Decomposition Of Operator Research Articles

Variable ansatz applied to spectral operator decomposition in a physical superconducting quantum device

Variable ansatz applied to spectral operator decomposition in a physical superconducting quantum device

On an inverse problem for a linearized system of Navier–Stokes equations with a final overdetermination condition

Abstract The theory of inverse problems is an actively studied area of modern differential equation theory. This paper studies the solvability of the inverse problem for a linearized system of Navier–Stokes equations in a cylindrical domain with a final overdetermination condition. Our approach is to reduce the inverse problem to a direct problem for a loaded equation. In contrast to the well-known works in this field, our approach is to find an equation for a loaded term whose solvability condition provides the solvability of the original inverse problem. At the same time, the classical theory of spectral decomposition of unbounded self-adjoint operators is actively used. Concrete examples demonstrate that the assertions of our theorems naturally develop and complement the known results on inverse problems. Various cases are considered when the known coefficient on the right-hand side of the equation depends only on time or both on time and a spatial variable. Theorems establishing new sufficient conditions for the unique solvability of the inverse problem under consideration are proved.

Nonlinear Through-the-Wall Detection Using DORT Applied With LFM Pulses

The detection of visually obstructed small nonlinear targets in a through-the-wall (TTW) environment is difficult because of linear scatterers and strong wall reflections. In this letter, harmonic-based nonlinear TTW detection is introduced as a method to effectively detect small nonlinear targets behind the wall. Nonlinear TTW detection allows for the suppression of linear scatterer responses and wall reflections by detecting only the harmonics generated from nonlinear targets. Here we apply the decomposition of the time-reversal operator (DORT) for the detection and separation of multiple nonlinear targets behind the wall. DORT extracts the eigenvalues and eigenvectors of nonlinear targets and allows for selective focusing (imaging) of each detected target. In addition, we utilize linear frequency modulated (LFM) pulses, which allow for the compression gain in the detection processing. To validate the proposed concept, we conducted simulation and measurements of various configurations of nonlinear TTW detection. The results demonstrate that nonlinear targets behind the wall can be effectively detected and located.

Operational control of probabilistic processes in a multi-stage specialized system

A process consisting of sequential operations is considered. For each operation, you can determine the currently available probability of its error-free (high-quality) execution. By the qualitative performance of the operation, we will understand the achievement of a given level of quality in quantitative proportion to the overall level. The purpose of the article is to build a quality management strategy in which a set of probabilities of quality levels not less than a certain threshold value is achieved, with minimal costs for carrying out appropriate measures. The article presents the decomposition of operations into activities; classification of activities into activities with conditionally linear or asymptotic response; quality assessment in a multi-level control system with response functions of various types; optimization problems and their algorithmic solutions.

Development of Direct-Ammonia Solid Oxide Fuel Cells (DA-SOFCs) and the Effect of Incorporating Internal Ammonia Decomposition Catalysts

Direct-ammonia solid oxide fuel cells (DA-SOFCs) are among the most efficient power sources that use ammonia. The high operating temperature of SOFCs enables ammonia decomposition reaction to proceed inside the anode compartment of SOFCs. Coupling exothermal SOFC operation and endothermal ammonia decomposition reaction increases the efficiency of the device. DA-SOFCs function as well as hydrogen-fueled ones over 750 °C. Recently we have reported that DA-SOFCs suffer significant performance loss under 650 °C, especially the high-performance cells. We discuss a range of phenomena taking place inside the anode compartment of DA-SOFCs: interconnect, current collector, and anode support. Interconnect and current collector have the capability of decomposing significant amounts of ammonia. Nevertheless, incorporating a catalyst is important at low operating temperatures.

Comparing metrics for mixed quantum states: Sjöqvist and Bures

It is known that there are infinitely many distinguishability metrics for mixed quantum states. This freedom, in turn, leads to metric-dependent interpretations of physically meaningful geometric quantities such as complexity and volume of quantum states. In this paper, we first present an explicit and unabridged mathematical discussion on the relation between the Sjoqvist metric and the Bures metric for arbitrary nondegenerate mixed quantum states, using the notion of decompositions of density operators by means of ensembles of pure quantum states. Then, to enhance our comprehension of the difference between these two metrics from a physics standpoint, we compare the formal expressions of these two metrics for arbitrary thermal quantum states specifying quantum systems in equilibrium with a reservoir at non-zero temperature. For illustrative purposes, we show the difference between these two metrics in the case of a simple physical system characterized by a spin-qubit in an arbitrarily oriented uniform and stationary external magnetic field in thermal equilibrium with a finite-temperature bath. Finally, we compare the Bures and Sjoqvist metrics in terms of their monotonicity property.

Translation invariant diagonal frame decomposition of inverse problems and their regularization

Solving inverse problems is central to a variety of important applications, such as biomedical image reconstruction and non-destructive testing. These problems are characterized by the sensitivity of direct solution methods with respect to data perturbations. To stabilize the reconstruction process, regularization methods have to be employed. Well-known regularization methods are based on frame expansions, such as the wavelet–vaguelette decomposition, which are well adapted to the underlying signal class and the forward model and furthermore allow efficient implementation. However, it is well known that the lack of translational invariance of wavelets and related systems leads to specific artifacts in the reconstruction. To overcome this problem, in this paper we introduce and analyze the translation invariant diagonal frame decomposition (TI-DFD) of linear operators as a novel concept generalizing the singular value decomposition. We characterize ill-posedness via the TI-DFD and prove that a TI-DFD combined with a regularizing filter leads to a convergent regularization method with optimal convergence rates. As illustrative example, we construct a wavelet-based TI-DFD for one-dimensional integration, where we also investigate our approach numerically. The results indicate that filtered TI-DFDs eliminate the typical wavelet artifacts when using standard wavelets and provide a fast, accurate, and stable solution scheme for inverse problems.

Resolution-Enhanced and Accurate Cascade Time-Reversal Operator Decomposition (C-DORT) Approach for Positioning Radiated Passive Intermodulation Sources

Attaining a high-resolution and accurate location for a radiated passive intermodulation source (R-PIMS) has been an increasingly interesting problem in modern multi-carrier wireless communication systems. For precisely positioning multiple closely spaced R-PIMSs, a novel imaging method called cascade decomposition of time-reversal operator (C-DORT) was developed. C-DORT constructs a new spectrum calculation by normalizing and multiplying the pseudo-spectrum at each sampled frequency together. The cascade process focuses the pseudo-spectrum at R-PIMS positions to form a highly brightened spectrum peak and to suppress the remained pseudo-spectrum to approximately zero, contributing to distinguishing the closely spaced R-PIMSs. The positioning performance of the positioning resolution, pseudo-spectrum width, positioning accuracy, and imaging robustness are analyzed by numerical simulations. Compared with the conventional central frequency decomposition of time-reversal operator (CF-DORT) and the time domain decomposition of time-reversal operator (TD-DORT) methods, the multiple R-PIMSs, spaced at a distance of diffraction limit, which is the spacing of 1/2 of a wavelength, are distinguished effectively in C-DORT. Additionally, the cross-range pseudo-spectrum full width at half maxima (CRPS-FWHM) is suppressed to the width of 1/4 of a wavelength by multiplication to improve the cross-range resolution in C-DORT. In addition, accurate positioning is obtained by providing the approximately zero positioning root mean square estimation (RMSE) at an SNR ranging from 0 dB to 10 dB. The results show that the proposed C-DORT improves the positioning accuracy and enhances the positioning resolution for locating an R-PIMS.

Multi-view Ensemble Clustering via Low-rank and Sparse Decomposition: From Matrix to Tensor

As a significant extension of classical clustering methods, ensemble clustering first generates multiple basic clusterings and then fuses them into one consensus partition by solving a problem concerning graph partition with respect to the co-association matrix. Although the collaborative cluster structure among basic clusterings can be well discovered by ensemble clustering, most advanced ensemble clustering utilizes the self-representation strategy with the constraint of low-rank to explore a shared consensus representation matrix in multiple views. However, they still encounter two challenges: (1) high computational cost caused by both the matrix inversion operation and singular value decomposition of large-scale square matrices; (2) less considerable attention on high-order correlation attributed to the pursue of the two-dimensional pair-wise relationship matrix. In this article, based on low-rank and sparse decomposition from both matrix and tensor perspectives, we propose two novel multi-view ensemble clustering methods, which tangibly decrease computational complexity. Specifically, our first method utilizes low-rank and sparse matrix decomposition to learn one common co-association matrix, while our last method constructs all co-association matrices into one third-order tensor to investigate the high-order correlation among multiple views by low-rank and sparse tensor decomposition. We adopt the alternating direction method of multipliers to solve two convex models by dividing them into several subproblems with closed-form solution. Experimental results on ten real-world datasets prove the effectiveness and efficiency of the proposed two multi-view ensemble clustering methods by comparing them with other advanced ensemble clustering methods.

Decomposition, trivially-acting symmetries, and topological operators

Trivially-acting symmetries in two-dimensional conformal field theory include twist fields of dimension zero which are local topological operators. We investigate the consequences of regarding these operators as part of the global symmetry of the theory. That is, we regard such a symmetry as a mix of topological defect lines (TDLs) and topological point operators (TPOs). TDLs related by a trivially-acting symmetry can join at a TPO to form non-trivial two-way junctions. Upon gauging, the local operators at those junctions can become vacua in a disjoint union of theories. Examining the behavior of the TPOs under gauging therefore allows us to refine decomposition by tracking the trivially-acting symmetries of each universe. Mixed anomalies between the TDLs and TPOs provide discrete torsion-like phases for the partition functions of these orbifolds, modifying the resulting decomposition. This framework also readily allows for the consideration of trivially-acting non-invertible symmetries.

Investigation of the dynamic vertical behaviour of a floating pile embedded in unsaturated poroelastic half-space

This paper proposes an analytical solution for investigating the dynamic vertical behaviour of a floating pile in unsaturated poroelastic half-space by considering the dynamic properties of the soil below the pile base. The pile shaft resistance is derived by applying Hankel integral transformation to the governing equations, and the pile base resistance is derived by introducing the potential function and operator decomposition methods. The dynamic interaction between the soil and the pile is considered by virtue of the impedance function. To account for the mixed boundary conditions of the pile-soil interface, an analytical solution for the vertical dynamic response of a floating pile is derived in the frequency domain. Additionally, by utilizing the inverse Fourier transform and convolution theorem, a quasi-analytical solution for the velocity response of the floating pile is derived in the time domain. Furthermore, the derived solution is verified through comparisons with existing solutions available in the literature. Finally, a parametric study is performed to investigate the influence of saturation and intrinsic permeability on the dynamic flexibility coefficient of porous unsaturated half-space, the influence of saturation on the dynamic impedance of the soil layer, and slenderness ratio on the vertical behaviour of floating pile and end-bearing pile.

Optimal compression of quantum many-body time evolution operators into brickwall circuits

Near term quantum computers suffer from a degree of decoherence which is prohibitive for high fidelity simulations with deep circuits. An economical use of circuit depth is therefore paramount. For digital quantum simulation of quantum many-body systems, real time evolution is typically achieved by a Trotter decomposition of the time evolution operator into circuits consisting only of two qubit gates. To match the geometry of the physical system and the CNOT connectivity of the quantum processor, additional SWAP gates are needed. We show that optimal fidelity, beyond what is achievable by simple Trotter decompositions for a fixed gate count, can be obtained by compiling the evolution operator into optimal brickwall circuits for the S=1/2S=1/2 quantum Heisenberg model on chains and ladders, when mapped to one dimensional quantum processors without the need of additional SWAP gates.

Differential privacy protection algorithm for network sensitive information based on singular value decomposition

In order to reduce the risk of data privacy disclosure and improve the effect of information privacy protection, a differential privacy protection algorithm for network sensitive information based on singular value decomposition is proposed. TF-IDF method is used to extract network sensitive information text. By comparing the word frequency of network sensitive information, high word frequency word elements in network information content are collected to obtain the mining results of network sensitive information text. According to the decision tree theory, the equal difference privacy budget allocation mechanism is improved to achieve equal difference privacy budget allocation. By discarding some small singular values and corresponding spectral vectors, the data can be disturbed, and the availability of the original data can be retained, so that it can truly represent the original data set structure. According to the results of equal difference privacy budget allocation and singular value decomposition disturbance, the data of high-dimensional network graph is reduced by random projection, singular value decomposition is performed on the reduced data, and Gaussian noise is added to the singular value. Finally, the matrix to be published is generated through the inverse operation of singular value decomposition to achieve differential privacy protection of network sensitive information. The experimental results show that the privacy protection quality of this algorithm is high and the data availability is effectively improved.

3D true-amplitude elastic wave-vector decomposition in heterogeneous anisotropic media

Wave-mode separation aims at separating the elastic wavefield into P- and S-wave modes in each subsurface grid. Several wave-mode separation methods require the calculation of polarization vectors of different wave modes. We have developed a wave-mode separation method based on an analytical decomposition operator and a low-rank approximation in the wavenumber domain. When dealing with a general anisotropic medium with low symmetry, the two S-wave modes suffer from the singularity problem, where strong planar artifacts are caused due to the discontinuous polarization vectors at the singularities. A weight function with an adjustable threshold parameter can be designed to mitigate the strong artifacts, thereby obtaining clean S-wave modes. Larger threshold results in a stronger suppression of the artifacts but at the expense of stronger damages to S-wave energy. Here, we provide a new way to deal with the planar artifacts. The S-wave modes separated using a zero threshold containing strong artifacts can be treated as a noisy wavefield. Applying a weighting function can be viewed as a denoising process. Choosing the threshold parameter in the weighting function will inevitably cause an amplitude loss of the signal. Thus, we can apply the local orthogonalization method to compensate for the amplitude loss. Considering the heterogeneity of the wavefield, we apply a nonstationary local orthogonalization method to obtain an accurate wave-mode separation. The final separated wavefields are true amplitude and thus kinematically and dynamically correct, which will benefit a variety of seismic forward and inverse problems. Several 3D synthetic examples demonstrate the performance of the new method.

Systematic study of one-loop realizations of d = 7 long-range 0νββ decay operators

We study the systematical one-loop decomposition of the dimension-7 long-range 0νββ decay operators. We find that there are 3 genuine one-loop topologies and 8 diagrams. The procedure to determine the SM quantum number assignments for both internal and external fields is presented. The Majorana neutrino mass in long-range 0νββ models is discussed. We also present a one-loop 0νββ decay model which produces Majorana neutrino mass at three-loop level. The phenomenological predictions for light neutrino mass and 0νββ decay half-life time including both mass mechanism and long-range contribution are studied.

The decomposition of neutron-antineutron oscillation operators

We study the systematic decomposition of the dimension nine neutron-antineutron oscillation operators at tree and one-loop levels. We discuss the topologies’ generation and the assignment of the chiral quarks. The completed lists of the decompositions are provided. We furthermore show an example that the neutron-antineutron oscillation occurs at one-loop level, with the tiny neutrino mass being generated via the scotogenic model and proton decay being evaded.

Extracting angle-domain common-image gathers in VTI media using ansiotropic-Helmholtz P/S wave-mode decomposition

Common-image gathers are extensively used in amplitude versus angle (AVA) and migration velocity analysis (MVA). The current state of methods for anisotropic angle gathers extraction use slant-stack, local Fourier transform or low-rank approximation, which requires much computation. Based on an anisotropic-Helmholtz P/S wave-mode decomposition method, we propose a novel and efficient approach to produce angle-domain common-image gathers (ADCIGs) in the elastic reverse time migration (ERTM) of VTI media. To start with, we derive an anisotropic-Helmholtz decomposition operator from the Christoffel equation in VTI media, and use this operator to derive the decomposed formulations for anisotropic P/S waves. Second, we employ the first-order Taylor expansion to calculate the normalized term of decomposed formulations and obtain the anisotropic-Helmholtz decomposition method, which generates the separated P/S wavefields with correct amplitudes and phases. Third, we develop a novel way that uses the anisotropic-Helmholtz decomposition operator to define the polarization angles for anisotropic P/S waves and substitute these angles to decomposing formulations. The polarization angles are then calculated directly from the separated vector P- and S-wavefields and converted to the phase angles. The ADCIGs are thusly produced by applying the phase angles to VTI ERTM. In addition, we develop a concise approximate expression of residual moveout (RMO) for PP-reflections of flat reflectors in VTI media, which avoids the complex transformations between the group angles and the phase angles. The approximate RMO curves show a good agreement with the exact solution and can be used as a tool to assess the migration velocity errors. As demonstrated by two selected examples, our ADCIGs not only produce the correct kinematic responses with regards to different velocity pertubatation, but also generate the reliable amplitude responses versus different angle. The final stacking images of ADCIGs data exhibit the identical imaging effect as that of VTI ERTM.

Composition and Decomposition of Positive Linear Operators (VIII)

In a series of papers, most of them authored or co-authored by H. Gonska, several authors investigated problems concerning the composition and decomposition of positive linear operators defined on spaces of functions. For example, given two operators with known properties, A and B, we can find the properties of the composed operator A∘B, such as the eigenstructure, the inverse, the Voronovskaja formula, and the second-order central moments. One motivation for studying composed operators is the possibility to obtain better rates of approximation and better Voronovskaja formulas. Our paper will address such problems involving compositions of some classical positive linear operators. We present general results as well as numerical experiments.

Bessel-Riesz Operators on Lebesgue Spaces and Morrey Spaces Defined in Measure Metric Spaces

The boundedness of Bessel–Riesz operators defined on Lebesgue spaces and Morrey spaces in measure metric spaces is discussed in this research study. The maximal operator and traditional dyadic decomposition are used to study the Bessel-Riesz operators. We investigate the interaction between the kernel and space parameters to get the results and see how this affects kernel-bound operators.

Vertical and horizontal square functions on a class of non-doubling manifolds

We consider a class of non-doubling manifolds M that are the connected sum of a finite number of N-dimensional manifolds of the form Rni×Mi. Following on from the work of Hassell and the second author [20], a particular decomposition of the resolvent operators (Δ+k2)−M, for M∈N⁎, will be used to demonstrate that the vertical square function operatorSf(x):=(∫0∞|t∇(I+t2Δ)−Mf(x)|2dtt)12 is bounded on Lp(M) for 1<p<nmin=mini⁡ni and weak-type (1,1). In addition, it will be proved that the reverse inequality ‖f‖p≲‖Sf‖p holds for p∈(nmin′,nmin) and that S is unbounded for p≥nmin provided 2M<nmin.Similarly, for M>1, this method of proof will also be used to ascertain that the horizontal square function operatorsf(x):=(∫0∞|t2Δ(I+t2Δ)−Mf(x)|2dtt)12 is bounded on Lp(M) for all 1<p<∞ and weak-type (1,1). Hence, for p≥nmin, the vertical and horizontal square function operators are not equivalent and their corresponding Hardy spaces Hp do not coincide.