that the introduced optimization procedure performs favorably compared to numerous SR approaches in terms of efficiency and accuracy. [ABSTRACT FROM AUTHOR]

Recently, a modified non-negative Tikhonov regularization method within the Bayesian frame has been successfully proposed, and demonstrated to have excellent compensation performance towards the signal distortion induced by long-distance transmission in coaxial cable. However, how the parameters of this modified method affect compensation performance has not yet been investigated and analyzed, which may in turn bring some obstacles in the practical utilization of this modified method. Herein, we carried out a systematic study on the effect of parameters on the signal compensation performance of this modified method. The bias constant D has almost no influence on the compensation performance if the unknown input signal satisfies the non-negative constraint. When the signal-to-noise ratio (SNR) is less than 40 dB, the compensation performance is continuously improved as the SNR increases. Once the SNR attains ∼40 dB, the compensated signal is nearly identical to the input signal. Further increment of SNR does not give rise to an improved compensation performance and the relative error between the compensated signal and input signal remains nearly unchanged. The sampling frequency interval Î" f of S21 should satisfy the restriction of Î" f = F s / n , with F s and n representing the sampling frequency of the digital oscilloscope and number of points of the output signal, respectively. The compensation performance is excellent initially, whereas it rapidly deteriorates once Î" f exceeds a certain value that is highly related to the length of the impulse response function. This study provides a beneficial reference for application of the modified signal compensation method. [ABSTRACT FROM AUTHOR]

Bearings in rotating machinery are prone to performance degradation and failures under complex working environments. The key issue of this work is to accurately extract the bearing fault induced transient signal from the noise signals. One of the effective methods to extract such transient components from noisy signals is sparse representation. However, commonly used sparse representation methods have the shortcomings of the underestimation of amplitude and insufficient estimation accuracy. To address these problems, we propose a generalized smoothing logarithmic (GSL) regularization based sparse representation method, which uses the Moreau envelope to perform generalized smoothing on the traditional logarithmic penalty function. Then the tunable Q-factor wavelet transform (TQWT) is adopted to construct sparse dictionaries for bearing faults since it is tight frame and matrix-free, which can ameliorate the computation speed of the proposed method. Simulation and experiments verify that the proposed GSL method is effective and applicable for bearing fault diagnosis. [ABSTRACT FROM AUTHOR]

In this paper, we propose an inexact Newton regularization combined with two-point gradient methods for nonlinear ill-posed problems. The basic idea of the proposed method is to linearize the equation around each outer iteration and subsequently apply a so-called two-point gradient method in the inner loop to accelerate the iterative process. Under suitable assumptions, we show that the iteration sequence generated by the proposed algorithm converges to a solution of the related problem in the noiseless situation. Furthermore, the stability and regularization properties of the proposed algorithm are analyzed in the noise-data case. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

When solving ill-posed inverse problems, a good choice of the prior is critical for the computation of a reasonable solution. A common approach is to include a Gaussian prior, which is defined by a mean vector and a symmetric and positive definite covariance matrix, and to use iterative projection methods to solve the corresponding regularized problem. However, a main challenge for many of these iterative methods is that the prior covariance matrix must be known and fixed (up to a constant) before starting the solution process. In this paper, we develop hybrid projection methods for inverse problems with mixed Gaussian priors where the prior covariance matrix is a convex combination of matrices and the mixing parameter and the regularization parameter do not need to be known in advance. Such scenarios may arise when data is used to generate a sample prior covariance matrix (e.g., in data assimilation) or when different priors are needed to capture different qualities of the solution. The proposed hybrid methods are based on a mixed Golub–Kahan process, which is an extension of the generalized Golub–Kahan bidiagonalization, and a distinctive feature of the proposed approach is that both the regularization parameter and the weighting parameter for the covariance matrix can be estimated automatically during the iterative process. Furthermore, for problems where training data are available, various data-driven covariance matrices (including those based on learned covariance kernels) can be easily incorporated. Numerical examples from tomographic reconstruction demonstrate the potential for these methods. [ABSTRACT FROM AUTHOR]

The robust tensor recovery problem consists in reconstructing a tensor from a sample of entries corrupted by noise, which has attracted great interest in a wide range of practical situations such as image processing and computer vision. In this paper, we study robust tensor recovery for third-order tensors with different degradations, which aims to recover a tensor from partial observations corrupted by Gaussian noise and sparse noise simultaneously. In contrast to traditional approaches based on the tensor nuclear norm penalty for the low-rank component and the tensor ℓ1 norm penalty for the sparse component, we propose a nonlocal robust low-rank tensor recovery model with nonconvex regularization (NRTRM) to explore the global low-rankness and nonlocal self-similarity of the underlying tensor. The NRTRM method is first to extract similar patched-tubes to form a third-order sub-tensor. Then a class of nonconvex low-rank penalties and nonconvex sparse penalties are employed to explore the low-rank component and the sparse corruptions for such sub-tensor, respectively. Moreover, a proximal alternating linearized minimization algorithm is developed to solve the resulting model in each group and its convergence is established under very mild conditions. Extensive numerical experiments on both multispectral images and video datasets demonstrate the superior performance of NRTRM in comparison with several state-of-the-art methods. [ABSTRACT FROM AUTHOR]

This paper investigates the inverse reaction coefficient problem for a time dependent nonlocal diffusion equation by utilizing the nonlocal flux measurement from an accessible part of region, which is a continuation and an extension of our recent work (Zheng and Ding 2020 Inverse Problems 36 035006). The uniqueness of inverse reaction coefficient problem is proved. The variational regularization method is proposed to overcome the ill-posedness of inverse problem, and an alternating iteration method is adopted to automatically select regularization parameters. Furthermore, we use Laplace approximation method to capture the statistics information of the solution. In particular, we prove the lower bound of confidence width is rigorously estimated to characterize the reliability of the method. Finally, numerical results indicate that the proposed method could yield an accurate estimate and efficient uncertainty quantification of the solution. [ABSTRACT FROM AUTHOR]

Multipath errors, as one of the manifest unmodelled error types introduced by code measurements, are absorbed in ionospheric observables, which results in ionospheric pollution. In this work, in order to cancel out random noise by an adjustment algorithm, the modelling of multipath errors derived from code measurement residuals in undifferentiated and uncombined precise point position (PPP) mode is realized by solving a regularization problem, in which Tikhonov regularization (i.e. the first order difference) is taken into account. Furthermore, the Thomas update algorithm is employed to find the solution and a bootstrap method is used to determine the optimal regularization parameter. After the multipath error model was constructed, it was used to extract the ionospheric observables, thereby enabling the correction of the code measurements. In this research, three ionosphere-retrieval schemes were designed, namely the the alignment of the carrier phase to the code (1) before and (2) after the multipath errors were corrected (CCBM and CCAM), and (3) a PPP solution (PPPS). Based on colocation experiments, the effects of removing errors arising from the multipath effect on the code were comparatively assessed. Large differences were found between the three schemes when used in a short-baseline experimental setting, but only small differences were found with a zero-baseline setup. In the short-baseline experiment, CCAM was more reliable than CCBM, with an improvement in accuracy of about 50%, and reached close agreement with PPPS. In addition, 14 continuously operating reference stations in Hong Kong were selected for ionosphere modelling with differential code bias (DCB) estimations. Similar results were seen, as follows: (1) compared with CCBM, CCAM and PPPS had better modelling accuracies, with an improvement of 60%; (2) CCAM and PPPS had better performance at five fitting stations with respect to CCBM, which resulted in improvements in the accuracies ranging from 60% to 80%; (3) taking five-day-average values of the satellite DCBs, the difference between Center for Orbit Determination in Europe (CODE) and CCAM was around 0.5 ns, which is consistent with the difference between CODE and PPPS, but about a 2 ns difference was verified between CODE and CCBM. [ABSTRACT FROM AUTHOR]

The non-convex regularization is a new approach for sparse recovery. A minimizer of the regularized function can be computed by applying the ST-(αℓ1 − βℓ2) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. However, the conventional PG method is limited to solve problems with the classical ℓ1 sparsity regularization. In this paper, we present two accelerated alternatives to the ST-(αℓ1 − βℓ2) algorithm by extending the PG method to the non-convex sparsity regularization. Moreover, we discuss a strategy to determine the radius R of the ℓ1-ball constraint by Morozov's discrepancy principle. Numerical results are reported to illustrate the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

We study linear inverse problems under the premise that the forward operator is not at hand but given indirectly through some input-output training pairs. We demonstrate that regularization by projection and variational regularization can be formulated by using the training data only and without making use of the forward operator. We study convergence and stability of the regularized solutions in view of Seidman (1980 J. Optim. Theory Appl. 30 535), who showed that regularization by projection is not convergent in general, by giving some insight on the generality of Seidman's nonconvergence example. Moreover, we show, analytically and numerically, that regularization by projection is indeed capable of learning linear operators, such as the Radon transform. [ABSTRACT FROM AUTHOR]

Choosing an appropriate regularization term is necessary to obtain a meaningful solution to an ill-posed linear inverse problem contaminated with measurement errors or noise. The ℓp norm covers a wide range of choices for the regularization term since its behavior critically depends on the choice of p and since it can easily be combined with a suitable regularization matrix. We develop an efficient algorithm that simultaneously determines the regularization parameter and corresponding ℓp regularized solution such that the discrepancy principle is satisfied. We project the problem on a low-dimensional generalized Krylov subspace and compute the Newton direction for this much smaller problem. We illustrate some interesting properties of the algorithm and compare its performance with other state-of-the-art approaches using a number of numerical experiments, with a special focus of the sparsity inducing ℓ1 norm and edge-preserving total variation regularization. [ABSTRACT FROM AUTHOR]

The classic regularization theory for solving inverse problems is built on the assumption that the forward operator perfectly represents the underlying physical model of the data acquisition. However, in many applications, for instance in microscopy or magnetic particle imaging, this is not the case. Another important example represent dynamic inverse problems, where changes of the searched-for quantity during data collection can be interpreted as model uncertainties. In this article, we propose a regularization strategy for linear inverse problems with inexact forward operator based on sequential subspace optimization methods (SESOP). In order to account for local modelling errors, we suggest to combine SESOP with the Kaczmarz' method. We study convergence and regularization properties of the proposed method and discuss several practical realizations. Relevance and performance of our approach are evaluated at simulated data from dynamic computerized tomography with various dynamic scenarios. [ABSTRACT FROM AUTHOR]

Recovering a signal from its Fourier magnitude is referred to as phase retrieval, which occurs in different fields of engineering and applied physics. This paper gives a new characterization of the phase retrieval problem. Particularly useful is the analysis revealing that the common gradient-based regularization does not restrict the set of solutions to a smaller set. Specifically focusing on binary signals, we show that a box relaxation is equivalent to the binary constraint for Fourier-types of phase retrieval. We further prove that binary signals can be recovered uniquely up to trivial ambiguities under certain conditions. Finally, we use the characterization theorem to develop an efficient denoising algorithm. [ABSTRACT FROM AUTHOR]

Fault diagnosis plays an important role in modern rotating machinery. Recently, some deep neural network-based domain adaptation methods have been successfully applied in cross-domain fault diagnosis problems. However, most of these methods only reduce the distribution discrepancy between different domains, without considering the geometry differences and misalignments across different domains. To this end, a new domain adaptation approach based on a deep neural network is proposed for mechanical fault diagnosis. There are three main contributions in total for the proposed approach. (i) ℓ2,1-norm based weight regularization is applied to reinforce the representative features of the original data. (ii) Manifold regularization is employed to further exploit the knowledge of marginal distributions and reduce the geometry differences between different domains. (iii) Subspace alignment is induced to reduce the misalignments of different domains. A gear dataset and a bearing dataset are selected to demonstrate the validity and robustness of the proposed approach. According to the experimental results, the proposed approach significantly outperforms the comparison approaches in mechanical fault diagnosis. [ABSTRACT FROM AUTHOR]

We consider the variational regularization for inverse problems in a general form. Based on the discrepancy principle, we propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level and is therefore purely data driven. Under variational source conditions, we obtain a posteriori error estimates. According to the Bakushinskii veto, convergence in the worst case scenario cannot be expected in general. However, by imposing certain conditions on the noisy data, we establish a convergence result for the heuristic rule. Applications of the results are addressed and numerical simulations are reported. [ABSTRACT FROM AUTHOR]

Low-dose x-ray computed tomography (LDCT) is desirable for reduced patient dose. This work develops new image reconstruction methods with deep learning (DL) regularization for LDCT. Our methods are based on the unrolling of a proximal forward–backward splitting (PFBS) framework with data-driven image regularization via deep neural networks. In contrast to PFBS-IR, which utilizes standard data fidelity updates via an iterative reconstruction (IR) method, PFBS-AIR involves preconditioned data fidelity updates that fuse the analytical reconstruction (AR) and IR methods in a synergistic way, i.e. fused analytical and iterative reconstruction (AIR). The results suggest that the DL-regularized methods (PFBS-IR and PFBS-AIR) provide better reconstruction quality compared to conventional methods (AR or IR). In addition, owing to the AIR, PFBS-AIR noticeably outperformed PFBS-IR and another DL-based postprocessing method, FBPConvNet. [ABSTRACT FROM AUTHOR]

This paper focuses on the data completion problem, which is well known to be an ill-posed inverse problem. We propose a dual regularization strategy without regularization parameter, based on the minimization of a functional which, instead of acting on the space of solutions, acts on the space of data. We prove the well-posedness of the minimization problem and the convergence of our regularized solution to the exact solution when the amount of noise on the data goes to 0. Moreover we prove that the regularized solution satisfies the well-known Morozov discrepancy principle. Our regularization strategy is closely related to the usual Kohn–Vogelius minimization strategy. In particular, we show that it allows not only to stably obtain a good reconstruction of the missing data of the Cauchy problem but also to determine the unique parameter of regularization for the Kohn–Vogelius strategy that satisfies the Morozov discrepancy principle. Finally, we present numerical results, in two and three dimensions, to highlight the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this work we consider a generalized bilevel optimization framework for solving inverse problems. We introduce fractional Laplacian as a regularizer to improve the reconstruction quality, and compare it with the total variation regularization. We emphasize that the key advantage of using fractional Laplacian as a regularizer is that it leads to a linear operator, as opposed to the total variation regularization which results in a nonlinear degenerate operator. Inspired by residual neural networks, to learn the optimal strength of regularization and the exponent of fractional Laplacian, we develop a dedicated bilevel optimization neural network with a variable depth for a general regularized inverse problem. We illustrate how to incorporate various regularizer choices into our proposed network. As an example, we consider tomographic reconstruction as a model problem and show an improvement in reconstruction quality, especially for limited data, via fractional Laplacian regularization. We successfully learn the regularization strength and the fractional exponent via our proposed bilevel optimization neural network. We observe that the fractional Laplacian regularization outperforms total variation regularization. This is specially encouraging, and important, in the case of limited and noisy data. [ABSTRACT FROM AUTHOR]

In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a linear vanishing damping term, which can be viewed not only as an extension of the asymptotical regularization, but also as a continuous analog of the Nesterov's acceleration scheme. New iterative regularization methods are derived from this continuous model in combination with appropriate numerical discretization schemes such as damped symplectic methods. The regularization property as well as convergence rates and acceleration effects under the Hölder-type source conditions of both continuous and discretized methods are proven. The second part of this paper is concerned with the application of the newly developed accelerated iterative regularization methods to the diffusion-based bioluminescence tomography, which is modeled as an inverse source problem in elliptic partial differential equations with both Dirichlet and Neumann boundary data. A novel mathematical formulation is proposed so that the discrepancy principle can be applied to stop the iteration procedure without the usage of Sobolev embedding constants. Several numerical examples, as well as acomparison with the state-of-the-art methods, are given to show the accuracy and the acceleration effect of the new methods. [ABSTRACT FROM AUTHOR]

We propose and study a data completion algorithm for recovering missing data from the knowledge of Cauchy data on parts of the same boundary. The algorithm is based on surface representation of the solution and is presented for the Helmholtz equation. This work is an extension of the data completion algorithm proposed by the two last authors where the case of data available of a closed boundary was studied. The proposed method is a direct inversion method robust with respect to noisy incompatible data. Classical regularization methods with discrepancy selection principles can be employed and automatically lead to a convergent schemes as the noise level goes to zero. We conduct 3D numerical investigations to validate our method on various synthetic examples. [ABSTRACT FROM AUTHOR]

Phase retrieval with prior information can be cast as a nonsmooth and nonconvex optimization problem. To decouple the signal and measurement variables, we introduce an auxiliary variable and reformulate it as an optimization with an equality constraint. We then solve the reformulated problem by graph projection splitting (GPS), where the two proximity subproblems and the graph projection step can be solved efficiently. With slight modification, we also propose a robust graph projection splitting (RGPS) method to stabilize the iteration for noisy measurements. Contrary to intuition, RGPS outperforms GPS with fewer iterations to locate a satisfying solution even for noiseless case. Based on the connection between GPS and Douglas–Rachford iteration, under mild conditions on the sampling vectors, we analyze the fixed point sets and provide the local convergence of GPS and RGPS applied to noiseless phase retrieval without prior information. For noisy case, we provide the error bound of the reconstruction. Compared to other existing methods, thanks for the splitting approach, GPS and RGPS can efficiently solve phase retrieval with prior information regularization for general sampling vectors which are not necessarily isometric. For Gaussian phase retrieval, compared to existing gradient flow approaches, numerical results show that GPS and RGPS are much less sensitive to the initialization. Thus they markedly improve the phase transition in noiseless case and reconstruction in the presence of noise respectively. GPS shows sharpest phase transition among existing methods including RGPS, while it needs more iterations than RGPS when the number of measurement is large enough. RGPS outperforms GPS in terms of stability for noisy measurements. When applying RGPS to more general non-Gaussian measurements with prior information, such as support, sparsity and TV minimization, RGPS either outperforms state-of-the-art solvers or can be combined with state-of-the-art solvers to improve their reconstruction quality. [ABSTRACT FROM AUTHOR]

We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data. [ABSTRACT FROM AUTHOR]

Several problems in magnetically confined fusion, such as the computation of exterior vacuum fields or the decomposition of the total magnetic field into separate contributions from the plasma and the external sources, are best formulated in terms of integral equation expressions. Based on Biot-Savart-like formulae, these integrals contain singular integrands. The regularization method commonly used to address the computation of various singular surface integrals along general toroidal surfaces is low-order accurate, and therefore requires a dense computational mesh in order to obtain sufficient accuracy. In this work, we present a fast, high-order quadrature scheme for the efficient computation of these integrals. Several numerical examples are provided demonstrating the computational efficiency and the high-order accurate convergence. A corresponding code for use in the community has been publicly released. [ABSTRACT FROM AUTHOR]

We consider the reconstruction of a diffusion coefficient in a quasilinear elliptic problem from a single measurement of overspecified Neumann and Dirichlet data. The uniqueness for this parameter identification problem has been established by Cannon and we therefore focus on the stable solution in the presence of data noise. For this, we utilize a reformulation of the inverse problem as a linear ill-posed operator equation with perturbed data and operators. We are able to explicitly characterize the mapping properties of the corresponding operators which allow us to apply regularization in Hilbert scales. We can then prove convergence and convergence rates of the regularized reconstructions under very mild assumptions on the exact parameter. These are, in fact, already needed for the analysis of the forward problem and no additional source conditions are required. Numerical tests are presented to illustrate the theoretical statements. [ABSTRACT FROM AUTHOR]

We use dimensional regularization to compute the one loop quantum gravitational contribution to the vacuum polarization on de Sitter background. Adding the appropriate Bogoliubov–Parasiuk–Hepp–Zimmermann counterterms gives a fully renormalized result which can be used to quantum-correct Maxwell’s equations. We use the Hartree approximation to argue that the electric field strengths of photons experience a secular suppression during inflation.Dedicated to Stanley Deser on the occasion of his 82nd birthday. [ABSTRACT FROM AUTHOR]

Sparsity optimization for linear least squares problems formulated as non-smooth regularization problems are considered in infinite-dimensional sequence spaces ℓp with p ∈ [0, 1]. Necessary optimality conditions in the format of a complementarity system are obtained. A monotonically convergent scheme is developed for the case p ∈ (0, 1]. For the case p = 0 a primal dual active set strategy based on the Lagrange multiplier rule is proposed and analyzed for special cases. [ABSTRACT FROM AUTHOR]

Many real-world applications are addressed through a linear least-squares problem formulation, whose solution is calculated by means of an iterative approach. A huge number of studies has been carried out in the optimization field to provide the fastest methods for the reconstruction of the solution, involving choices of adaptive parameters and scaling matrices. However, in the presence of an ill-conditioned model and real data, the need for a regularized solution instead of the least-squares one changed the point of view in favour of iterative algorithms able to combine fast execution with stable behaviour with respect to the restoration error. In this paper we analyse some classical and recent gradient approaches for the linear least-squares problem by looking at their way of filtering the singular values, showing in particular the effects of scaling matrices and non-negative constraints in recovering the correct filters of the solution. An original analysis of the filtering effect for the image deblurring problem with Gaussian noise on the data is also provided. [ABSTRACT FROM AUTHOR]

The process of covariant conserved charge of gravitational theory, which is covariant under general coordinate and local Lorentz transformations, has been applied to many tetrad fields, which reproduce Kerr-Ads spacetime, to calculate their conserved charges. It is shown that this process gives an infinite value of the conserved charges for Kerr—Ads spacetime. Therefore, the method of “regularization through relocalization", i.e., modification of the Lagrangian of the gravitational field through a total derivative, is used. This method gaves a finite and a consistent result of total energy and angular momentum for Kerr—Ads spacetime. [ABSTRACT FROM AUTHOR]

We present an exhaustive class of states with quantum backflow--the phenomenon in which a state consisting entirely of positive momenta has negative current and the probability flows in the opposite direction to the momentum. They are characterized by a general function of momenta subject to very weak conditions. Such a family of states is of interest in the light of a recent experimental proposal to measure backflow. We find one particularly simple state which has surprisingly large backflow--about 41% of the lower bound on flux derived by Bracken and Melloy. We study the eigenstates of a regularized current operator and we show how some of these states, in a certain limit, lead to our class of backflow states. This limit also clarifies the correspondence between the spectrum of the regularized current operator, which has just two non-zero eigenvalues in our chosen regularization, and the usual current operator. [ABSTRACT FROM AUTHOR]

We provide the full set of renormalization group functions for the renormalization of QCD in the minimal MOM scheme to four loops for the colour group SU(Nc). [ABSTRACT FROM AUTHOR]

We study the moduli space of the spectral curves y2 = W' (z)2 + f (z) which characterize the vacua of N = 1 U(n) supersymmetric gauge theories with an adjoint Higgs field and a polynomial tree level potential W(z). The integrable structure of the Whitham equations is used to determine the spectral curves from their moduli. An alternative characterization of the spectral curves in terms of critical points of a family of polynomial solutions W to Euler-Poisson- Darboux equations is provided. The equations for these critical points are a generalization of the planar limit equations for one-cut random matrix models. Moreover, singular spectral curves with higher order branch points turn out to be described by degenerate critical points of W. As a consequence we propose a multiple scaling limit method of regularization and show that, in the simplest cases, it leads to the Painlevè-I equation and its multi-component generalizations. [ABSTRACT FROM AUTHOR]

This paper extends the linear regularization scheme known as the approximate inverse to unbounded linear operators on Banach spaces. The principle of feature reconstruction is adapted from bounded operators to the unbounded scenario and, in addition, a new situation is examined where the data need to be pre-processed to fit into the mathematical model. In all these cases, invariance and regularization properties are surveyed and established for the example of fractional differentiation. Numerical results confirm the derived characteristics of the presented methods. [ABSTRACT FROM AUTHOR]

This paper deals with an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the fractional order in the 1D time-fractional diffusion equation with smooth initial functions by using boundary measurements. The uniqueness results for the inverse problem are proved on the basis of the inverse eigenvalue problem, and the Lipschitz continuity of the solution operator is established. A modified optimal perturbation algorithm with a regularization parameter chosen by a sigmoid-type function is put forward for the discretization of the minimization problem. Numerical inversions are performed for the diffusion coefficient taking on different functional forms and the additional data having random noise. Several factors which have important influences on the realization of the algorithm are discussed, including the approximate space of the diffusion coefficient, the regularization parameter and the initial iteration. The inversion solutions are good approximations to the exact solutions with stability and adaptivity demonstrating that the optimal perturbation algorithm with the sigmoid-type regularization parameter is efficient for the simultaneous inversion. [ABSTRACT FROM AUTHOR]

In this work we investigate the spectral signature of Navier-Stokes-Voigt (NSV) viscoelastic fluid flows by employing numerical simulations of a singular dyadic shell model. Our results clearly show that as the relaxation time is increased above a threshold, the inertial range is reduced, conserving part of the large-scale statistics. These results differ drastically from the two power-law scenarios observed in a previous work, where the NSV model was studied via Sabra shell model simulations instead.We also show that the additional elastic term regularizes the singular dyadic model, which is the main reason behind this reduction of degrees of freedom. The results of this work aim at proposing the NSV regularization as a sub-grid model. [ABSTRACT FROM AUTHOR]

The extended-domain-eigenfunction method (EDEM) proposed for solving elliptic boundary value problems on annular-like domains requires an inversion process. The procedure thus represents an ill-posed problem, whose numerical solution involves an ill-conditioned system of equations. In this paper, the ill-posed nature of EDEM is studied and numerical solutions based on regularization schemes are considered. It is shown that the EDEM solution methodology lends itself naturally to a formulation in terms of the well-known iterative Landweber method and the more general and faster converging semiiterative regularization schemes. Theoretical details and numerical results of the regularization schemes are presented for the case of the two-dimensional Laplace operator on annular domains. [ABSTRACT FROM AUTHOR]

In Alvarez et al (2006 Inverse Problems22 69–88), we studied the direct and inverse problem of deep bed filtration for a model proposed by Herzig et al (1970 Indus. Eng. Chem.65 8–35), which describes particle retention in a porous medium under injection of water with solid inclusions. However, many questions were left unanswered. Here most of these issues are solved for an alternate model of deep bed filtration also from [10]. Both models depend on the filtration function, which must be recovered from experimental data histories by solving an inverse problem. The main issues solved here are: (i) we establish the stability of the inverse problem under adequate conditions, leading to a robust numerical procedure; (ii) we present a numerical method for the direct problem that allows us to solve the inverse problem by parameter estimation in several spatial dimensions. We impose conditions on the experimental data preprocessing that allow us to recover analytic filtration functions with adequate physical behavior. [ABSTRACT FROM AUTHOR]

In standard inverse problems, the task is to solve an operator equation from given noisy data. However, sometimes also the operator is not known exactly. Therefore we propose a method that allows errors both in the operator and the data. In particular, we consider operator equations where the operator can be characterized by a function. For the stable reconstruction we propose the use of a Tikhonov-type functional with a generalized misfit term and an additional penalty term which promotes sparsity. Using an appropriate parameter choice rule for the two regularization parameters, we prove convergence and convergence rates for the method, and provide a first numerical example. [ABSTRACT FROM AUTHOR]

Variational sparsity regularization based on ℓ1-norms and other nonlinear functionals has gained enormous attention recently, both with respect to its applications and its mathematical analysis. A focus in regularization theory has been to develop error estimation in terms of regularization parameter and noise strength. For this sake, specific error measures such as Bregman distances and specific conditions on the solution such as source conditions or variational inequalities have been developed and used. In this paper we provide, for a certain class of ill-posed linear operator equations, a convergence analysis that works for solutions that are not completely sparse, but have a fast-decaying nonzero part. This case is not covered by standard source conditions, but surprisingly can be treated with an appropriate variational inequality. As a consequence, the paper also provides the first examples where the variational inequality approach, which was often believed to be equivalent to appropriate source conditions, can indeed go farther than the latter. [ABSTRACT FROM AUTHOR]

We introduce a classification of simple, regular, closed symmetric operators with deficiency indices (1, 1) according to a geometric criterion that extends the classical notions of entire operators and entire operators in the generalized sense due to M G Krein. We show that these classes of operators have several distinctive properties, some of them related to the spectra of their canonical selfadjoint extensions. In particular, we provide necessary and sufficient conditions on the spectra of two canonical self-adjoint extensions of an operator for it to belong to one of our classes. Our discussion is based on some recent results in the theory of de Branges spaces. [ABSTRACT FROM AUTHOR]

The classical dynamics of M-dimensional extended objects arising from stationary points of the world volume swept out in space time is discussed from various points of view. An introduction to the Hamiltonian mechanics of bosonic compact M(em)branes is given, emphasing the diversity of the different formulations and gauge choices. For moving hypersurfaces, a graph description--including its nonlinear realization of Lorentz invariance--and hydrodynamic formulations (in light-cone coordinates as well as when choosing the time coordinate of a Lorentz observer as the dependent variable) are presented. A matrix regularization for M = 2 (existing for all topologies) is explained in detail for the 2-sphere, as well as multilinear formulations for M > 2. The recently found dynamical symmetry that exists for all M and related reconstruction algebras are covered, just as some explicit solutions of the level-set equations. [ABSTRACT FROM AUTHOR]

In this paper, we give a regularized directional derivative-based Newton method for solving the inverse singular value problem (ISVP). The proposed method is also globalized by employing the directional derivative-based Wolfe line search conditions. Under some mild assumptions, the global and quadratic convergence of our method is established. To improve the practical effectiveness, we also propose a hybrid method for solving the ISVP.We show that the hybrid method converges quadratically and globally in the sense that a stationary point of a merit function for the ISVP is computed. Numerical tests demonstrate that the proposed hybrid method is very effective for solving the ISVP with distinct and multiple singular values. [ABSTRACT FROM AUTHOR]

I describe the logical basis of the method that I developed in 1962 and 1963 to define a quantum operator corresponding to the observable particle number of a quantized free scalar field in a spatially-flat isotropically expanding (and/or contracting) universe. This work also showed for the first time that particles were created from the vacuum by the curved spacetime of an expanding spatially-flat Friedmann-Lemâitre-Robertson-Walker (FLRW) universe. The same process is responsible for creating the nearly scaleinvariant spectrum of quantized perturbations of the inflaton scalar field during the inflationary stage of the expansion of the universe. I explain how the method that I used to obtain the observable particle number operator involved adiabatic invariance of the particle number (hence, the name adiabatic regularization) and the quantum theory of measurement of particle number in an expanding universe. I also show how I was led in a surprising way, to the discovery in 1964 that there would be no particle creation by these spatially-flat FLRW universes for free fields of any integer or half-integer spin satisfying field equations that are invariant under conformal transformations of the metric. The methods I used to define adiabatic regularization for particle number were based on generally-covariant concepts like adiabatic invariance and measurement that were fundamental and determined results that were unique to each given adiabatic order. This article is part of a special issue of Journal of Physics A:Mathematical and Theoretical in honour of StuartDowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'. [ABSTRACT FROM AUTHOR]

We review a few rigorous and partly unpublished results on the regularization of the stress-energy in quantum field theory on curved spacetimes: (1) the symmetry of the Hadamard/Seeley-DeWitt coefficients in smooth Riemannian and Lorentzian spacetimes, (2) the equivalence of the local ζ -function and the Hadamard-point-splitting procedure in smooth static spacetimes and (3) the equivalence of theDeWitt-Schwinger- and theHadamard-point-splitting procedure in smooth Riemannian and Lorentzian spacetimes. This article is part of a special issue of Journal of Physics A:Mathematical and Theoretical in honour of StuartDowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'. [ABSTRACT FROM AUTHOR]

We review an exact analytical resolution method for general one-dimensional quantal anharmonic oscillators: stationary Schrödinger equations with polynomial potentials. It is an exact form of WKB treatment involving 'spectral' (usual) versus 'classical' (newer) zeta-regularizations in parallel. The central results are a set of Bohr-Sommerfeld-like but exact quantization conditions, directly drawn from Wronskian identities, and appearing to extend the Betheansatz formulae of integrable systems. Such exact quantization conditions do not just select the eigenvalues; some evaluate the spectral determinants, and others the wavefunctions, for the spectral parameter in general position. This article is part of a special issue of Journal of Physics A:Mathematical and Theoretical in honour of StuartDowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'. [ABSTRACT FROM AUTHOR]

To estimate a parameter in an elliptic boundary value problem, the method of equation error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement). The estimated parameter converges to the exact value as the measured data converge to the exact value, provided Tikhonov regularization is used to control the instability inherent in the problem. The error in the estimated solution can be bounded in an appropriate quotient norm; estimates can be derived for both the underlying (infinite-dimensional) problem and a finite-element discretization that can be implemented in a practical algorithm. Numerical experiments demonstrate the efficacy and limitations of the method. [ABSTRACT FROM AUTHOR]

The problem of the high-frequency electromagnetic field coupling to an overhead electrical transmission line is an important issue in power signal transmission and here it is dealt with the regularization theory of ill-posed problems. The integro-differential equation describing the field-induced current into a line of finite length lying over a perfectly conducting soil is solved combining the use of Fourier series expansion and the Landweber algorithm. More precisely, the iterative regularization procedure is applied in order to obtain the value of the Fourier coefficients. [ABSTRACT FROM AUTHOR]

The Fourier method is a rather effective and very simple regularization method for solving some ill-posed problems. The known works on this method are all limited to the a priori choice of the regularization parameter. In this paper, we will systematically consider the a posteriori choice of the regularization parameter, and the corresponding error estimates between the exact solution and its approximation are given. Numerical examples show the effectiveness of the a posteriori method, and the comparisons of the numerical effect between the a posteriori and the a priori Fourier methods are also taken into account. [ABSTRACT FROM AUTHOR]

We consider approximate cloaking from a regularization viewpoint introduced in Kohn et al (2008 Inverse Problems 24 015016) for EIT and further investigated in Kohn et al (2010 Commun. Pure Appl. Math. 63 0973-1016) and Liu (2009 Inverse Problems 25 045006) for the Helmholtz equation. The cloaking schemes given by Kohn et al and Liu are shown to be (optimally) within | ln ρ|-1 in 2D and ρ in 3D of perfect cloaking, where ρ denotes the regularization parameter. In this paper, we show that by employing a soundhard layer right outside the cloaked region, one could (optimally) achieve ρN in RN, N ≥ 2, which significantly enhances the near-cloak. We then develop a cloaking scheme by making use of a lossy layer with well-chosen parameters. The lossy-layer cloaking scheme is shown to possess the same cloaking performance as the one with a sound-hard layer. Moreover, it is shown that the lossy layer could be taken as a finite realization of the sound-hard layer. Numerical experiments are also presented to assess the cloaking performances of all the cloaking schemes for comparisons. [ABSTRACT FROM AUTHOR]

In this paper we explore classification techniques for ill-posed problems. Two classes are linearly separable in some Hilbert space X if they can be separated by a hyperplane. We investigate stable separability, i.e. the case where we have a positive distance between two separating hyperplanes. When the data in the space Y is generated by a compact operator A applied to the system states ψ &b.epsis; X, we will show that in general we do not obtain stable separability in Y even if the problem in X is stably separable. In particular, we show this for the case where a nonlinear classification is generated from a non-convergent family of linear classes in X. We apply our results to the problem of quality control of fuel cells where we classify fuel cells according to their efficiency. We can potentially classify a fuel cell using either some external measured magnetic field or some internal current. However we cannot measure the current directly since we cannot access the fuel cell in operation. The first possibility is to apply discrimination techniques directly to the measured magnetic fields. The second approach first reconstructs currents and then carries out the classification on the current distributions. We show that both approaches need regularization and that the regularized classifications are not equivalent in general. Finally, we investigate a widely used linear classification algorithm Fisher's linear discriminant with respect to its ill-posedness when applied to data generated via a compact integral operator. We show that the method cannot stay stable when the number of measurement points becomes large. [ABSTRACT FROM AUTHOR]

This paper is concerned with a novel regularization technique for solving linear ill-posed operator equations in Hilbert spaces from data that are corrupted by white noise. We combine convex penalty functionals with extreme-value statistics of projections of the residuals on a given set of sub-spaces in the image space of the operator. We prove general consistency and convergence rate results in the framework of Bregman divergences which allows for a vast range of penalty functionals. Various examples that indicate the applicability of our approach will be discussed. We will illustrate in the context of signal and image processing that the presented method constitutes a locally adaptive reconstruction method. [ABSTRACT FROM AUTHOR]