Approximate Inverse Research Articles

A Butterfly-Accelerated Volume Integral Equation Solver for Broad Permittivity and Large-Scale Electromagnetic Analysis.

A butterfly-accelerated volume integral equation (VIE) solver is proposed for fast and accurate electromagnetic (EM) analysis of scattering from heterogeneous objects. The proposed solver leverages the hierarchical off-diagonal butterfly (HOD-BF) scheme to construct the system matrix and obtain its approximate inverse, used as a preconditioner. Complexity analysis and numerical experiments validate the O(N log 2 N) construction cost of the HOD-BF-compressed system matrix and O(N 1.5 log N) inversion cost for the preconditioner, where N is the number of unknowns in the high-frequency EM scattering problem. For many practical scenarios, the proposed VIE solver requires less memory and computational time to construct the system matrix and obtain its approximate inverse compared to a matrix-accelerated VIE solver. The accuracy and efficiency of the proposed solver have been demonstrated via its application to the EM analysis of large-scale canonical and real-world structures comprising of broad permittivity values and involving millions of unknowns.

Inversion formulae for ray transforms in vector and tensor tomography

Ray transforms for vector or tensor fields have been studied for a long time, also in connection with Doppler or polarization tomography. In contrast to the classical Radon transform or x-ray transform their known inversion formulae are rather complicated, and represented, for example, using series expansions or singular value decompositions. Recently, applications of the approximate inverse have been studied. In this paper we present a new class of inversion formulae for compactly supported fields similar to those of the x-ray transform. The derivation is based on two observations. First, we apply Derevtsov’s formula relating the ray transforms of vector or tensor fields to a derivative of the x-ray transform of a generating potential, [3]. Second, we apply the calculation of derivatives of a scalar function from its x-ray transform, as it is realized in feature reconstruction, [15, 16]. Generalizations complete the presentation.

The Dirichlet-type Laplace transforms

We show that it is possible to define extensions of the Laplace transform that use a general Dirichlet series as a kernel. These transforms, denoted by DLTs, further generalize those, considered in previous papers, in which the kernels were related to Laguerre-type exponentials or Bell polynomials. Computational techniques, exploiting expansions in Laguerre polynomials, and using Tricomi's method, have been considered. Since it turns out that the transforms considered are obtained as linear combinations of ordinary Laplace transforms, it is also possible to define an approximation of the relevant inverse transforms. Numerical experiments, performed with the algebra program Mathematica, show that the introduced technique is fast and efficient.

Local Multiple Traces Formulation for electromagnetics: Stability and preconditioning for smooth geometries

We consider the time-harmonic electromagnetic transmission problem for the unit sphere. Appealing to a vector spherical harmonics analysis, we prove the first stability result of the local multiple traces formulation (MTF) for electromagnetics, originally introduced by Hiptmair and Jerez-Hanckes (2012) for the acoustic case, paving the way towards an extension to general piecewise homogeneous scatterers. Moreover, we investigate preconditioning techniques for the local MTF scheme and study the accumulation points of induced operators. In particular, we propose a novel second-order inverse approximation of the operator. Numerical experiments validate our claims and confirm the relevance of the preconditioning strategies proposed.

Output feedback model predictive control for Hammerstein model with bounded disturbance

This paper proposes two model predictive control (MPC) methods for a Hammerstein model with both unmeasurable state and bounded disturbance. First, a dynamic output feedback MPC is designed such that the dynamics of the state estimation error is decoupled from that of the estimated state. Hence, the separation principle holds and a part of the controller parameters can be designed off-line. Second, a case is considered where Hammerstein nonlinearity is not exactly inverted, that is, a reserved static nonlinearity exists in the closed-loop system. This reserved nonlinearity is merged into a polytopic description in combination with the linear dynamic part. The control move is then parameterized as a feedback law followed by the approximate inverse of Hammerstein nonlinearity. The recursive feasibility and closed-loop stability of both approaches are guaranteed. Numerical examples are given in order to show effectiveness of the proposed approaches.

A comparison of two data analysis approaches for quantitative magnetic resonance imaging

Magnetic resonance imaging (MRI) is a medical imaging technique which is widely used in clinical routine. Standard imaging methods lead to so-called contrast-weighted images. The contrast arises from several tissue-related parameters such as the relaxation times T 1 and T 2. The aim in quantitative MRI (qMRI) is an estimation of these quantitative parameters. Magnetic resonance fingerprinting (MRF) is a promising technique in qMRI that allows for the simultaneous determination of multiple tissue-related parameters within a short acquisition time. The conventional MRF method utilizes an approximate inverse Fourier transformation for the highly undersampled data in the Fourier domain, leading to aliasing errors in the reconstructed magnetization courses. Computationally expensive statistical MRF modeling approaches overcome this issue by modeling the data directly in the Fourier domain. However, this leads to a non-convex and large-scale optimization task that is challenging to solve and requires expensive calculations. We compare two recent approaches, namely the conventional MRF method and a statistical MRF modeling approach based on Bayesian statistics in terms of their accuracy, reliability and computational costs. The comparison is carried out for simulated data with known ground truth for different signal-to-noise ratios, in the presence of errors in the physical model, and for several Fourier domain sampling schemes. It is demonstrated that a residual analysis can help to decide if the conventional MRF method is sufficient or if the complex Bayesian Fourier domain modeling approach can lead to a significant improvement.

SynthIA: A Synthetic Inversion Approximation for the Stokes Vector Fusing SDO and Hinode into a Virtual Observatory

Abstract Both NASA’s Solar Dynamics Observatory (SDO) and the JAXA/NASA Hinode mission include spectropolarimetric instruments designed to measure the photospheric magnetic field. SDO’s Helioseismic and Magnetic Imager (HMI) emphasizes full-disk, high-cadence, and good-spatial-resolution data acquisition while Hinode’s Solar Optical Telescope Spectro-Polarimeter (SOT-SP) focuses on high spatial resolution and spectral sampling at the cost of a limited field of view and slower temporal cadence. This work introduces a deep-learning system, named the Synthetic Inversion Approximation (SynthIA), that can enhance both missions by capturing the best of each instrument’s characteristics. We use SynthIA to produce a new magnetogram data product, the Synthetic Hinode Pipeline (SynodeP), that mimics magnetograms from the higher-spectral-resolution Hinode/SOT-SP pipeline, but is derived from full-disk, high-cadence, and lower-spectral-resolution SDO/HMI Stokes observations. Results on held-out data show that SynodeP has good agreement with the Hinode/SOT-SP pipeline inversions, including magnetic fill fraction, which is not provided by the current SDO/HMI pipeline. SynodeP further shows a reduction in the magnitude of the 24 hr oscillations present in the SDO/HMI data. To demonstrate SynthIA’s generality, we show the use of SDO/Atmospheric Imaging Assembly data and subsets of the HMI data as inputs, which enables trade-offs between fidelity to the Hinode/SOT-SP inversions, number of observations used, and temporal artifacts. We discuss possible generalizations of SynthIA and its implications for space-weather modeling. This work is part of the NASA Heliophysics DRIVE Science Center at the University of Michigan under grant NASA 80NSSC20K0600E, and will be open-sourced.

New Class of Rank 1 Update for Solving Unconstrained Optimization Problem

The focus of this article is to add a new class of rank one of modified Quasi-Newton techniques to solve the problem of unconstrained optimization by updating the inverse Hessian matrix with an update of rank 1, where a diagonal matrix is the first component of the next inverse Hessian approximation, The inverse Hessian matrix is generated by the method proposed which is symmetric and it satisfies the condition of modified quasi-Newton, so the global convergence is retained. In addition, it is positive definite that guarantees the existence of the minimizer at every iteration of the objective function. We use the program MATLAB to solve an algorithm function to introduce the feasibility of the proposed procedure. Various numerical examples are given`.

Depth-resolved residual stress evaluation from X-ray diffraction measurement data using the approximate inverse method

Abstract The paper deals with the depth determination of residual stress states from diffraction data. First an historical overview of the known approaches is given. Then we apply the approximate inverse method to this problem. This method is known to be very efficient and stable with respect to noise-contaminated data. It is even possible to prove convergence and it allows an error estimate of the calculated depth resolved residual stress profile.

Bernstein and Bernstein-Like Inequalities for Modulation Spaces

It is shown that the modulation spaces Mwp can be characterized by the approximation behavior of their elements using Local Fourier bases. In analogy to the Local Fourier bases, we show that the modulation spaces can also be characterized by the approximation behavior of their elements using Gabor frames. We derive direct and inverse approximation theorems that describe the best approximation by linear combinations of N terms of a given function using its modulates and translates.

Numerical approximation for a nonlinear variable-order fractional differential equation via a collocation method

We study a numerical approximation for a nonlinear variable-order fractional differential equation via a collocation method. Due to the lack of monotonicity of the discretization coefficients of the variable-order fractional derivative in standard approximation schemes, existing numerical analysis techniques do not apply directly. By an approximate inversion technique, the proposed model is transformed as a second kind Volterra integral equation, based on which a collocation method under uniform or graded mesh is developed and analyzed. In particular, the mesh grading parameter we used is consistent with respect to the regularity of the solutions, and the convergence rates are characterized in terms of the initial value of the variable order, which serve as improvements compared with the existing results in the literature.

On [formula omitted] matrix-based approximate inverse preconditioning technique for diagonal-plus-Toeplitz linear systems from spatial fractional diffusion equations

In this paper, we firstly explore the special structure of the discretized linear systems from the spatial fractional diffusion equations. The coefficient matrices of the resulting discretized systems have a diagonal-plus-Toeplitz structure. Because the resulting Toeplitz matrix is symmetric positive definite (SPD), then we can employ the τ matrix to approximate it. By making use of the piecewise interpolation polynomials, we propose a new approximate inverse preconditioner to handle the diagonal-plus-Toeplitz coefficient matrices. The τ matrix-based approximate inverse (TAI) preconditioning technique can be implemented very efficiently by using discrete sine transforms(DST). Theoretically, we have proved that the spectrum of the resulting preconditioned matrices are clustered around one. Thus, Krylov subspace methods with the proposed preconditioners converge very fast. To demonstrate the efficiency of the new preconditioners, numerical experiments are implemented. The numerical results show that with the proper interpolation node numbers, the performance of the τ-matrix based preconditioning technique is better than the other tested preconditioners.

Optimization of Homomorphic Comparison Algorithm on RNS-CKKS Scheme

The sign function can be adopted to implement the comparison operation, max function, and rectified linear unit (ReLU) function in the Cheon&#x2013;Kim&#x2013;Kim&#x2013;Song (CKKS) scheme; hence, several studies have been conducted to efficiently evaluate the sign function in the CKKS scheme. Recently, Lee <i>et al.</i> (IEEE Trans. Depend. Sec. Comp.) proposed a practically optimal approximation method for the sign function in the CKKS scheme using a composition of minimax approximate polynomials. In addition, Lee <i>et al.</i> proposed a polynomial-time algorithm that finds the degrees of component polynomials that minimize the number of non-scalar multiplications. However, homomorphic comparison/max/ReLU functions using Lee <i>et al.</i>&#x2019;s approximation method have not been successfully implemented in the residue number system variant CKKS (RNS-CKKS) scheme. In addition, the degrees of component polynomials found by Lee <i>et al.</i>&#x2019;s algorithm are not optimized for the RNS-CKKS scheme because the algorithm does not consider that the running time of non-scalar multiplication depends significantly on the ciphertext level in the RNS-CKKS scheme. In this study, we propose a fast algorithm for the inverse minimax approximation error, which is a subroutine required to find the optimal set of degrees of component polynomials. The proposed algorithm facilitates determining the optimal set of degrees of component polynomials with higher degrees than in the previous study. In addition, we propose a method to find the degrees of component polynomials optimized for the RNS-CKKS scheme using the proposed algorithm for the inverse minimax approximation error. We successfully implement the homomorphic comparison, max function, and ReLU function algorithms on the RNS-CKKS scheme with a low comparison failure rate (<inline-formula> <tex-math notation="LaTeX">$&lt; 2^{-15}$ </tex-math></inline-formula>), and provide various parameter sets according to the precision parameter <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>. We reduce the depth consumption of the homomorphic comparison, max function, and ReLU function algorithms by one depth for several values of <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>. In addition, the numerical analysis demonstrates that the homomorphic comparison, max function, and ReLU function algorithms using the degrees of component polynomials found by the proposed algorithm reduce the running time by 6&#x0025;, 7&#x0025;, and 6&#x0025; on average, respectively, compared with those using the degrees of component polynomials found by Lee <i>et al.</i>&#x2019;s algorithm.

Variable Structure Control of Second Order Systems With Nonlinear Uncertain Discontinuous Input Functions Using Approximate Inverses

We consider second order nonlinear systems in control canonical form, where the functional relation between the control variable and the system acceleration is nonlinear, uncertain and discontinuous. We approximate the nonlinear uncertain and discontinuous function with an invertible one, and use variable structure control theory to suppress the error of the approximate inverse. We rewrite the finite time reaching law or the terminal reaching law into the form of inequalities and use these inequalities to design a control law that drives the system state to the sliding surface in finite time, and one that ensures the convergence of the system speed and position errors to a narrow, adjustable width band around zero. The stability of the control system is also analyzed and proven. We demonstrate how to ensure the sufficiency of the available control effort, by the proper design of control parameters. The method is then validated on a numerical model of a nonlinear system with nonlinear, uncertain and discontinuous input function.

An Approach to Calculate Inverse of LR-Trapezoidal Fuzzy Matrix

In this paper, we devised a numerical algorithm for calculating the inverse of trapezoidal fuzzy numbers matrix. By analysing the existence of fuzzy inverse, the sufficient conditions are restricted. Furthermore, through calculating the approximate fuzzy inverse, the algorithm can be widely used in solving fully fuzzy linear system and it is validated by some given examples.

Improved Iterative Inverse Matrix Approximation Algorithm for Zero Forcing Precoding in Large Antenna Arrays

Precoding algorithms are used in massive multiple-input multiple-output (mMIMO) communication systems to ensure effective signal transmission. The zero-forcing (ZF) is one of the most common linear precoding algorithms used in MIMO and mMIMO systems. ZF precoding is complex to implement because it requires direct matrix inversion of the gram matrix. Iterative algorithms have been proposed for approximating matrix inversions. However, iterative algorithms require initial conditions and pre-computations to converge to the optimal transmitted signal vector. This paper proposes a new improved iterative algorithm that guarantees convergence under any circumstances without dependency on any optimized initial parameter or condition. The proposed algorithm is based on a three-step iterative and iterative generalized inverse matrix approximation algorithm. The proposed algorithm was verified using a new correlated channel model that included mutual coupling effects and gain and phase variances caused by radio frequency elements at a base station (BS). The computational complexity of the proposed algorithm was then computed. This study analyzes and compares the bit error rate (BER) performance of the proposed algorithm with that of prominent existing algorithms. Moreover, the sum-rate performance of the proposed algorithm was analyzed. Simulations were performed under both correlated and uncorrelated channel conditions, for comparison and analysis. The simulation results demonstrate that the proposed algorithm outperforms the compared algorithms in terms of the convergence and convergence rates.

Bach-Einstein Gravitational Field Equations as a Perturbation of Einstein Gravitational Field Equations

The Bach equations are a version of higher-order gravitational field equations, exactly they are of fourth-order. In 4-dimensions the Bach-Einstein gravitational field equations are treated here as a perturbation of Einstein’s gravity. An approximate inversion formula is derived which admits a comparison of the two field theories. An application to these theories is given where the gravitational Lagrangian is expressed linearly in terms of R, R2, |Ric|2, where the Ricci tensor Ric = Rαβdxαdxβ is inserted in some formulas which are of geometrical or physical importance, such as; Raychaudhuri equation and Tolman’s formula.

Inverse Result of Approximation for the Max-Product Neural Network Operators of the Kantorovich Type and Their Saturation Order

In this paper, we consider the max-product neural network operators of the Kantorovich type based on certain linear combinations of sigmoidal and ReLU activation functions. In general, it is well-known that max-product type operators have applications in problems related to probability and fuzzy theory, involving both real and interval/set valued functions. In particular, here we face inverse approximation problems for the above family of sub-linear operators. We first establish their saturation order for a certain class of functions; i.e., we show that if a continuous and non-decreasing function f can be approximated by a rate of convergence higher than 1/n, as n goes to +∞, then f must be a constant. Furthermore, we prove a local inverse theorem of approximation; i.e., assuming that f can be approximated with a rate of convergence of 1/n, then f turns out to be a Lipschitz continuous function.

Accurate approximants of inverse Brillouin functions

We present an investigation to improve the polynomial part of an analytical approximation of inverse Brillouin functions, recently proposed in the literature. For the analytical approximation, we employ a sextic polynomial model. To determine the best fit coefficients of the model, we apply a simple statistical analysis. We obtained the same accuracy as the only case presented in the literature, but with three orders of magnitude fewer points involved in the statistical analysis, which translates into a faster convergence of the coefficients. We extend the investigation to the most interesting cases for physical applications. The method can be applied to any other desired inverse Brillouin functions or polynomial approximations of special functions.

Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

Direct and inverse approximation theorems are proved in the Besicovitch-Stepanets spaces $B{\mathcal S}^{p}$ of almost periodic functions in terms of the best approximations of functions and their generalized moduli of smoothness.