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This paper outlines a novel approach for simulating general nonlinear circuits in the time-domain. The proposed approach can be considered as the generalization of the numerical inversion Laplace transform (NILT) which has been used for circuits with only linear elements. The new approach enables the well-known advantages of NILT such the guaranteed numerical stability and the high-order approximation, to be carried to the domain of nonlinear circuit. A numerical example is given to demonstrate the validity of the proposed method.

Full-waveform inversion (FWI) serves as a useful tool to quantitatively investigate the properties of the subsurface. Presently, three-dimensional (3D) elastic FWI uses a finite-difference time-domain (FDTD) approach in numerical simulation. However, such an FDTD scheme often includes only second-order temporal approximations, causing errors in temporal dispersion in the case of a large time-stepping size. Such temporal dispersion will affect the inversion results and reduce the inversion quality. We introduce a unique 3D elastic FWI using a temporal fourth-order finite-difference approximation. A new quasi-stress–velocity elastic equation is solved by the temporal fourth-order and spatial arbitrary even-order FDTD method, and a novel inversion procedure for the convolutional objective function based on this equation is derived. The multiscale strategy is used to enhance the robustness of our algorithm. The forward modeling and FWI examples presented here demonstrate that our method can achieve modeling and inversion with a high degree of accuracy.

International audience; The augmented Lagrangian (AL) method provides a flexible and efficient framework for solving extended-space full-waveform inversion (FWI), a constrained nonlinear optimization problem whereby we seek model parameters and wavefields that minimize the data residuals and satisfy the wave-equation constraint. The AL-based wavefield reconstruction inversion, also known as iteratively refined wavefield reconstruction inversion, extends the search space of FWI in the source dimension and decreases the sensitivity of the inversion to the initial model accuracy. Furthermore, it benefits from the advantages of the alternating direction method of multipliers, such as generality and decomposability for dealing with nondifferentiable regularizers, e.g., total variation regularization, and large-scale problems, respectively. In practice, any extension of the method aiming at improving its convergence and decreasing the number of wave-equation solves would have great importance. To achieve this goal, we recast the method as a general fixed-point iteration problem, which enables us to apply sophisticated acceleration strategies such as Anderson acceleration. The accelerated algorithm stores a predefined number of previous iterates and uses their linear combination together with the current iteration to predict the next iteration. We investigate the performance of our accelerated algorithm on a simple checkerboard model and the benchmark Marmousi II and 2004 BP salt models through numerical examples. These numerical results confirm the effectiveness of our algorithm in terms of convergence rate and the quality of the final estimated model.

This study focuses on advancing the inversion of aerosol data measured by a cascade impactor. We aim to find and validate a comprehensive and robust mathematical model for reconstructing a particle...

The Gabor quaternionic offset linear canonical transform (GQOLCT) is defined as a generalization of the quaternionic offset linear canonical transform (QOLCT). In this paper, we investigate the 2D GQOLCT. A new definition of the GQOLCT is provided along with its several important properties, such as boundedness, orthogonality relation, Plancherel and inversion formulas, are derived based on the spectral representation of the GQOLCT. Further, we establish a version of Lieb’s and logarithmic inequalities. Finally we will prove a type of the Heisenberg inequality by using local uncertainty principle.

In this paper, we introduce a new two-dimensional fractional Stockwell transform using the kernel of the coupled fractional Fourier transform. We establish that the two-dimensional fractional Stockwell transform satisfies all the expected properties including Parseval identity and inversion formula. We also characterize the range of the fractional Stockwell transform on $$\mathscr {L}^2(\mathbb {R}^2)$$ and prove a convolution theorem of the transform. Finally, we prove the uncertainty principles for the coupled fractional Fourier transform as well as for the two-dimensional fractional Stockwell transform.

In this paper we present an algorithm for solving quasi-pentadiagonal Toeplitz linear systems, that is based on Du et al. method (Du et al., 2014) and the Sherman–Morrison–Woodbury inversion formula. All algorithms have been implemented in Matlab and numerical experiments are given in order to illustrate the validity and efficiency of our algorithm. An application of the proposed algorithm to the Lax–Wendroff scheme is also considered.

We present an approach to studying and predicting the spatio-temporal progression of infectious diseases. We treat the problem by adopting a partial differential equation (PDE) version of the Susceptible, Infected, Recovered, Deceased (SIRD) compartmental model of epidemiology, which is achieved by replacing compartmental populations by their densities. Building on our recent work (Computat Mech 66:1177, 2020), we replace our earlier use of global polynomial basis functions with those having local support, as epitomized in the finite element method, for the spatial representation of the SIRD parameters. The time dependence is treated by inferring constant parameters over time intervals that coincide with the time step in semi-discrete numerical implementations. In combination, this amounts to a scheme of field inversion of the SIRD parameters over each time step. Applied to data over ten months of 2020 for the pandemic in the US state of Michigan and to all of Mexico, our system inference via field inversion infers spatio-temporally varying PDE SIRD parameters that replicate the progression of the pandemic with high accuracy. It also produces accurate predictions, when compared against data, for a three week period into 2021. Of note is the insight that is suggested on the spatio-temporal variation of infection, recovery and death rates, as well as patterns of the population's mobility revealed by diffusivities of the compartments. Supplementary information The online version contains supplementary material available at 10.1007/s11831-021-09643-1.

Denoisers can help solve inverse problems via a recently proposed framework known as regularization by denoising (RED). The RED approach defines the regularization term of the inverse problem via explicit denoising engines. Simultaneous source separation techniques, themselves being a combination of inversion and denoising methods, provide a formidable field to explore RED. We investigate the applicability of RED to simultaneous-source data processing and introduce a deblending algorithm called REDeblending (RDB). The formulation permits developing deblending algorithms in which the user can select any denoising engine that satisfies RED conditions. Two popular denoisers are tested, but the method is not limited to them: frequency-wavenumber thresholding and singular spectrum analysis. We offer synthetically blended data examples to showcase the performance of RDB.

We study the full-waveform inversion (FWI) problem for the recovery of velocity model/image in acoustic media. FWI is formulated as a typical nonlinear optimization problem, many regularization tec...

The even dimensional Clifford-Fourier transforms have been studied in detail in the last decade. However, the research on the odd dimension is hard to start because the closed expressions and the bounds of these kernels are not obtained. In this paper, we prove that the odd-dimensional Clifford-Fourier kernels are polynomially bounded as in the even-dimensional cases. The crucial ingredient in our proof is the closed expression of these kernels in the Laplace domain. With these bounds, various analytic properties can be established by carrying over the proof for even dimensions, such as inversion theorem and uncertainty principles.

An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $\Psi$-fractional calculus. The operational calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.

The attenuated X-ray transform arises from the image reconstruction in single photon emission computed tomography. The theory of attenuated X-ray transforms is so far incomplete and many questions remain open. In this paper, we are pursuing the inversion of the attenuated X-ray transforms with nonnegative varying attenuation functions μ. By the symmetric attenuated X-ray transform A μ on plane and the method of convolution-backprojection, we obtain the inversion formulas of the attenuated X-ray transforms with nonnegative attenuation functions μ, satisfying some decaying and smoothing conditions on plane.

We have developed an inversion method for recovery of the geometry of an arbitrary number of geologic units using a regularized least-squares framework. The method addresses cases in which each geologic unit can be modeled using a constant physical property. Each geologic unit or group assigned the same physical property value is modeled using the signed distance to its interface with other units. We invert for this quantity and recover the location of interfaces between units using the level-set method. We formulate and solve the inverse problem in a least-squares sense by inverting for such signed distances. The sensitivity matrix to perturbations of the interfaces is obtained using the chain rule, and model mapping from the signed distance is used to recover the physical properties. Exploiting the flexibility of the framework that we develop allows any number of rock units to be considered. In addition, it allows the design and use of regularization incorporating prior information to encourage specific features in the inverted model. We apply this general inversion approach to gravity data favoring minimum adjustments of the interfaces between rock units to fit the data. The method is first tested using noisy synthetic data generated for a model compoed of six distinct units, and several scenarios are investigated. It is then applied to field data from the Yerrida Basin (Australia) where we investigate the geometry of a prospective greenstone belt. The synthetic example demonstrates the proof of concept of the proposed methodology, whereas the field application provides insights into, and potential reinterpretation of, the tectonic setting of the area.

Analyzing well-testing data by the type-curve matching is a modern well-testing analysis method and is widely used in the petroleum and gas industry. By improving accuracy of type curve, we can get more accurate results from analyzing well-testing data, which provide a scientific base for development of oil, gas and water resources. By solving percolation equations, we can obtain type curves. The Laplace transformation methods are often used to solve them. In this paper, we improve the accuracy of type curve by improving the numerical inverse Laplace transformation (NILT) based on infinite series. We combine the NILT based on infinite series with Levin convergence acceleration and determine necessary parameters through numerical experiments to improve accuracy and speed. To verify this method, we compare the improved method with the Stehfest method using some functions such as trigonometric function. Type curves for analysis of well-testing data for the homogeneous reservoir with elastic outer boundary and a dual porosity reservoir are plotted and compared by using the improved numerical inversion and the Stehfest numerical inversion, respectively. These results show that type curves plotted by the improved method are less in vibration and fluctuation than ones plotted by the Stehfest method.

In 3D gravity inversion, a regularization technique must be introduced in order to deal with the non-uniqueness and stability of the inversion process. For this purpose, stabilizing functionals based on minimum norm and maximum smoothness have been utilized as the regularization item in the objective function of gravity inversion, but yield a smoothed distribution of subsurface density which does not give a clear delineation of the boundaries of blocky geological units. Although some functionals such as minimum support and minimum gradient support functionals have been applied to focusing inversion of potential field data, these functionals need to be provided with a suitable focusing parameter for successful inversion. In this paper, we have developed a focusing 3D inversion of gravity data based on an arctangent stabilizing functional. To deal with the numerical solution to the gravity inversion problem, the arctangent-function-based stabilizing functional is first reformulated in pseudo-quadratic form as a weighting matrix. The Gauss Newton (GN) minimization scheme is then employed to perform an optimization process. For the stabilizing functional introduced in this study, there is no need to determine in advance two optimum parameters involved in the stabilizing functional. A test on synthetic examples demonstrates that the boundaries of the anomalous bodies recovered are sharper and the density values are also closer to the true model. We also apply this approach to field gravity data collected from the San Nicolas deposit in Mexico, illustrating that the inversion result shows good consistency with the a priori information available.

This paper deals with the so-called Radon inversion problem formulated in the following way: Given a $$p>0$$ p > 0 and a strictly positive function H continuous on the unit circle $${\partial {\mathbb {D}}}$$ ∂ D , find a function f holomorphic in the unit disc $${\mathbb {D}}$$ D such that $$\int _0^1|f(zt)|^pdt=H(z)$$ ∫ 0 1 | f ( z t ) | p d t = H ( z ) for $$z \in {\partial {\mathbb {D}}}$$ z ∈ ∂ D . We prove solvability of the problem under consideration. For $$p=2$$ p = 2 , a technical improvement of the main result related to convergence and divergence of certain series of Taylor coefficients is obtained.

In this paper, we study the inverse problem of the generalized X-ray transform in n-dimensional Schwartz space. We extend Natterer's inversion formula to the generalized X-ray transform. Moreover, ...

The Gaver–Stehfest algorithm is widely used for numerical inversion of the Laplace transform. In this paper we provide the first rigorous study of the rate of convergence of the Gaver–Stehfest algorithm. We prove that Gaver–Stehfest approximations converge exponentially fast if the target function is analytic in a neighbourhood of a point and they converge at a rate $o(n^{-k})$ if the target function is $(2k+3)$-times differentiable at a point.

Discrete analogs of the index transforms, involving Bessel and Lommel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.

In this paper, we study conics, which are invariant under the hyperbolic inversion with respect to the absolute of an extended hyperbolic plane $H^2$ of curvature radius $\rho$, $\rho \in \mathbb R_+$. They are called the hyperbolic Raisa Orbits of the second order. We prove that each hyperbolic Raisa Orbits of the second order in $H^2$ belongs to one of four conics types of this plane. These types are as follows: the bihyperbolas of one sheet; the hyperbolas; the hyperbolic parabolas of one sheet and two branches; the elliptic cycles of radius $\pi \rho / 4$. The family of all hyperbolic Raisa Orbits from the family of all bihyperbolas of one sheet (or all hyperbolas) defined exactly up to motions, is one-parametric. The family of all hyperbolic Raisa Orbits from the family of all hyperbolic parabolas of one sheet and two branches (or all elliptic cycles) contains a unique conic defined exactly up to motions.

We have developed a MATLAB-based inversion program, b-spline polynomial approximation using the differential evolution algorithm (SPODEA), to recover the concealed basement geometry under heterogeneous sedimentary basins. Earlier inversion techniques used the discretized subsurface interface topography into a grid of juxtaposed elementary prisms to estimate the basement depth of a basin. Such discretization leads to the failure of the depth profile continuity and requires a higher number of inversion parameters for achieving the desired accuracy. The novel approach of SPODEA overcomes such limitations of earlier inversion techniques. SPODEA is based on the segment-wise b-spline optimization technique to estimate the basement depth by using high-order polynomials. Moreover, it can achieve an optimal misfit with minimal parametric information, which reduces the computational expense. Our inversion approach uses the differential evolution algorithm, which provides real parametric optimization and uses b-splines for accurate estimation of continuous depth profiles. The efficiency of our algorithm was determined with two complex synthetic sedimentary basin models comprised of constant and depth-varying density distributions. Furthermore, the uncertainty analysis of our inversion technique is evaluated by incorporating white Gaussian noise into the synthetic models. Finally, the utility of SPODEA is evaluated by inverting gravity anomalies for two different real sedimentary basins. It produces geologically reasonable outcomes that are in close agreement with basement structures from previously reported results.

With the use of the theory of generalized inversion of operators and integral operators, we obtain a criterion of solvability and determine the general form of solutions of a linear boundary-value problem for the integrodifferential equation with degenerate kernel in a Banach space.

Interpretation and inversion of microgravity anomalies belong to important tasks of near-surface geophysics, mostly in cavities detection in engineering, environmental and archaeological applications. One of the mostly used concepts of inversion in applied gravimetry is based on the approximation of the model space by means of 2D or 3D elementary sources with the aim to estimate their densities by means of the solution of a corresponding linear equation system. There were published several approaches trying to obtain correct and realistic results, which describe real parameters of the sources. In the proposed contribution we analyse the properties of two additional functionals, which describe additional properties of the searched solution – namely so-called L2-smoothing and minimum support focusing stabilizers. For the inversion itself, we have used the regularized conjugate gradient method. We have studied properties of these two stabilizers in the case of one synthetic model and one real-world dataset (microgravity data from St. Nicholas church in Trnava). Results have shown that proposed algorithm with the minimum support stabilizer can generate satisfactory model results, from which we can describe real geometry, dimensions and physical properties of interpreted cavities.

The measuring problem of calibrating the cosmological distance scale is considered from the perspective of applicability conditions for regression analysis. The rank inversion and statistical inhomogeneity of information on SN Ia supernovae, used in the works of 1998–1999 and 2004–2007 to detect the “accelerating expansion of the Universe” and as an “extraordinary evidence” of its existence, respectively, are demonstrated to be the reason for the discrepancy and inconsistency of the obtained parametric estimates of the Friedman–Robertson–Walker model. Although the use of lack-of-fit tests for cosmological distance scale models reduces the above negative effects, the fact remains that the cosmological distance scale based on the redshift has neither metric nor ordinal status.

Before performing the inversion process, the original measured data set is often transformed (corrected, smoothed, Fourier-transformed, interpolated etc.). These preliminary transformations may make the original (statistically independent) noisy measurement data correlated. The noise correlation on transformed data must be taken into account in the parameter fitting procedure (inversion) by proper derivation of likelihood function. The covariance matrix of transformed data system is no longer diagonal, so the likelihood based metrics, which determines the fitting process is also changed as well as the results of inversion. In the practice, these changes are often neglected using the “customary” estimation procedure (simple least square method) resulting wrong uncertainty estimation and sometimes biased results. In this article the consequence of neglected correlation is studied and discussed by decomposing the inversion functional to “customary” and additional part which represents the effect of correlation. The ratio of two components demonstrates the importance and justification of the inversion method modification.

The use of Rayleigh wave dispersion curve for estimating shear wave velocity (Vs) is a common task in the field of engineering geophysics. However, because of the nonlinear nature of Rayleigh wave dispersion curves, using a proper technique in the inversion procedure in order to find adequate model parameters is a challenging problem. In this study, a comparative study is performed between linear and nonlinear inversion methods. Between different techniques, a new nonlinear searching algorithm, i.e., the modified cuckoo search (MCS) algorithm was introduced to invert the Rayleigh wave dispersion curves. Also, the generalized inverse (GI) method was used as a linear searching algorithm applied to the construction of the Vs profile. The proposed algorithms were tested by different synthetic (noisy and noise free) datasets. The results show that the MCS algorithm has more exploration ability in comparison with the original cuckoo search. Then, an actual dataset was inverted by the introduced inversion methods. The results show that the performance of the MCS is better than that of the GI method. MCS is a fast, powerful and user-friendly technique in the field of dispersion curve inversion.

Relations between Laplace’s spheroidal harmonics associated with different spheroidal coordinates are derived. The transition matrices for the functions of the 1st kind are lower triangular and are related by inversion. The matrices for the functions of the 2nd kind are the transposed ones for the functions of the 1st kind. The series for the functions of the 1st kind are finite, and those for the 2nd kind are infinite. In the latter case the region of convergence is considered. Using the derived relations, the rigid solution to the electrostatic problem for the multi-layered scatterers with nonconfocal spheroidal boundaries of the layers is obtained and the Rayleigh approximation is constructed, as well as an approximate approach to a similar light scattering problem, which provides reliable results far beyond the range of applicability of the Rayleigh approximation, is suggested.

The nearly analytic discrete (NAD) method is a kind of finite difference method with advantages of high accuracy and stability. Previous studies have investigated the NAD method for simulating wave propagation in the time-domain. This study applies the NAD method to solving three-dimensional (3D) acoustic wave equations in the frequency-domain. This forward modeling approach is then used as the “engine” for implementing 3D frequency-domain full waveform inversion (FWI). In the numerical modeling experiments, synthetic examples are first given to show the superiority of the NAD method in forward modeling compared with traditional finite difference methods. Synthetic 3D frequency-domain FWI experiments are then carried out to examine the effectiveness of the proposed methods. The inversion results show that the NAD method is more suitable than traditional methods, in terms of computational cost and stability, for 3D frequency-domain FWI, and represents an effective approach for inversion of subsurface model structures.

Based on the transmittance matrix ( T -matrix) approach, an efficient and stable rigorous coupled-wave analysis (RCWA) method for ultrathin multilayer gratings without solving eigenvalues is proposed in this letter. The numerical instability caused by the inversion of an ill-conditioned matrix in the traditional T -matrix approach is solved by implementing the proposed method. Besides, compared with the traditional RCWA method, the proposed method does not need to solve the eigenvalues and eigenvectors of the eigenmatrix, which avoids the numerical problems and time-consuming problems caused by this process. A numerical example shows that the results of CST simulations are consistent with the results obtained by the proposed method.

We propose a novel direct sampling method (DSM) for the effective and stable inversion of the Radon transform. The DSM is based on a generalization of the important almost orthogonality property in classical DSMs to fractional order Sobolev duality products and to a new family of probing functions. The fractional order duality product proves to be able to greatly enhance the robustness of the reconstructions in some practically important but severely ill-posed inverse problems associated with the Radon transform. We present a detailed analysis to better understand the performance of the new probing and index functions, which are crucial to stable and effective numerical reconstructions. The DSM can be computed in a very fast and highly parallel manner. Numerical experiments are carried out to compare the DSM with a popular existing method, and to illustrate the efficiency, stability, and accuracy of the DSM.

In the context of holography, entanglement entropy can be studied either by i) extremal surfaces or ii) bit threads, i.e., divergenceless vector fields with a norm bound set by the Planck length. In this paper we develop a new method for metric reconstruction based on the latter approach and show the advantages over existing ones. We start by studying general linear perturbations around the vacuum state. Generic thread configurations turn out to encode the information about the metric in a highly nonlocal way, however, we show that for boundary regions with a local modular Hamiltonian there is always a canonical choice for the perturbed thread configurations that exploits bulk locality. To do so, we express the bit thread formalism in terms of differential forms so that it becomes manifestly background independent. We show that the Iyer-Wald formalism provides a natural candidate for a canonical local perturbation, which can be used to recast the problem of metric reconstruction in terms of the inversion of a particular linear differential operator. We examine in detail the inversion problem for the case of spherical regions and give explicit expressions for the inverse operator in this case. Going beyond linear order, we argue that the operator that must be inverted naturally increases in order. However, the inversion can be done recursively at different orders in the perturbation. Finally, we comment on an alternative way of reconstructing the metric non-perturbatively by phrasing the inversion problem as a particular optimization problem., Comment: 51 pages, 7 figures and 2 appendices. v2: Matches published version

In this paper, we analyze the effect which the choice of a friction model has on tippe top inversion in the case where the resulting action of all dissipative forces is described not only by the force applied at the contact point, but also by the additional rolling resistance torque. We show that the possibility or impossibility of tippe top inversion depends on the existence of specific integrals of the motion of the system. In this paper, we consider an example of the law of rolling resistance by which the area integral is preserved in the system. We examine in detail the case where the action of all dissipative forces reduces to the horizontal rolling resistance torque. This model describes fast rotations of the top between two horizontal smooth planes. For this case, we find permanent rotations of the system and analyze their linear stability. The stability analysis suggests that no tippe top inversion is possible under fast rotations between two planes.

Explicit time advancement for continuous finite elements requires the inversion of a global mass matrix. For spectral element simulations on quadrilaterals and hexahedra, there is an accurate appro...

In this paper we consider a generalization to the q-calculus theory in the space of q-integrable functions. We introduce q-delta sequences and develop q-convolution products to derive certain q-convolution theorem. By using the concept of q-delta sequences, we establish various axioms and set up q-spaces of generalized functions named q-Boehmian spaces. The new assigned spaces of q-generalized functions are acceptable and compatible with the classical spaces of the ordinary functions. Consequently, we extend the generalized q-Sumudu transform to the sets of q-Boehmian spaces. On top of that, we nominate the canonical q-embeddings between the q-integrable sets of functions and the q-integrable sets of q-Boehmians. Furthermore, we address the general properties of the generalized q-Sumudu transform and its inversion formula in some detail.

We introduce the directional short-time Fourier transform for which we prove a new inversion formula. We also prove for this transform several uncertainty principles as Heisenberg inequalities, Far...

Given complex parameters $x$, $\nu$, $\alpha$, $\beta$ and $\gamma \notin -\mathbb{N}$, consider the infinite lower triangular matrix $\mathbf{A}(x,\nu;\alpha, \beta,\gamma)$ with elements $$ A_{n,k}(x,\nu;\alpha,\beta,\gamma) = \displaystyle (-1)^k\binom{n+\alpha}{k+\alpha} \cdot F(k-n,-(\beta+n)\nu;-(\gamma+n);x) $$ for $1 \leqslant k \leqslant n$, depending on the Hypergeometric polynomials $F(-n,\cdot;\cdot;x)$, $n \in \mathbb{N}^*$. After stating a general criterion for the inversion of infinite matrices in terms of associated generating functions, we prove that the inverse matrix $\mathbf{B}(x,\nu;\alpha, \beta,\gamma) = \mathbf{A}(x,\nu;\alpha, \beta,\gamma)^{-1}$ is given by \begin{align} B_{n,k}(x,\nu;\alpha, \beta,\gamma) = & \; \displaystyle (-1)^k\binom{n+\alpha}{k+\alpha} \; \cdot \nonumber \\ & \; \biggl [ \; \frac{\gamma+k}{\beta+k} \, F(k-n,(\beta+k)\nu;\gamma+k;x) \; + \nonumber \\ & \; \; \; \frac{\beta-\gamma}{\beta+k} \, F(k-n,(\beta+k)\nu;1+\gamma+k;x) \; \biggr ] \nonumber \end{align} for $1 \leqslant k \leqslant n$, thus providing a new class of linear inversion formulas. Functional relations for the generating functions of related sequences $S$ and $T$, that is, $T = \mathbf{A}(x,\nu;\alpha, \beta,\gamma) \, S \Longleftrightarrow S = \mathbf{B}(x,\nu;\alpha, \beta,\gamma) \, T$, are also provided., Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1904.08283, arXiv:1909.09694

We introduce a continuous k-Hankel Gabor transform acting on a Hilbert space deforming $$L^2(\mathbb R)$$ . We prove a Plancherel formula and $$L^2$$ -inversion formulas for it. We also prove several uncertainty principles for this transform such as Heisenberg type inequalities and Faris–Price type uncertainty principle.

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.

In this paper, we have introduced and studied finite $$(p,q)$$ -Mellin transform which extends the known theory of finite q-Mellin transform. Also, in this paper, we study finite $$(p,q)$$ -Mellin transform in the framework of symmetric calculus. This study is new even for the special case $$p=q$$ . In both the cases, we provide several basic properties, inversion formulas, the appropriate convolution product and the Parseval type relations.

Hypersingular integrals have appeared as effective tools for inversion of multidimensional potential-type operators such as Riesz, Bessel, Flett, parabolic potentials, etc. They represent (at least formally) fractional powers of suitable differential operators. In this paper the family of the so-called “truncated hypersingular integral operators” $\mathbf{D}_{\varepsilon }^{\alpha }f$ D ε α f is introduced, that is generated by the modified Poisson semigroup and associated with the Flett potentials F α φ = ( E + − Δ ) − α φ ($0 0 < α < ∞ , $\varphi \in L_{p}(\mathbb{R}^{n})$ φ ∈ L p ( R n ) ). Then the relationship between the order of “$L_{p}$ L p -smoothness” of a function f and the “rate of $L_{p}$ L p -convergence” of the families $\mathbf{D}_{\varepsilon }^{\alpha } \mathcal{F}^{\alpha }f$ D ε α F α f to the function f as $\varepsilon \rightarrow 0^{+}$ ε → 0 + is also obtained.

This paper provides a novel inversion free variant of the fixed point iteration strategy to obtain a maximal positive definite solution for the nonlinear matrix equation X + A ⁎ X − 1 A = Q . The conditions for the convergence of the method and for the existence of solution of this nonlinear matrix equation are derived. At the end, numerical examples are also presented to show the behavior of the proposed technique.

Inverse statistical problems associated with the inversion of some linear homogeneous transformation arise in areas such as tomography, plasma physics, optics, etc. As a rule, noise is present in observations, and it is necessary to apply some regularization methods. Recently, methods of threshold processing of wavelet expansion coefficients have become popular. When using threshold processing methods, it is usually assumed that the number of decomposition coefficients is fixed and the noise distribution is Gaussian. This model has been well studied in the literature, and the optimal threshold values have been calculated for different classes of signal functions. However, in some situations, the sample size is not known in advance and has to be modeled by some random variable. In this paper, a model with a random number of observations is considered, and it is shown that the limit distribution of the mean-square risk estimate is a shift-scale mixture of normal laws.

In the present contribution, we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems, in which the forward problems are modeled by some forms of partial differential equations. The method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of sub-inverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest, and then carries out the inversion on the coarsest grid by the homotopy method. The strategy may give a rapidly and globally convergent method. As a practical application, this method is used to solve the nonlinear inverse problem of a nonlinear convection–diffusion equation, which is the saturation equation within the two-phase porous media flow. We demonstrate the effectiveness and merits of the multigrid–homotopy method on two actual model problems.

A linearized rock physics inversion method is proposed to deal with two important issues, rock physical model and inversion algorithm, which restrict the accuracy of rock physics inversion. In this method, first, the complex rock physics model is expanded into Taylor series to get the first-order approximate expression of the inverse problem of rock physics; then the damped least square method is used to solve the linearized rock physics inverse problem directly to get the analytical solution of the rock physics inverse problem. This method does not need global optimization or random sampling, but directly calculates the inverse operation, with high computational efficiency. The theoretical model analysis shows that the linearized rock physical model can be used to approximate the complex rock physics model. The application of actual logging data and seismic data shows that the linearized rock physics inversion method can obtain accurate physical parameters. This method is suitable for linear or slightly non-linear rock physics model, but may not be suitable for highly non-linear rock physics model. Key words: rock physics inversion, linearization, physical parameters, rock physics model, Taylor expansion

With the aim to circumvent the limitations of the special affine Fourier transform, we introduce a novel time–frequency transform namely the windowed special affine Fourier transform. We initiate our investigation by studying some fundamental properties of the proposed transform such as orthogonality relation, inversion formula and characterization of the range by employing the machinery of special affine Fourier transforms and operator theory. Continuing our endeavour, we propose a discrete analogue of the proposed windowed special affine Fourier transform and obtain the corresponding reconstruction formula. Besides, some potential applications of this new transform including windowed series expansion, Poisson summation formula, Paley–Wiener criterion and uncertainty principles are also given.