Adjoint Operator Research Articles

Improved cosmic microwave background (de-)lensing using general spherical harmonic transforms

Deep cosmic microwave background polarization experiments allow a very precise internal reconstruction of the gravitational lensing signal in principle. For this aim, likelihood-based or Bayesian methods are typically necessary, where very large numbers of lensing and delensing remappings on the sphere are sometimes required before satisfactory convergence. We discuss here an optimized piece of numerical code in some detail that is able to efficiently perform both the lensing operation and its adjoint (closely related to delensing) to arbitrary accuracy, using nonuniform fast Fourier transform technology. Where applicable, we find that the code outperforms current widespread software by a very wide margin. It is able to produce high-resolution maps that are accurate enough for next-generation cosmic microwave background experiments on the timescale of seconds on a modern laptop. The adjoint operation performs similarly well and removes the need for the computation of inverse deflection fields. This publicly available code enables de facto efficient spherical harmonic transforms on completely arbitrary grids, and it might be applied in other areas as well.

Singular value decomposition for longitudinal, transverse and mixed ray transforms of 2D tensor fields

The operators of longitudinal, transverse and mixed ray transforms acting on two-dimensional symmetric tensor fields of arbitrary degree m in an unit disk are considered in the article. The singular value decompositions of the operators for a parallel scheme of data acquisition are constructed. Orthogonal bases in original spaces and image spaces are constructed using harmonic, Jacobi and Gegenbauer polynomials. Based on the obtained decompositions the polynomial expressions for the (pseudo)inverse and adjoint operators are obtained.

Positive Operator Frame for Hilbert C*-modules

Positive Operator Frame for Hilbert C*-modules

Non-self-adjoint model statement of boundary problem of acoustics. Part 1

Based on the mathematical definition of a non-self-adjoint operator and the physical meaning of a specific boundary value problem, a non-self-adjoint model statement of boundary value problems in acoustics is formulated. As an example, boundary value problems for the reflection of a plane wave and a spherical wave at the interface between two liquid media are considered. A new definition of the reflection coefficient of a spherical wave is introduced. In the region of subcritical angles of incidence, the new definition takes into account the appearance in the total sound field of converging recoil waves corresponding to the eigenfunctions of the adjoint operator. In the region of supercritical angles of incidence, the participation of the eigenfunctions of two conjugate operators in the total sound field forms a nonzero power flux through the interface and its transformation into the vortex component of the intensity vector at the total internal reflection horizon. The definition of the total internal reflection horizon is given. Experimental data are presented that confirm the new definition of the reflection coefficient and the physical correctness of the non-self-adjoint model statement.

Source-independent least-squares reverse time migration in vertical transversely isotropic media based on the Student’s t-distribution

Abstract Seismic anisotropy exists in various types of strata and should be considered in seismic imaging schemes. Seismic imaging algorithms based on isotropic assumption neglect the effects of anisotropy on seismic data, which cause migration artifacts and waveform distortion. To correct the effects of anisotropy on seismic wave propagation, we propose an imaging algorithm that performs least-squares reverse time migration in vertical transversely isotropic acoustic media. We derive the following operators to implement this algorithm, the de-migration operator, its adjoint migration operator, and the corresponding gradient. However, an inaccurate estimated source wavelet will introduce error in the seismic simulation, and thus increase the mismatch between observed and synthetic data for least-squares reverse time migration. In addition, the noises, especially the noises with abnormal amplitudes in the seismic data, damage the inversion convergence and reduce the imaging resolution. To improve the image quality, we propose to use convolved wavefields between observed and synthetic data so that such a mismatch can be independent of the source wavelets. Also, we use the Student’s t-distribution instead of L2 norm in our inversion scheme to better handle the seismic noise. Its implementation only modifies the gradient of the conventional least-square reverse time migration scheme. Our numerical tests show a clear improvement using our proposed imaging algorithm when compared with the conventional isotropic migration scheme for the anisotropic data. Also, the synthetic examples demonstrate the feasibility and effectiveness of our proposed source-independent algorithm using the Student’s t-distribution.

Syndetic Sensitivity and Mean Sensitivity for Linear Operators

We study syndetic sensitivity and mean sensitivity for linear dynamical systems. For the syndetic sensitivity aspect, we obtain some properties of syndetic sensitivity for adjoint operators and left multiplication operators. We also show that there exists a linear dynamical system (X×Y,T×S) such that (X×Y,T×S) is cofinitely sensitive but (X,T) and (Y,S) are not syndetically sensitive. For the mean sensitivity aspect, we show that if (Y,S) is sensitive and not mean sensitive, where Y is a complex Banach space, the spectrum of T meets the unit circle. We also obtain some results regarding mean sensitive perturbations.

An algebraic treatment of the Pastro polynomials on the real line

The properties of the Pastro polynomials on the real line are studied with the help of a triplet of q q -difference operators. The q q -difference equation and recurrence relation these polynomials obey are shown to arise as generalized eigenvalue problems involving the triplet of operators, with the Pastro polynomials as solutions. Moreover, a discrete biorthogonality relation on the real line for the Pastro polynomials is obtained and is then understood using adjoint operators. The algebra realized by the triplet of q q -difference operators is investigated.

Regularity and Neumann problems for operators with real coefficients satisfying Carleson conditions

In this paper, we continue the study of a class of second order elliptic operators of the form L=div(A∇⋅) in a domain above a Lipschitz graph in Rn, where the coefficients of the matrix A satisfy a Carleson measure condition, expressed as a condition on the oscillation on Whitney balls. For this class of operators, it is known (since 2001) that the Lq Dirichlet problem is solvable for some 1<q<∞. Moreover, further studies completely resolved the range of Lq solvability of the Dirichlet, Regularity, Neumann problems in Lipschitz domains, when the Carleson measure norm of the oscillation is sufficiently small.We show that there exists preg>1 such that for all 1<p<preg the Lp Regularity problem for the operator L=div(A∇⋅) is solvable. Furthermore 1preg+1q⁎=1 where q⁎>1 is the number such that the Lq Dirichlet problem for the adjoint operator L⁎ is solvable for all q>q⁎.Additionally when n=2, there exists pneum>1 such that for all 1<p<pneum the Lp Neumann problem for the operator L=div(A∇⋅) is solvable. Furthermore 1preg+1q⁎=1 where q⁎>1 is the number such that the Lq Dirichlet problem for the operator L1=div(A1∇⋅) with matrix A1=A/det⁡A is solvable for all q>q⁎.

On the Joint A-Numerical Radius of Operators and Related Inequalities

In this paper, we study p-tuples of bounded linear operators on a complex Hilbert space with adjoint operators defined with respect to a non-zero positive operator A. Our main objective is to investigate the joint A-numerical radius of the p-tuple.We established several upper bounds for it, some of which extend and improve upon a previous work of the second author. Additionally, we provide several sharp inequalities involving the classical A-numerical radius and the A-seminorm of semi-Hilbert space operators as applications of our results.

C*-Isomorphisms Associated with Two Projections on a Hilbert C*-Module

Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of $${\cal L}(H)$$ , where $${\cal L}(H)$$ denotes the set of all adjointable operators on H. The C*-subalgebras of $${\cal L}(H)$$ generated by elements in {P − P ∧ Q, Q − P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation $$(\tilde \pi ,X)$$ of i(P, Q, H) such that $$\matrix{{\tilde \pi (P - P \wedge Q) = \pi (P) - \pi (P) \wedge \pi (Q),} \hfill \cr {\tilde \pi (Q - P \wedge Q) = \pi (Q) - \pi (P) \wedge \pi (Q).} \hfill \cr } $$ When (P, Q) is semi-harmonious, that is, $$\overline {{\cal R}(P + Q)} $$ and $$\overline {{\cal R}(2I - P - Q)} $$ are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I − Q, I − P, H) are unitarily equivalent via a unitary operator in $${\cal L}(H)$$ . A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I − Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C*-modules are also provided.

Imagify: Reconstruction of High- Resolution Images from Degraded Images

Abstract: Image Denoising simply uses photos with noise and the U-Net architecture. In this study, we demonstrate that the unet design, which is based on convolutional and deconvolutional neural networks or transpose convolutional neural networks, is quite effective at reducing image noise. The assignment falls under abroad category of problems on distribution of posterior probabilities P(theta | X) represents the probability of theparameter theta given the evidence X. These techniques yield great results, but their practical application might be problematic due to issues like the expensive forward and adjoint operators' computations and the challenging hyper parameter selection.

On the Adjoint Problem in a Domain with Deviation Out from the Characteristic for the Mixed Parabolic-Hyperbolic Equation with the Fractional Order Operator

В настоящей статье доказана классическая, сильная разрешимость и вольтерровость сопряженной задачи c отходом от характеристики для уравнения смешанного параболо-гиперболического типа с оператором дробного порядка в смысле Герасимова-Капуто. Целью исследования является решение сопряженной задачи для уравнения смешанного параболо-гиперболического типа дробного порядка. Учитывая, свойств операторов дробного порядка найдены сопряженный оператор и применены постановки сопряженной задачи. Для исследования поставленной задачи в параболической частью смешанной области решается первой краевой задачи для уравнения параболического типа дробного порядка в смысле Герасимова-Капуто. Используя, свойств функции Райта получено функциональное соотношение на линии перехода. Точно также решая, задачи Коши гиперболической частью смешанной области находим функциональное соотношение. Следовательно, поставленная задача эквивалентным образом сводится к интегральному уравнению Вольтерра второго рода со слабой особенностью. Согласно теории интегральных уравнений Вольтерра второго рода доказывается однозначной разрешимость полученного уравнения. Кроме того, используя методы операторов интегро — дифференцирования дробного порядка, теории специальных функций, априорных оценок, теория интегральных уравнений доказываются теоремы единственности, существования и вольтерровость сопряженной задачи в области с отходом от характеристики для уравнения смешанного типа дробного порядка. Полученные результаты новые и отличаются от результатов М.А. Садыбекова и А.С. Бердышева. In this article, it was proved the classical, strong solvability and Volterra property of the adjoint problem with departure from the characteristic for an equation of mixed parabolic-hyperbolic type with a fractional order operator in the sense of Gerasimov-Caputo. The aim of the research is to solve the conjugate problem for the equation of a mixed parabolic-hyperbolic type of fractional order. Taking into account the properties of fractional order operators, the adjoint operator is found and the statements of the adjoint problem are applied. To study the formulated problem in the parabolic part of the mixed domain, the first boundary value problem for a parabolic type equation of fractional order in the sense of Gerasimov-Caputo is solved. Using the properties of the Wright function, a functional relation is obtained on the transition line. In the same way, solving the Cauchy problem with the hyperbolic part of the mixed domain, we find a functional relation. Consequently, the problem posed reduces in an equivalent way to a Volterra integral equation of the second kind with a weak singularity. According to the theory of Volterra integral equations of the second kind, the unique solvability of the resulting equation is proved. In addition, using the methods of integro-differentiation operators of fractional order, the theory of special functions, a priori estimates, the theory of integral equations, uniqueness, existence and Volterra theorems for the adjoint problem in a domain with deviation out of the characteristic for a mixed-type equation of fractional order are proved. The results obtained are new and differ from the results of M. A. Sadybekov and A. S. Berdyshev.

Probability of Initiation in Neutron Transport

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed, and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modeled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi, and by Pazy and Rabinowitz.

Near-Surface Seismic Arrival Time Picking with Transfer and Semi-Supervised Learning

The understanding of subsurface information on the Earth is crucial in numerous fields such as economics of oil and gas, geophysical exploration, archaeology and hydro-geophysics, particularly in a context of climate change. The methodology consists in reconstructing the seismic velocity model of the near surface, that contains information about the basement structure, by solving the inverse problem and resolving the related complex nonlinear systems with the data collected from seismic experiments and measurements. In the last few years, many deep neural networks have been proposed to simplify the seismic inversion problem based, for instance, on automatic differentiation of the adjoint operator, or on automatic arrival time picking. However, such approaches require a large amount of labeled training data, which are hardly available in real applications. We present here a deep learning approach for arrival time picking, aimed to deal with unlabeled data. The main building blocks are transfer learning, as well as a semi-supervised learning strategy where the pseudo-labels are greedily computed with robust regression, and classification algorithms. The hybrid method showcases very high scores when evaluating on synthetic data, and its application to a real dataset containing a limited amount of labeled data shows the computational efficiency and very accurate results.

Properties of Real Fuzzy Hilbert Adjoint Operator

Riesz representation theorem is established for fuzzy real sesquilinear form. Existence theorem and some properties of fuzzy real Hilbert adjoint operator are discussed.

On Discrete Weighted Lorentz Spaces and Equivalent Relations between Discrete ℓp-Classes

In this paper, we study some relations between different weights in the classes Bp,Bp*,Mp and Mp* that characterize the boundedness of the Hardy operator and the adjoint Hardy operator. We also prove that these classes generate the same weighted Lorentz space Λp. These results will be proven by using the properties of classes Bp,Bp*,Mp and Mp*, including the self-improving properties and also the properties of the generalized Hardy operator Hp, the adjoint operator Sq and some fundamental relations between them connecting their composition to their sum.

Building adjoint operators for least-squares migration using the acoustic wave equation

Seismic imaging can be solved as an inversion problem, which can be implemented as a least-squares migration (LSM). Compared with conventional migration algorithms, an LSM can produce imaging results with enhanced illumination and resolution. However, solving an inversion problem faces difficulties in convergence, stability, and computational efficiency. To address these issues, efforts have been spent on examining the key elements in an LSM, including the modeling operators, the migration operators, the inversion solvers, etc. Advanced modeling operators are developed to accurately simulate the seismic data. Innovative migration algorithms are implemented to precisely compute the subsurface image. Fast inversion solvers are used to improve computation efficiency. Although all these techniques are robust to improve the LSMs, they often are studied independently. It is not often considered whether these elements are properly combined when used in an LSM. We have constructed LSMs with adjoint modeling and migration operators, and we develop algorithms to prepare input shot data for these LSMs.

Riesz Transform for a Flow Laplacian on Homogeneous Trees

We prove the L^p-boundedness, for p in (1,infty ), of the first order Riesz transform associated to the flow Laplacian on a homogeneous tree with the canonical flow measure. This result was previously proved to hold for p in (1,2] by Hebisch and Steger, but their approach does not extend to p>2 as we make clear by proving a negative endpoint result for p = infty for such operator. We also consider a class of “horizontal Riesz transforms” corresponding to differentiation along horocycles, which inherit all the boundedness properties of the Riesz transform associated to the flow Laplacian, but for which we are also able to prove a weak type (1, 1) bound for the adjoint operators, in the spirit of the work by Gaudry and Sjögren in the continuous setting. The homogeneous tree with the canonical flow measure is a model case of a measure-metric space which is nondoubling, of exponential growth, does not satisfy the Cheeger isoperimetric inequality, and where the Laplacian does not have spectral gap.

Forward and Adjoint Modeling of Three Viscoacoustic Equations Based on Rheological Models Using DEVITO

Disregarding the effect of seismic signal attenuation in gas and hydrocarbon zones leads to a final image with low resolution. However, a range of equations called viscoacoustics overcome such limitations. We use three secondorder equations based on Maxwell, Kelvin-Voigt (KV), and Standard Linear Solid (SLS) models. We analyze the dissipation and dispersion effects on each of them through seismograms. We also perform Reverse Time Migration (RTM) using the exact adjoint operators (Q-RTAM). All numerical experiments were implemented using Devito - a domain-specific language (DSL) and code generation framework to design highly optimized finite difference kernels for use in inversion methods.

Alternative fan-beam backprojection and adjoint operators

Abstract We present in this work alternative analytic formulations for the fan-beam tomographic backprojection operation and its associated adjoint transform in standard (equiangular) and linear (equidistant) detector geometries. The proposed formulations are obtained from a recent backprojection theorem in parallel tomography. Such formulations are written as a Bessel–Neumann series in the frequency domain that can be implemented as an O ⁢ ( N 2.3729 ) {O(N^{2.3729})} matrix multiplication. Proofs are provided together with numerical simulations compared with conventional fan-beam O ⁢ ( N 3 ) {O(N^{3})} backprojection representations showing more robustness when dealing with highly noisy data.