Approximate Inverse Research Articles

Theorems on Direct and Inverse Approximation by Algebraic Polynomials and Piecewise Polynomials in the Spaces $${{H}^{m}}(a,b)$$ and $$B_{{2,q}}^{s}(a,b)$$

Theorems on Direct and Inverse Approximation by Algebraic Polynomials and Piecewise Polynomials in the Spaces $${{H}^{m}}(a,b)$$ and $$B_{{2,q}}^{s}(a,b)$$

On the Optimality of Inverse Gaussian Approximation for Lognormal Channel Models

On the Optimality of Inverse Gaussian Approximation for Lognormal Channel Models

PCAIME: Principal Component Analysis-Enhanced Approximate Inverse Model Explanations Through Dimensional Decomposition and Expansion

PCAIME: Principal Component Analysis-Enhanced Approximate Inverse Model Explanations Through Dimensional Decomposition and Expansion

Determination of breakpoints and the magnitude of the jump of the original according to its Laplace image

The application of the integral Laplace transform for a wide class of problems leads to a simpler equation for the image of the desired original. At the next step, the inversion problem arises, i. e., finding the original by its image. As a rule, it is not possible to carry out this step analytically. The problem arises of using approximate inversion methods. In this case, the approximate solution is represented as a linear combination of the image and its derivatives at a number of points of the complex half-plane in which the image is regular. However, the original, unlike the image, may even have break points. Of undoubted interest is the task of developing methods for determining the possible break points of the original and the magnitude of the original jump at these points. The proposed methods use the values of high-order image derivatives in order to obtain a satisfactory accuracy of approximate solutions. Methods for accelerating the convergence of the obtained approximations are indicated. The results of numerical experiments illustrating the effectiveness of the proposed methods are presented.

Inverse approximation theorem using Bernstein’s Durrmeyer operator in terms of 2nd modulus of smoothness

Many inverse theorems have been proven using the ordinary modulus of smoothness but no one proved the invers theorems for approximation by Bernstein’s Durrmeyer operator in terms 2nd modulus of smoothness.

Generalizing reduction‐based algebraic multigrid

AbstractAlgebraic multigrid (AMG) methods are often robust and effective solvers for solving the large and sparse linear systems that arise from discretized PDEs and other problems, relying on heuristic graph algorithms to achieve their performance. Reduction‐based AMG (AMGr) algorithms attempt to formalize these heuristics by providing two‐level convergence bounds that depend concretely on properties of the partitioning of the given matrix into its fine‐ and coarse‐grid degrees of freedom. MacLachlan and Saad (SISC 2007) proved that the AMGr method yields provably robust two‐level convergence for symmetric and positive‐definite matrices that are diagonally dominant, with a convergence factor bounded as a function of a coarsening parameter. However, when applying AMGr algorithms to matrices that are not diagonally dominant, not only do the convergence factor bounds not hold, but measured performance is notably degraded. Here, we present modifications to the classical AMGr algorithm that improve its performance on matrices that are not diagonally dominant, making use of strength of connection, sparse approximate inverse (SPAI) techniques, and interpolation truncation and rescaling, to improve robustness while maintaining control of the algorithmic costs. We present numerical results demonstrating the robustness of this approach for both classical isotropic diffusion problems and for non‐diagonally dominant systems coming from anisotropic diffusion.

Schröder-Based Inverse Function Approximation

Schröder approximations of the first kind, modified for the inverse function approximation case, are utilized to establish general analytical approximation forms for an inverse function. Such general forms are used to establish arbitrarily accurate analytical approximations, with a set relative error bound, for an inverse function when an initial approximation, typically with low accuracy, is known. Approximations for arcsine, the inverse of x − sin(x), the inverse Langevin function and the Lambert W function are used to illustrate this approach. Several applications are detailed. For the root approximation of a function, Schröder approximations of the first kind, based on the inverse of a function, have an advantage over the corresponding generalization of the standard Newton–Raphson method, as explicit analytical expressions for all orders of approximation can be obtained.

Comparison of four different methods for massive MIMO detection and VLSI design

The use of Multiple-input-multiple-output (MIMO) technology has proven to be successful. To make MIMO more accessible, it is necessary to simplify the processor operation. This paper proposes four approaches for achieving this goal: symmetric successive overrelaxation (SSOR), Gauss-Seidel (GS), approximate matrix inversion based on Neumann series, and Conjugate Gradients. The performance and VLSI of these methods are compared to determine their effectiveness.

Kinematical Fluctuations Vary with Galaxy Surface Mass Density

The Galaxy inner parts are generally considered to be optically symmetric, as well as kinematically symmetric for most massive early-type galaxies. At the lower-mass end, many galaxies contain lots of small patches in their velocity maps, causing their kinematics to be nonsmooth in small scales and far from symmetry. These small patches can easily be mistaken for measurement uncertainties and have not been well discussed. We used the comparison of observations and numerical simulations to demonstrate the small patches existence beyond uncertainties. For the first time we have found that the fluctuation degrees have an approximate inverse loglinear relation with the galaxy stellar surface mass densities. This tight relation among galaxies that do not show obvious optical asymmetry that traces environmental perturbations indicates that stellar motion in galaxies has inherent asymmetry besides external environment influences. The degree of the kinetic asymmetry is closely related to and constrained by the intrinsic properties of the host galaxy.

A numerical method for inverse Helmholtz problem based on approximate inverse of a matrix

A numerical method for inverse Helmholtz problem based on approximate inverse of a matrix

Seismic differential semblance-oriented migration velocity analysis — Status and the way forward

Differential semblance optimization (DSO) is a seismic waveform inversion approach. It has been developed to remove the limitation of conventional full-waveform inversion as a velocity model building tool when only reflected waves are available. It has connections with migration velocity analysis (MVA) techniques defined in the image domain to build macrovelocity models, with a split between the macromodel and the small-scale heterogeneities. Among the various DSO approaches, DSO-MVA has unique characteristics for velocity model building of complex structures, while ensuring the convergence to the global minimum in a local optimization approach: the wavefields are computed with the two-way wave equation operators, focusing panels are built in a survey-sinking mode, and the quality of the macrovelocity model is evaluated on common-image gathers in the subsurface-offset domain through a differential annihilator. We review the significant aspects that have been better understood over the past 15 years and discuss the way forward. The first identified issue has been the imprint of the small-scale heterogeneities on the macromodel update that is revealed with the use of wave equation-based Green’s functions and an ad hoc solution is subsequently developed by slightly modifying the definition of the objective function. A second important aspect has been the development of efficient approaches to compute the required quantitative migration based on approximate inverses to replace the previously used migrations. The issue of correctly interpreting the amplitudes has also moved into one of an improved model parameterization, i.e., from constant-density acoustics media to full acoustic media as a path toward fully viscoelastic media. Finally, we discuss the way forward with the challenges of extending DSO-MVA beyond the single scattering approximation, for processing diving waves and multiples, and also the challenge of numerical implementation, which remains critical even in two dimensions, for applicability to large-scale real data.

Implementation of dynamics inversion algorithms in active vibration control systems: Practical guidelines

It is common that excessive vibrations problems arise in modern slender, low-damping, lightweight structures such as footbridges or long-span floors subjected to human actions. The integration of inertial vibration controllers has been extensively used to mitigate the problems associated with vibration serviceability of such structures. Among them, the active vibration absorber constitutes the most sophisticated and effective alternative. Nonetheless, the inherent dynamics of the actuators used in the practical implementations of these systems may negatively affect their performance, compromising the overall system stability and performance. In this work, the practical application of dynamics inversion techniques to the force control of electrodynamic proof-mass actuators employed as active vibration absorption systems in lightweight structures subjected to human-induced vibrations is presented. The main idea behind the dynamics inversion approach is to find an approximate inverse of the actuator system which, upon implementation on a real-time controller, approximately cancels out actuator dynamics leading to an improved force tracking, with the subsequent enhancement in the performance of any force-based vibration control algorithm. The dynamics inversion has been applied to the classical direct velocity feedback approach, leading to a novel vibration control algorithm denominated velocity feedback with dynamics inversion. Additionally, the stability of the suggested method has been studied and practical guidelines, including a step-by-step design procedure, have been provided for researchers willing to apply the proposed algorithm. The goodness of the suggested method has been assessed by means of a tests campaign carried over an existing glass fiber reinforced polymer footbridge, which fulfills the static requirements at Serviceability and Ultimate Limit States, but exhibits excessive vibrations when its first resonance frequency is excited. Two types of tests have been performed: a walking load and a vandal bouncing excitation exerted at the midspan of the footbridge. In both cases, the experimental results show that the new procedure outperforms the classical velocity feedback approach thanks to the enhanced force tracking, leading to a remarkable reduction in the vibrations experimented by the structure under study.

Automatic reinforcement for robust model‐free neurocontrol of robots without persistent excitation

SummaryModel‐based adaptive control suffers over parametrization from the many adaptive parameters compared to the order of system dynamics, leading to sluggish tracking with a poor adaptation transient without robustness. Likewise, adaptive model‐free neurocontrol that relies on the Stone–Weierstrass theorem also suffers from similar problems in addition to over‐fitting to approximate inverse dynamics. This article proposes a novel reinforced adaptive mechanism to guarantee a transient and robustness for the model‐free adaptive control of nonlinear Lagrangian systems. Inspired by the symbiosis of Actor‐Critic (AC) architecture and integral sliding modes, the reinforced stage neural network, analogous to the critic, injects excitation signals to reinforce the parametric learning of the adaptive stage neural network, analogous to the actor to improve the approximation of inverse dynamics. The underlying integral sliding surface error drives improved learning onto a low‐dimensional invariant manifold to guarantee local exponential convergence of tracking errors. Lyapunov stability substantiates the robustness with an improved transient response. Our proposal stands for a hybrid approach between AC and neurocontrol, where the reinforced stage does not require a value function nor reward to provide automatic reinforcement to the adaptive stage parametric adaptation. Dynamic simulations are presented for a nonlinear robot manipulator under different conditions.

Adaptive neural approximated inverse control for photovoltaic power generation servo systems with all states constrained

For PV power generation servo systems with motor hysteresis, an all-state constrained decentralized adaptive approximation inversion control strategy is suggested in order to further increase the tracking precision of photovoltaic (PV) panels to the sun and the efficiency of solar energy utilization. Firstly, the hysteresis phenomenon in the servo motors of photovoltaic panels is considered, and the hysteresis-approximate inverse compensator is first studied in the servo motors of photovoltaic panels. Secondly, to deal with the constrained states such as rotor angle, angular velocity, and stator current in practical servo motors, and to ensure all states are strictly confined within each set of constraints with pre-set tracking performance, the asymmetric barrier Lyapunov functions and the error transformed functions are designed. Finally, to test the effectiveness of the proposed control strategy, a hardware-in-the-loop simulation platform and a photovoltaic servo system for PV power generation are built.

Restricted Search Space Exploration With Refinement for Symbol Detection in Uplink Massive MIMO

Massive multiple-input multiple-output (MIMO) is a promising technique to overcome the explosive increase in the demand for high-speed data, quality of service and energy efficiency for fifth generation (5 G) and beyond wireless systems. Low-complexity symbol detection in massive MIMO is one of the challenging issues for their practical realization. The existing approximate matrix inversion based and matrix inversion less iterative symbol detection techniques suffer significant performance loss when the number of users increases in the system. Therefore, this article proposes a low-complexity restricted search space exploration based ordered iterative detection (REOD) technique to achieve superior performance in large user massive MIMO systems. The proposed low-complexity REOD technique iteratively prunes the discrepancy term associated with the previous estimation and further explores an improved estimate confined to a predefined search space. Convergence of the proposed REOD technique is analytically justified. An analytical expression for the approximate upper bound on the bit error rate is derived and corroborated by simulations. The obtained results prove the viability of the proposed REOD technique compared to the several existing state-of-the-art uplink massive MIMO detection techniques, both in terms of error performance and computational complexity.

Data-Driven Inverse Kinematics Approximation of a Delta Robot with Stepper Motors

The Delta robot is a parallel robot that is over-actuated and has a highly nonlinear dynamic model, which poses a significant challenge to its control design. The inverse kinematics that maps the motor angles to the position of the end effector is highly nonlinear and extremely important for the control design of the Delta robot. It has been experimentally shown that geometry-based inverse kinematics is not accurate enough to capture the dynamics of the Delta robot due to manufacturing component errors, measurement errors, joint flexibility, backlash, friction, etc. To address this issue, we propose a neural network model to approximate the inverse kinematics of the Delta robot with stepper motors. The neural network model is trained with randomly sampled experimental data and implemented on the hardware in an open-loop control for trajectory tracking. Extensive experimental results show that the neural network model achieves excellent performance in terms of the trajectory tracking of the Delta robot under different operation conditions, and outperforms the geometry-based inverse kinematics model. A critical numerical observation indicates that neural networks trained with the specific trajectory data fall short of anticipated performance due to a lack of data. Conversely, neural networks trained on random experimental data capture the rich dynamics of the Delta robot and are quite robust to model uncertainties compared to geometry-based inverse kinematics.

Determination of Heat Transfer Coefficient in Casting: A Comparison of Analytical, Inverse and Trial-and-Error Methods

The Heat Transfer Coefficient (HTC) in casting is critical input in numerical simulations. However, it depends on alloy composition, melt temperature, mold preheating temperature and local casting modulus, hence it is difficult to determine a priori. This work uses temperature measurements obtained by thermocouples during investment casting of a multiple section reference part. HTC was determined by three different methods: (a) analytical, formulating a detailed expression for the HTC in time steps for a one-dimensional approximation (b) inverse, following a well-known iterative algorithm proposed in literature and (c) "trial-and-error" simulation runs for different HTCs based on the judgment of the analyst. In conclusion, the analytical method proved to be fast, but yields very approximate HTC. The inverse method achieves accurate temperature evolution, but it might be unrealistic in physics terms. The trial-and-error method is flexible and may be accurate yet cumbersome and uncertain, unless automated, for instance through a Genetic Algorithm.

Gravity inversion by second order approximation applied to the Styrian Basin (Austria)

AbstractAn approximative second order gravity inversion scheme by a truncated power series expansion is applied to derive the thickness of the Neogene, mostly clastic, sedimentary section of the Styrian Basins in South–East Austria, which are sub‐basins of the Pannonian Basin System. These sub‐basins with a derived thickness of up to 4 km are of interest for geothermal exploitation because of the increased geothermal gradient and heat flow observed in the Pannonian Basin in general and a geothermal gradient of 4°–5°/100 m measured in some wells in the Styrian Basin. The Styrian Basin also has been an area for hydrocarbon exploration in the past 50 years, with oil and gas show encountered in several exploration wells and one sub‐commercial gas discovery. The Miocene and Plio–Pleistocene volcanism in the Styrian Basin caused by Miocene crustal thinning is discussed in terms of the influence to the gravity inversion taking the aeromagnetic field into account. The volcanism is of relevance for the geothermal prospectivity but poses problems for the single layer‐based gravity inversion scheme. Results are discussed from a computational side comparing observed and calculated gravity fields but also the match with well data is discussed. In terms of gravity inversion methodology, the presented can be viewed as an approximative fast‐track approach.

Theoretical study on dynamic stress redistribution around circular tunnel with different unloading paths

This study presents a solution for stress redistribution induced by dynamic excavation of a rock tunnel in hydrostatic and non-hydrostatic stress states. Considering rock as an elastic medium, the analytical solution of dynamic excavation unloading was derived in the Laplace transform space utilizing the wave function expansion method and mode decomposition, and transformed into the time domain using the approximate numerical inversion method. The effects of the unloading path (linear, cosine, and exponential), unloading rate, and initial stress state on stress redistribution were analyzed. The analytical results indicated that a high unloading rate led to a high stress concentration and oscillations around the tunnel, and instantaneous dynamic unloading caused 20–30% stress concentration. When dynamic unloading is complete, the stress state around the tunnel converges to the Kirsch solution. The 3D numerical results indicated that deformation of the tunnel boring face increased as the initial stress increased. Moreover, the accumulation and release of strain energy and stress state change paths were analyzed and discussed. The results of this study provide a theoretical basis for understanding the stress adjustment and failure of surrounding rock induced by excavation of deep-buried openings in hydrostatic and non-hydrostatic stress states.

Hölder classes in the 𝐿^{𝑝} norm on a chord-arc curve in ℝ³

The Hölder classes L p α ( L ) L_p^{\alpha } (L) in the L p ( L ) L_p(L) norm on a chord-arc curve L L in R 3 \mathbb {R}^3 are defined and direct and inverse approximation theorems are proved for functions from these classes by functions harmonic in a neighborhood of the curve. The approximation is estimated in the L p ( L ) L^p(L) norm, the direct theorem is proved for a certain subclass of L p α ( L ) L^\alpha _p(L) and the inverse theorem covers the entire Hölder class.