Inverse Operator Research Articles

A sequential inversion with outer iterations for the velocity and the intrinsic attenuation using an efficient wavefield inversion

Full-waveform inversion (FWI) has become a popular method to retrieve high-resolution subsurface model parameters. It is a highly nonlinear optimization problem based on minimizing the misfit between the observed and predicted data. For intrinsically attenuating media, wave propagation experiences significant loss of energy. Thus, for better data fitting, it is sometimes crucial to consider attenuation in FWI. Viscoacoustic FWI aims at achieving a joint inversion of the velocity and attenuation models. However, multiparameter FWI imposes additional challenges including expanding the null space and facing parameter trade-off issues. Theoretically, an ideal way to mitigate the trade-off issue in multiparameter FWI is to apply the inverse Hessian operator to the parameter gradients. However, it is often not practical to calculate the full Hessian and its matrix inverse because this will be extremely expensive. To improve the computational efficiency and mitigate the trade-off issue, we have used an efficient wavefield inversion (EWI) method to invert for the velocity and the intrinsic attenuation. This approach is implemented in the frequency domain, and the velocity, in this case, is complex-valued in the viscoacoustic EWI. We evaluate a sequential update strategy for the velocity and the intrinsic attenuation, and we repeat the separate optimizations, which we refer to as outer iterations, until the convergence is achieved. Because viscoacoustic EWI is able to recover an accurate velocity model, the velocity update leakage to the [Formula: see text] model is largely reduced. We determine the effectiveness of this approach using synthetic data generated for the viscoacoustic Marmousi and Overthrust models. To further demonstrate the validity of our approach, we generate data in the time domain using a viscoelastic wave equation solver and obtain reasonable inversion results in the frequency domain using the viscoacoustic approximation.

On Application Oriented Fuzzy Numbers for Imprecise Investment Recommendations

The subtraction of fuzzy numbers (FNs) is not an inverse operator to FNs addition. The family of all oriented FNs (OFNs) may be considered as symmetrical closure of all the FNs family in that the subtraction is an inverse operation to addition. An imprecise present value is modelled by a trapezoidal oriented FN (TrOFN). Then, the expected discount factor (EDF) is a TrOFFN too. This factor may be applied as a premise for invest-making. Proposed decision strategies are dependent on a comparison of an oriented fuzzy profit index and the specific profitability threshold. This way we get an investment recommendation described as a fuzzy subset on the fixed rating scale. Risk premium measure is a special case of profit index. Further in the paper, the Sharpe’s ratio, the Jensen’s ratio, the Treynor’s ratio, the Sortino’s ratio, Roy’s criterion and the Modiglianis’ coefficient are generalised for the case when an EDF is given as a TrOFN. In this way, we get many different imprecise recommendations. For this reason, an imprecise recommendation management module is described. Obtained results show that the proposed theory can be used as a theoretical background for financial robo-advisers. All theoretical considerations are illustrated by means of a simple empirical case study.

О методе расчета модуля непрерывности обратного оператора и его модификаций с приложением к нелинейным задачам геоэлектрики

Рассматриваются априорные оценки неоднозначности (погрешности) приближенных решений условно-корректных нелинейных обратных задач, основанные на модуле непрерывности обратного оператора и его модификациях. Установлена связь модуля непрерывности обратного оператора с разрешающей способностью геофизического метода. Показано, что в классе кусочно-постоянных решений, определенных на заданной сетке параметризации, модуль непрерывности обратного оператора и его модификации монотонно возрастают с увеличением размерности сетки. Предложен метод построения оптимальной сетки параметризации, которая имеет максимальную размерность при условии, что модуль непрерывности обратного оператора не превышает заданной величины. Представлен численный алгоритм расчета модуля непрерывности обратного оператора и его модификаций с использованием алгоритмов Монте-Карло, исследуются вопросы сходимости алгоритма. Предлагаемый метод применим также для расчета классических апостериорных оценок погрешности. Приводятся численные примеры для нелинейных обратных задач геоэлектрики. The article considers a priori estimates of the ambiguity (error) of approximate solutions of conditionally correct nonlinear inverse problems based on the modulus of continuity of the inverse operator and its modifications. It is shown that in the class of piecewise constant solutions defined on a given parametrization grid, the modulus of continuity of the inverse operator and its modifications monotonously increase with increasing mesh dimension. A method is proposed for constructing an optimal parameterization grid that has a maximum dimension provided that the modulus of continuity of the inverse operator does not exceed a given value. A numerical algorithm for calculating the modulus of continuity of the inverse operator and its modifications using Monte Carlo algorithms is presented; questions of convergence of the algorithm are investigated. The proposed method is also applicable for calculating classical posterior error estimates. Numerical examples are given for nonlinear inverse problems of geoelectrics.

Analytical inversion of one-dimensional operators of diffraction of electromagnetic waves in Sobolev spaces

The paper develops a method for analytical inversion in Sobolev spaces on a segment of one-dimensional integral diffraction operators that depend on a parameter. The method is based on positive definiteness of operators and the presence of an orthonormal basis. For a special case, the operator of diffraction of an H – polarized wave on a band or on the surface of a vibratory antenna, the inverse operator is explicitly constructed. It is possible to study the physical properties of surface currents using the inverse operator, as well as to construct an effective numerical method for solving integral diffraction equations.

Approximate Controllability for Stochastic Fractional Hemivariational Inequalities of Degenerate Type

AbstractThis paper is mainly concerned with stochastic fractional hemivariational inequalities of degenerate (or Sobolev) type in Caputo and Riemann-Liouville derivatives with order (1, 2), respectively. Based upon some properties of fractional resolvent family and generalized directional derivative of a locally Lipschitz function, some sufficient conditions are established for the existence and approximate controllability of the aforementioned systems. Particularly, the uniform boundedness for some nonlinear terms, the existence and compactness of certain inverse operator are not necessarily needed in obtained approximate controllability results.

Design of Time-Vertex Node-Variant Graph Filters

This paper addresses the node-variant graph filter on the joint time-vertex domain. An efficient algorithm is proposed to design the filter that can provide a good approximation to desirable linear operators for the time-varying signals on graphs. As a case study, the application of the designed filter on the signal denoise is addressed, where the filter is designed to approximate an inverse filtering operator. Numerical experiments conducted on the synthetic and real-world datasets demonstrate that the proposed filter is superior to the polynomial counterpart.

The rationale for the use of response functions in formalizing the tasks of economic management of the region

The paper presents studies of linear models of economic dynamics of the Neumann-Gale type, taking into account their possible stationarity, presents an analysis of existing classification approaches to the concept of optimality, presents their advantages and comparative characteristics, it is noted that the first type model open connects the concept of optimality with discounted maximization total utility. The first considers a closed system, the technological description of which includes the reproduction of all the resources necessary for development, including labor. Such a system has no external goals; its natural end in itself is development at the maximum pace. This is the most abstract and idealized scheme, but on the other hand it was it that made it possible to develop such fundamental concepts as equilibrium, a ray of (Neumann) balanced growth. Later, the apparatus of the closed model was replenished with the concepts of “direct and inverse Bellman operators”, “effective functional” (“potential”) of the model, etc. The second approach involves explicit accounting for consumption. Here the description becomes open, consumption is derived from the "technology" and described using the utility function. A new approach to the concept of “optimal development strategy” is proposed, a detailed analysis of the corresponding model is given. The article consists of three sections. 1 - staging part; 2 - analysis of the model with illustrative examples; 3 - conjugate (dual) model. The last section contains the main result on the connection of the optimal trajectories of the direct and dual problems. The paper provides an overview of literary sources in the subject area, as well as an economic interpretation of the results.

Dynamic development model using a temporary consumer scale

The paper presents studies of linear models of economic dynamics of the Neumann-Gale type, taking into account their possible stationarity, presents an analysis of existing classification approaches to the concept of optimality, presents their advantages and comparative characteristics, it is noted that the first type model - open - connects the concept of optimality with discounted maximization total utility. The first considers a closed system, the technological description of which includes the reproduction of all the resources necessary for development, including labor. Such a system has no external goals; its natural end in itself is development at the maximum pace. This is the most abstract and idealized scheme, but on the other hand it was it that made it possible to develop such fundamental concepts as equilibrium, a ray of (Neumann) balanced growth. Later, the apparatus of the closed model was replenished with the concepts of “direct and inverse Bellman operators”, “effective functional” (“potential”) of the model, etc. The second approach involves explicit accounting for consumption. Here the description becomes open, consumption is derived from the "technology" and described using the utility function. A new approach to the concept of “optimal development strategy” is proposed, a detailed analysis of the corresponding model is given. The article consists of three sections. 1 - staging part; 2 - analysis of the model with illustrative examples; 3 - conjugate (dual) model. The last section contains the main result on the connection of the optimal trajectories of the direct and dual problems. The paper provides an overview of literary sources in the subject area, as well as an economic interpretation of the results.

A methodological framework for inverse-modeling of propagating cortical activity using MEG/EEG

The prevailing view on the dynamics of large-scale electrical activity in the human cortex is that it constitutes a functional network of discrete and localized circuits. Within this view, a natural way to analyse magnetoencephalographic (MEG) and electroencephalographic (EEG) data is by adopting methods from network theory. Invasive recordings, however, demonstrate that cortical activity is spatially continuous, rather than discrete, and exhibits propagation behavior. Furthermore, human cortical activity is known to propagate under a variety of conditions such as non-REM sleep, general anesthesia, and coma. Although several MEG/EEG studies have investigated propagating cortical activity, not much is known about the conditions under which such activity can be successfully reconstructed from MEG/EEG sensor-data. This study provides a methodological framework for inverse-modeling of propagating cortical activity. Within this framework, cortical activity is represented in the spatial frequency domain, which is more natural than the dipole domain when dealing with spatially continuous activity.We define angular power spectra, which show how the power of cortical activity is distributed across spatial frequencies, angular gain/phase spectra, which characterize the spatial filtering properties of linear inverse operators, and angular resolution matrices, which summarize how linear inverse operators leak signal within and across spatial frequencies. We adopt the framework to provide insight into the performance of several linear inverse operators in reconstructing propagating cortical activity from MEG/EEG sensor-data. We also describe how prior spatial frequency information can be incorporated into the inverse-modeling to obtain better reconstructions.

Scattering theory in homogeneous Sobolev spaces for Schrödinger and wave equations with rough potentials

We study the scattering theory for the Schrödinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of this paper is concerned with an inverse-square potential in both of subcritical and critical constant cases, which is a particular model of scaling-critical singular perturbations. In the subcritical case, the existence of the wave and inverse wave operators defined on a range of homogeneous Sobolev spaces is obtained. In particular, we have the scattering to a free solution in the homogeneous energy space for both of the Schrödinger and wave equations. In the critical case, it is shown that the solution is asymptotically a sum of an n-dimensional free wave and a rescaled two-dimensional free wave. The second half of this paper is concerned with a generalization to a class of strongly singular decaying potentials. We provide a simple criterion in an abstract framework to deduce the existence of wave operators defined on a homogeneous Sobolev space from the existence of the standard ones defined on a base Hilbert space.

Regularization of 1/X2 potential in general case of deformed space with minimal length

In the general case of deformed Heisenberg algebra leading to the minimal length, we present a definition of the inverse square position operator. Our proposal is based on the functional analysis of the square of the position operator. Using this definition, a particle in the field of the inverse square potential is studied. We obtain analytical and numerical solutions for the energy spectrum of the considerable problem in different cases of deformation function. We conclude that the energy spectrum weakly depends on the choice of deformation function.

Generalized Inverse Operator for an Integrodifferential Operator in the Banach Space

By using the theory of generalized inversion of operators in Banach spaces, we establish conditions for the generalized invertibility of integrodifferential operators with degenerate kernels in Banach spaces. We obtain formulas for the construction of bounded projectors and the bounded generalized inverse operator for an integrodifferential operator with degenerate kernel in a Banach space. The results of investigations are illustrated by an example.

Direct approach for the characteristic function of a dissipative operator with distributional potentials

The main aim of this paper is to investigate the spectral properties of a singular dissipative differential operator with the help of its Cayley transform. It is shown that the Cayley transform of the dissipative differential operator is a completely non-unitary contraction with finite defect indices belonging to the class $$C_{0}.$$ Using its characteristic function and the spectral properties of the resolvent operator, the complete spectral analysis of the dissipative differential operator is obtained. Embedding the Cayley transform to its natural unitary colligation, a Caratheodory function is obtained. Moreover, the truncated CMV matrix is established which is unitary equivalent to the Cayley transform of the dissipative differential operator. Furthermore, it is proved that the imaginary part of the inverse operator of the dissipative differential operator is a rank-one operator and the model operator of the associated dissipative integral operator is constructed as a semi-infinite triangular matrix. Using the characteristic function of the dissipative integral operator with rank-one imaginary component, associated Weyl functions are established.

A Perception-Based Inverse Tone Mapping Operator for High Dynamic Range Video Applications

The drastic visual improvements introduced by High Dynamic Range (HDR) technologies open new markets for a wide range of industries. Among them, a significant opportunity is offered to owners of legacy Standard Dynamic Range (SDR) content that can be converted to the new standard to take advantage of the enhanced capabilities of the HDR displays. Similarly, since SDR broadcasting infrastructure will continue to be around for the time being, such conversion process is becoming an obvious necessity. To this end, different approaches have tried to efficiently convert SDR images and videos to HDR format, a procedure well known as inverse Tone Mapping. In this paper, we propose a novel high visual quality video inverse Tone Mapping Operator (iTMO) that addresses the inadequacies of the state-of-the-art methods, resulting in high visual quality HDR videos that match the capabilities of the HDR technology. Our approach is based on human visual perception and employs a segmentation method according to the Human Visual System (HVS) sensitivity to brightness changes in different regions of the frame and constructs the mapping curve using the brightness distribution information of these regions. Our iTMO uses a hybrid approach to achieve an optimal balance between the overall contrast and brightness of the output HDR frame by maximizing a weighted sum of contrast and brightness difference between input SDR and generated HDR frame. The proposed iTMO works equally well for all levels of brightness, eliminating any visual artifacts by dynamically maintaining changes at non-perceivable levels. Subjective and objective evaluations validated the superior visual performance of our proposed iTMO over state-of-the-art methods.

Some Fractional Operators with the Generalized Bessel–Maitland Function

In this paper, we aim to determine some results of the generalized Bessel–Maitland function in the field of fractional calculus. Here, some relations of the generalized Bessel–Maitland functions and the Mittag-Leffler functions are considered. We develop Saigo and Riemann–Liouville fractional integral operators by using the generalized Bessel–Maitland function, and results can be seen in the form of Fox–Wright functions. We establish a new operator Zν,η,ρ,γ,w,a+μ,ξ,m,σϕ and its inverse operator Dν,η,ρ,γ,w,a+μ,ξ,m,σϕ, involving the generalized Bessel–Maitland function as its kernel, and also discuss its convergence and boundedness. Moreover, the Riemann–Liouville operator and the integral transform (Laplace) of the new operator have been developed.

Multidomain pseudospectral integration preconditioning matrices for the advection and the diffusion operators

Single domain spectral/pseudospectral integration preconditioning matrices have been shown to be effective operators for solving differential equations. In this study we extend the integration precondition methodology to a multidomain computational framework, and construct global inverse matrices for the advection operator discretized by multidomain Gauss-Lobatto-Legendre pseudospectral methods. These inverse operators can be used as solution operators for solving first order boundary value problems even with piecewise continuous variable coefficients. Moreover, they are effective integration preconditioning matrices for the second order diffusion operator, in the sense that a model diffusion problem can be solved by an iterative method with the number of iteration steps being proportional to the number of subdomains but independent of the degree of the solution polynomial. Numerical experiments were conducted and we observed the performance of the inverse operators as expected.

Seismic-gather wavelet-stretching correction based on multiwavelet decomposition algorithm

Normal moveout correction is crucial in seismic data processing, but it generates a wavelet-stretching effect, especially on the larger offset or incident-angle seismic data. Wavelet stretching reduces the dominant frequency of seismic data. The greater the incident angle or offset, the lower dominant the frequency becomes. This is an unfavorable effect to amplitude variation with offset analysis. Therefore, we have introduced a wavelet stretching correction method based on the multiwavelet decomposition (MWD) algorithm. First, it decomposes the near-offset pilot trace and all the far-offset seismic traces in the same gather into a series of wavelets via the MWD algorithm. Then, the dominant frequencies of wavelets in the far-offset seismic traces are replaced by those corresponding wavelets in the pilot trace. Finally, the wavelets after the stretching correction are used to reconstruct the seismic trace. The model and field-data processing results show that this method can not only effectively reduce the wavelet stretching effect but it can also maintain the amplitude of each wavelet as invariant during the stretching correction procedure. Because only the frequencies of the decomposed wavelets are used, and no inverse wavelet operators is introduced, the wavelet stretching correction method does not distort the amplitude information.

A Multilevel Approach for Stability Conditions in Fractional Time Diffusion Problems

The Caputo definition of fractional derivatives introduces solution to the difficulties appears in the numerical treatment of differential equations due its consistency in differentiating constant functions. In the same time the memory and hereditary behaviors of the time fractional order derivatives (TFODE) still common in all definitions of fractional derivatives. The use of properties of companion matrices appears in reformulating multilevel schemes as generalized two level schemes is employed with the Gerschgorin disc theorems to prove stability condition. Caputo fractional derivatives with finite difference representations is considered. Moreover the effect of using the inverse operator which transmit the memory and hereditary effects to other terms is examined. The theoretical results is applied to a numerical example. The calculated solution has a good agreement with the exact solution.

Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part I

Within the framework of the Duffin-Kemmer-Petiau (DKP) formalism with a deformation, an approach to the construction of the path integral representation in parasuperspace for the Green’s function of a spin-1 massive particle in external Maxwell’s field is developed. For this purpose a connection between the deformed DKP-algebra and an extended system of the parafermion trilinear commutation relations for the creation and annihilation operators {a}_k^{pm } and for an additional operator a0 obeying para-Fermi statistics of order 2 based on the Lie algebra mathfrak{so} (2M + 2) is established. The representation for the operator a0 in terms of generators of the orthogonal group SO(2M correctly reproducing action of this operator on the state vectors of Fock space is obtained. An appropriate system of the parafermion coherent states as functions of para-Grassmann numbers is introduced. The procedure of the construction of finite-multiplicity approximation for determination of the path integral in the relevant phase space is defined through insertion in the kernel of the evolution operator with respect to para-supertime of resolutions of the identity. In the basis of parafermion coherent states a matrix element of the contribution linear in covariant derivative {hat{D}}_{mu } to the time-dependent Hamilton operator hat{mathrm{mathscr{H}}}left(tau right) , is calculated in an explicit form. For this purpose the matrix elements of the operators a0, {a}_0^2 , the commutators [a0, {a}_n^{pm } ], [ {a}_0^2,{a}_n^{pm } ], and the product hat{A} [a0, {a}_n^{pm } ] with hat{A} ≡ exp( -ifrac{2pi }{3}{a}_0 ), were preliminary defined.

Temporal large-eddy simulation based on direct deconvolution

In this work, we propose an approach for Temporal Large-Eddy Simulation (TLES) with direct deconvolution. In contrast to previous approaches such as the Temporal Approximate Deconvolution Model (TADM) by Pruett et al. [“A temporal approximate deconvolution model for large-eddy simulation,” Phys. Fluids 18, 028104 (2006)], the non-filtered velocity field is recovered using the differential form of the filter operation rather than from a truncated series expansion of the inverse filter operator. This direct deconvolution is used to obtain formal closure of an analytic evolution equation of the temporal residual-stress tensor. Thus, the Temporal Direct Deconvolution Model (TDDM) has advantages relative to the TADM in being both more accurate and requiring less computational effort. As for the TADM, a secondary regularization term based on selective frequency damping is employed. The TDDM was implemented in the spectral element code Nek5000 (Argonne National Laboratory, NEK5000 Version 17.0, 2019, https://nek5000.mcs.anl.gov) to simulate two canonical incompressible flows as three test cases: an a priori test case of Homogeneous Isotropic Turbulence (HIT) at Reλ = 50, a greatly coarsened a posteriori HIT case at Reλ = 190, and an a posteriori highly anisotropic turbulent channel flow at Reτ = 180. Analyses of the energy spectrum, the mean flow, the root-mean-square of the velocity fluctuations, and the Reynolds stresses are presented. The results demonstrate a significant improvement compared to no-model solutions regarding the mean flow in the turbulent channel and the energy spectrum in the HIT case, while the computational cost is reduced dramatically compared to the direct numerical simulation.