Approximation Operators Research Articles

Ptychographic estimation of pure multiqubit states in a quantum device

Quantum ptychography is a method for estimating an unknown pure quantum state by subjecting it to overlapping projections, each one followed by a projective measurement on a single prescribed basis. Here, we present a comprehensive study of this method applied for estimating n-qubit states in a circuit-based quantum computer, including numerical simulations and experiments carried out on an IBM superconducting quantum processor. The intermediate projections are implemented through Pauli measurements on one qubit at a time, which sets the number of ptychographic circuits to 3n (in contrast to the 3n circuits for standard Pauli tomography); the final projective measurement in the computational basis is preceded by the quantum Fourier transform (QFT). Due to the large depth and number of two-qubit gates of the QFT circuit, which is unsuitable for noisy devices, we also test the approximate QFT (AQFT) and separable unitary operations. Using the QFT and AQFT of degree 2, we obtained high estimation fidelities in all tests with separable and entangled states for up to three and four qubits, respectively; on the other hand, the separable unitaries in this scenario provided good estimations only for separable states, in general. Our results compare favorably with recent results in the literature, and we discuss further alternatives to make the ptychographic method scalable for the current noisy devices.

One-way reflection waveform inversion with depth-dependent gradient preconditioning

Summary Reflection Waveform Inversion (RWI) is a technique that uses pure reflection data to estimate subsurface background velocity, relying on evolving seismic images. Conventional RWI operates in a cyclic workflow, with two key components in each cycle—migration and reflection tomography. Conventional RWI may result in suboptimal background velocity estimation, partly due to limited or unresolved resolution within each component in each cycle. While gradient preconditioning with the reciprocal of Hessian information helps resolve this issue in both components of RWI, it becomes impractical for a large number of model parameters. One-way reflection waveform inversion (ORWI) is a reflection waveform inversion technique in which the forward modeling scheme operates in one direction (downward and then upward) via virtual parallel depth levels within the medium. Leveraging the ORWI framework, we decompose and reduce the linear Hessian operator (also known as the approximate Hessian or Gauss-Newton Hessian) into multiple smaller sub-operators. In particular, the diagonal blocks of the mono-frequency approximate Hessian operators, each corresponding to a single depth level within the medium, are extracted and inverted to precondition the corresponding mono-frequency gradients in both the migration and reflection tomography components of ORWI. This depth-dependent gradient preconditioning transforms standard ORWI into a high-resolution, yet computationally feasible version aimed at addressing suboptimal velocity estimation, referred to as high-resolution ORWI (HR-ORWI). The effectiveness of the proposed approach is demonstrated through successful applications to synthetic data examples.

Approximate single precision floating point adder for low power applications

With an increasing demand for power-hungry data-intensive computing, design methodologies with low power consumption are increasingly gaining prominence in the industry. Most of the systems operate on critical and noncritical data both. An attempt to generate a precision result results in excessive power consumption and results in a slower system. For noncritical data, approximate computing circuits significantly reduce the circuit complexity and hence power consumption. In this paper, a novel approximate single precision floating point adder is proposed with an approximate mantissa adder. The mantissa adder is designed with three 8-bit full adder blocks. In this paper, a detailed mathematical background, and proposed design approach in terms of the circuit configuration and truth tables are discussed. Additionally, a concept of switching between exact computing and approximate computing is analysed considering an approximate carry look-ahead adder. The delay and power consumption for the exact operating mode and approximate operation mode considering varied window sizes is observed. Performance of the approximate computation is compared against exact computation and varied approximate computing approaches.

Fatigue Damage and Reliability Assessment of Wind Turbine Structure During Service Utilizing Real-Time Monitoring Data

Under the action of wind load, a wind turbine tower will produce alternating stress, which leads to fatigue failure. According to the mean wind speed at the wind turbine impeller collected from the SCADA system, the mean wind speed of the simulation point is calculated by using the wind speed exponential model formula. Davenport spectra are used to simulate the pulsating wind speed time series. The wind spectrum is obtained using the harmonic superposition method. Subsequently, the wind speed time series and wind load time series at the simulation point are calculated. Structural modeling of a 5 MW wind turbine tower is performed in ABAQUS 2021. The modal shape and natural frequency are obtained by modal analysis to verify the rationality of the model. Subsequently, wind loads are applied to the model, and structural stress time history is obtained by transient modal dynamics analysis. The stress time history of the maximum stress area of the tower structure is extracted, and the rain flow counting method is applied to it to obtain the stress spectrum. The Weibull distribution of the stress spectrum is fitted, the mean and variance of the total damage in one day are calculated, and the fatigue reliability analysis of the maximum stress area of the tower structure is carried out. And the nonlinear fatigue cumulative damage analysis of the region is carried out. This work has implications for fatigue reliability studies for approximate operating conditions.

Approximate Observation Weighted ℓ2/3 SAR Imaging under Compressed Sensing.

Compressed Sensing SAR Imaging is based on an accurate observation matrix. As the observed scene enlarges, the resource consumption of the method increases exponentially. In this paper, we propose a weighted ℓ2/3-norm regularization SAR imaging method based on approximate observation. Initially, to address the issues brought by the precise observation model, we employ an approximate observation operator based on the Chirp Scaling Algorithm as a substitute. Existing approximate observation models typically utilize ℓq(q = 1, 1/2)-norm regularization for sparse constraints in imaging. However, these models are not sufficiently effective in terms of sparsity and imaging detail. Finally, to overcome the aforementioned issues, we apply ℓ2/3 regularization, which aligns with the natural image gradient distribution, and further constrain it using a weighted matrix. This method enhances the sparsity of the algorithm and balances the detail insufficiency caused by the penalty term. Experimental results demonstrate the excellent performance of the proposed method.

Jacobian-free Newton-Krylov method for multilevel nonlocal thermal equilibrium radiative transfer problems

Context. The calculation of the emerging radiation from a model atmosphere requires knowledge of the emissivity and absorption coefficients, which are proportional to the atomic level population densities of the levels involved in each transition. Due to the intricate interdependence of the radiation field and the physical state of the atoms, iterative methods are required in order to calculate the atomic level population densities. A variety of different methods have been proposed to solve this problem, which is known as the nonlocal thermodynamical equilibrium (NLTE) problem. Aims. Our goal is to develop an efficient and rapidly converging method to solve the NLTE problem under the assumption of statistical equilibrium. In particular, we explore whether the Jacobian-Free Newton-Krylov (JFNK) method can be used. This method does not require an explicit construction of the Jacobian matrix because it estimates the new correction with the Krylov-subspace method. Methods. We implemented an NLTE radiative transfer code with overlapping bound-bound and bound-free transitions. This solved the statistical equilibrium equations using a JFNK method, assuming a depth-stratified plane-parallel atmosphere. As a reference, we also implemented the Rybicki & Hummer (1992) method based on linearization and operator splitting. Results. Our tests with the Fontenla, Avrett and Loeser C model atmosphere (FAL-C) and two different six-level Ca II and H I atoms show that the JFNK method can converge faster than our reference case by up to a factor 2. This number is evaluated in terms of the total number of evaluations of the formal solution of the radiative transfer equation for all frequencies and directions. This method can also reach a lower residual error compared to the reference case. Conclusions. The JFNK method we developed poses a new alternative to solving the NLTE problem. Because it is not based on operator splitting with a local approximate operator, it can improve the convergence of the NLTE problem in highly scattering cases. One major advantage of this method is that it is expected to allow for a direct implementation of more complex problems, such as overlapping transitions from different active atoms, charge conservation, or a more efficient treatment of partial redistribution, without having to explicitly linearize the equations.

Some properties of new sequence spaces based on Riordan numbers

In this paper, we define a new class of sequence spaces via Riordan numbers and prove their topological properties, and inclusion relations, obtain Schauder basis, and describe α,β and γ duals of them. We have given conditions under which there is matrix transformation between those new sequence spaces and the well-known classical sequence spaces. In the last part, we are given some results related to some special operator classes, such as approximable operators, nuclear operators, and ideal operators.

SuperUDF: Self-Supervised UDF Estimation for Surface Reconstruction.

Learning-based surface reconstruction based on unsigned distance functions (UDF) has many advantages such as handling open surfaces. We propose SuperUDF, a self-supervised UDF learning which exploits a learned geometry prior for efficient training and a novel regularization for robustness to sparse sampling. The core idea of SuperUDF draws inspiration from the classical surface approximation operator of locally optimal projection (LOP). The key insight is that if the UDF is estimated correctly, the 3D points should be locally projected onto the underlying surface following the gradient of the UDF. Based on that, a number of inductive biases on UDF geometry and a pre-learned geometry prior are devised to learn UDF estimation efficiently. A novel regularization loss is proposed to make SuperUDF robust to sparse sampling. Furthermore, we also contribute a learning-based mesh extraction from the estimated UDFs. Extensive evaluations demonstrate that SuperUDF outperforms the state of the arts on several public datasets in terms of both quality and efficiency. Code will be released after accteptance.

Efficient least-squares reverse time migration with the approximate asymptotic inverse operator in place of the adjoint operator

Least-squares reverse-time migration (LSRTM) has emerged as a method to generate high-quality images by progressively decreasing the misfit between the observed data and the modeled data, in order to achieve an inverse effect of the modeling operator. In the construction of a inversion problem from a modeling operator, the standard procedure begins by seeking the adjoint operator. However, the adjoint operator is not the inverse operator, and thus it often needs numerous iterations to approach the inverse effect. Alternatively, we can seek the approximate asymptotic inverse operator as the migration operator. This allows us to achieve the inverse effect with fewer iterations. The approximate inverse operator lies in three aspects: the back-propagation wavefield, the bond betweeen the reflectivity function and the scattering function, and the weighting factor [Formula: see text], where [Formula: see text] is the incidence angle. The adjoint operator utilizes the observed data or data residual as sources to generate the back-propagation wavefield, whereas the approximate inverse operator treats these as boundary conditions. This subtle difference has implications for the amplitude perservation of the inverted images. By binding the reflectivity function and the scattering function through the factor, we can effectively employ a single modeling operator, based the scattering function, as the demigration operator for the reflectivity function. This strategy enables the simultaneous inversion for the scattering and reflectivity functions, effectively bypassing the challenges typically associated with direct modeling through the reflectivity function itself. Numerical examples indicate that 3 to 5 iterations may be adequate to produce high-quality images.

On Approximation Operators Involving Tricomi Function

On Approximation Operators Involving Tricomi Function

Transition metals-based electrocatalysts on super-flat substrate for perovskite photovoltaic hydrogen production with 13.75% solar to hydrogen efficiency

Harnessing the inexhaustible solar energy for water splitting is regarded one of the most promising strategies for hydrogen production. However, sluggish kinetics of oxygen evolution reaction (OER) and expensive photovoltaics have hindered commercial viability. Here, an adhesive-free electrodeposition process is developed for in-situ preparation of earth-abundant electrocatalysts on super-flat indium tin oxide (ITO) substrate. NiFe hydroxide exhibited prominent OER performance, achieving an ultra-low overpotential of 236 mV at 10 mA/cm2 in alkaline solution. With the superior OER activity, we achieved an unassisted solar water splitting by series connected perovskite solar cells (PSCs) of 2 cm2 aperture area with NiFe/ITO//Pt electrodes, yielding overall solar to hydrogen (STH) efficiency of 13.75 %. Furthermore, we upscaled the monolithic facility to utilize perovskite solar module for large-scale hydrogen production and maintained an approximate operating current of 20 mA. This creative strategy contributes to the decrease of industrial manufacturing expenses for perovskite-based photovoltaic-electrochemical (PV-EC) hydrogen production, further accelerating the conversion and utilization of carbon-free energy.

Generalized Yosida inclusion problem involving multi-valued operator with XOR operation

Abstract In this article, we study a generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation. It is shown that the generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation is equivalent to a fixed point equation. We have proved that the generalized Yosida approximation operator is Lipschitz continuous. Finally, we prove an existence and convergence result for our problem.

On the approximation to fractional calculus operators with multivariate Mittag-Leffler function in the kernel

Several numerical techniques have been developed to approximate Riemann–Liouville (R - L) and Caputo fractional calculus operators. Recently linear positive operators have been started to use to approximate fractional calculus operators such as R - L, Caputo, Prabhakar and operators containing bivariate Mittag-Leffler functions. In the present paper, we first define and investigate the fractional calculus properties of Caputo derivative operator containing the multivariate Mittag-Leffler function in the kernel. Then we introduce approximating operators by using the modified Kantorovich operators for the approximation to fractional integral and Caputo derivative operators with multivariate Mittag-Leffler function in the kernel. We study the convergence properties of the operators and compute the degree of approximation by means of modulus of continuity and Hölder continuous functions. The obtained results corresponds to a large family of fractional calculus operators including R- L, Caputo and Prabhakar models.

Measure-operator convolutions and applications to mixed-state Gabor multipliers

For the Weyl-Heisenberg group, convolutions between functions and operators were defined by Werner as a part of a framework called quantum harmonic analysis. We show how recent results by Feichtinger can be used to extend this definition to include convolutions between measures and operators. Many properties of function-operator convolutions carry over to this setting and allow us to prove novel results on the distribution of eigenvalues of mixed-state Gabor multipliers and derive a version of the Berezin-Lieb inequality for lattices. New results on the continuity of Gabor multipliers with respect to lattice parameters, masks and windows as well as their ability to approximate localization operators are also derived using this framework.

Some algebras and logics from quasiorder-generated covering-based approximation spaces

In A. Kumar, & M. Banerjee [(2012). Definable and rough sets in covering-based approximation spaces. In T. Li. (eds.), Rough sets and knowledge technology (pp. 488–495). Springer-Verlag], A. Kumar, & M. Banerjee [(2015). Algebras of definable and rough sets in quasi order-based approximation spaces. Fundamenta Informaticae, 141(1), 37–55], authors proposed a pair of lower and upper approximation operators based on granules generated by quasiorders. This work is an extension of algebraic results presented therein. A characterisation has been presented for those quasiorder-generated covering-based approximation spaces whose corresponding collections of definable and rough sets form Stone algebras. The notion of rough lattice was proposed in A. Kumar, & M. Banerjee [(2015). Algebras of definable and rough sets in quasi order-based approximation spaces. Fundamenta Informaticae, 141(1), 37–55], A. Kumar [(2020). A Study of Algebras and Logics of Rough Sets Based on Classical and Generalized Approximation Spaces. In Transactions on Rough Sets XXII, LNCS (Vol. 12485, pp. 123–251). Springer]. Some special rough lattices are introduced in this work, viz. rough Stone algebra, ∼ 1 -complemented and ∼ 2 -complemented rough lattices. Representations of these algebras in terms of rough sets are obtained. Moreover, logics for these algebras are shown to be sound and complete with respect to rough set semantics.

Nonlinear adaptive pose motion control of a servicer spacecraft in approximation with an accelerated tumbling target

Removing a limited number of large debris can significantly reduce space debris risks. These bodies are generally exposed to extreme environmental disturbance torques or consecutive accidents due to their large wet area, which causes them to experience accelerated high-rate tumbling motion. The existing literature has adequately explored the approximation operations with non-cooperative targets exhibiting 3-axis tumbling motion. However, the research gap lies in the lack of attention given to addressing this approximation for targets undergoing accelerated motion. Agile, accurate, and large-angle maneuvers are three common necessities for safely capturing such targets. Changes in the moment of inertia brought on by fuel slushing cannot be disregarded during such a maneuver. To deal with nonlinearities, adverse coupling effects, actuator saturation constraints, time-varying moment of inertia, and external disturbances that worsen during accelerated agile large-angle maneuvers, a novel adaptive control approach is developed in this paper. The controller's main advantage is its adjustable desired acceleration, which maintains its performance even when dealing with accelerated motion. The control law is directly synthesized from the nonlinear relative equations of motion, without any linearization or simplification of the system dynamics, making it robust to a variety of orbital elements and target behaviors. Adaptation laws are extracted from the Lyapunov stability theorem in a way that guarantees asymptotic stability. Moreover, control actuator roles (delay, saturation, and allocation) are accounted for in modeling and simulation. Finally, a comprehensive numerical simulation based on three different realistic and strict scenarios is carried out to demonstrate the effectiveness and performance of the proposed control approach. The controller's robustness against time-varying dynamic parameters (sharp and sudden change, smooth and slow change, and periodic change) is extensively demonstrated through simulation.

A sequence decomposition genetic algorithm for petrophysical inversion with spatial correlation

Predicting petrophysical properties based on seismic attributes is essential for accurate subsurface characterization. Typically, a forward model is first established to relate the petrophysical properties to geophysical measurements, and then the petrophysical properties are inverted from the actual measurements. Imposing prior constraints of the spatial relationship is also desirable to regularize the ill-posed inverse problem. However, petrophysical inversion involving spatial correlation presents a high-dimensional optimization problem with multiple local extrema. This problem can be challenging to solve, particularly when discrete variables (such as lithofacies) and continuous variables (such as porosity and fluid saturation) are considered together. Therefore, we develop a global optimization algorithm that uses the Markov chain decomposition and a genetic algorithm for a joint discrete-continuous petrophysical inversion with spatial correlation. The innovation of this algorithm is to spatially decouple the sequence optimization problem into multiple low-dimensional subproblems and overcome the initial value dependence to converge to the global optimal solution. Furthermore, an approximation operation is introduced to expedite the optimization algorithm. Numerical experiments show that after algorithmic acceleration, the spatially correlated inversion requires the same computational time as a simple spatially uncorrelated inversion and achieves significantly better accuracy than the latter. Finally, we apply the method to a case study in East China, wherein the lithofacies, clay content, porosity, and water saturation are inferred from the data of the wave velocities and density. The inversion results around the wells generally match the well-logging data, verifying the feasibility of the new method.

Interpolation of set-valued functions

Abstract Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently. In the first stage, we develop an efficient algorithm for computing an interpolant to $F$, inspired by the ‘metric polynomial interpolation’, which is based on the theory in Dyn et al. (2014, Approximation of Set-Valued Functions: Adaptation of Classical Approximation Operators. Imperial College Press). By this theory, a ‘metric polynomial interpolant’ is a collection of polynomial interpolants to all the ‘metric chains’ of the given samples of $F$. For set-valued functions whose graphs have nonempty interior, the collection of these ‘metric chains’ can be infinite. Our algorithm computes a small finite subset of ‘significant metric chains’, which is sufficient for approximating $F$. For the class of Lipschitz continuous functions with samples at the roots of the Chebyshev polynomials of the first kind, we prove that the error incurred by our computed interpolant decays with increasing number of interpolation points in the same rate as in the case of interpolation by the metric polynomial interpolant. This is also demonstrated by our numerical examples. For the class of set-valued functions whose graphs have smooth boundaries, we extend our algorithm to achieve a high-precision detection of the points of topology change, followed by a high-order approximation of the boundaries of the graph of F. We further discuss the case of set-valued functions whose graphs have ‘holes’ with Hölder-type singularities at the points of change of topology. To treat this case we apply some special approximation ideas near the singular points of the holes. We analyze the approximation order of the algorithm, including the error in approximating the points of change of topology, and show by several numerical examples the capability of obtaining high-order approximation of the holes.

Best Approximation and Inverse Results for Neural Network Operators

In the present paper we considered the problems of studying the best approximation order and inverse approximation theorems for families of neural network (NN) operators. Both the cases of classical and Kantorovich type NN operators have been considered. As a remarkable achievement, we provide a characterization of the well-known Lipschitz classes in terms of the order of approximation of the considered NN operators. The latter result has inspired a conjecture concerning the saturation order of the considered families of approximation operators. Finally, several noteworthy examples have been discussed in detail

Construction of New Operators by Composition of Integral-Type Operators and Discrete Operators

In this paper, we propose some new positive linear approximation operators, which are obtained from a composition of certain integral type operators with certain discrete operators. It turns out that the new operators can be expressed in discrete form. We provide representations for their coefficients. Furthermore, we study their approximation properties and determine their moment generating functions, which may be useful in finding several other convergence results in different settings.