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In this study, to approximate nabla sequential differential equations of fractional order, a class of discrete Liouville–Caputo fractional operators is discussed. First, some special functions are re-called that will be useful to make a connection with the proposed discrete nabla operators. These operators exhibit inherent symmetrical properties which play a crucial role in ensuring the consistency and stability of the method. Next, a formula is adopted for the solution of the discrete system via binomial coefficients and analyzing the Riemann–Liouville fractional sum operator. The symmetry in the binomial coefficients contributes to the precise approximation of the solutions. Based on this analysis, the solution of its corresponding continuous case is obtained when the step size p 0 tends to 0. The transition from discrete to continuous domains highlights the symmetrical nature of the fractional operators. Finally, an example is shown to testify the correctness of the presented theoretical results. We discuss the comparison of the solutions of the operators along with the numerical example, emphasizing the role of symmetry in the accuracy and reliability of the numerical method. [ABSTRACT FROM AUTHOR]

Systematic sampling method has been used in a finite population for estimation of the population characteristic. In this article, we are trying to extend this method when the population size is countably infinite. In finite population, a unit has equal probability of being chosen in most of the cases except PPS. However, this cannot be so in infinite population. In that case, we approximate infinite population by large finite population and then use the traditional finite population method. It is true that the original population have some distribution, but without knowing that we are using systematic sampling method and based on that we are estimating population mean. We want to see the effectiveness of this systematic manner if the population has some specified distribution. Here, based on Poisson assumption, we provide some approximation methods and see performances in each case. However, this is to be done for some important distributions also. Here, the problem has been viewed in three ways and out of that the first method performs well, where the Poisson probabilities has been assigned to all possible samples rather than any truncation of the infinite population by finite population. [ABSTRACT FROM AUTHOR]

Simple and accurate closed-form approximations to the probability density function of the sum of independent identically distributed random variables are derived. We propose a unified approach capable of providing the pdf of the sum when the summands involve the generalized Nakagami-m, generalized Gamma, Rayleigh and Weibull random variables. Unlike others work, the approximations do not use any infinite series or require any complicated special functions. Numerical results show that the new approximations have satisfactory accuracies. As an application, the outage probability and the M-QAM average SER performance of pre-detection EGC diversity are presented. [ABSTRACT FROM AUTHOR]

This article is devoted to the history of development and main research areas of the scientific school in the field of pattern recognition, image processing and analysis, and artificial intelligence and machine learning, founded in the early 1990s at the Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University (SINP MSU) by Prof. Igor' Georgievich Persiantsev. For many years Persiantsev was the permanent leader of this scientific school; he laid down the basic principles and approaches to scientific research that still guide his disciples to this day. During this time, more than 30 people became students of Persiantsev's school, who carried out scientific work under his leadership or under the leadership of his disciples, defended their candidate's dissertations or diploma at the Faculty of Physics, Lomonosov Moscow State University. The article provides a brief historical background and an overview of the areas of research and major works published over more than 30 years (from 1992 to 2023) by Persiantsev and his disciples. [ABSTRACT FROM AUTHOR]

The envelope theory is an easy-to-use approximation method to obtain eigensolutions for some quantum many-body systems, in particular in the domain of hadronic physics. Even if the solutions are reliable and an improvement procedure exists, the method can lack accuracy for some systems. In a previous work, two hypotheses were proposed to explain the low precision: the presence of a divergence in the potential or the lack of a variational character for peculiar interactions. In the present work, different systems are studied to test these hypotheses. These tests show that the presence of a divergence does indeed cause less accurate results, while the lack of a variational character reduces the impact of the improvement procedure.

In this study, to approximate nabla sequential differential equations of fractional order, a class of discrete Liouville–Caputo fractional operators is discussed. First, some special functions are re-called that will be useful to make a connection with the proposed discrete nabla operators. These operators exhibit inherent symmetrical properties which play a crucial role in ensuring the consistency and stability of the method. Next, a formula is adopted for the solution of the discrete system via binomial coefficients and analyzing the Riemann–Liouville fractional sum operator. The symmetry in the binomial coefficients contributes to the precise approximation of the solutions. Based on this analysis, the solution of its corresponding continuous case is obtained when the step size p0 tends to 0. The transition from discrete to continuous domains highlights the symmetrical nature of the fractional operators. Finally, an example is shown to testify the correctness of the presented theoretical results. We discuss the comparison of the solutions of the operators along with the numerical example, emphasizing the role of symmetry in the accuracy and reliability of the numerical method.

Constrained partially observable Markov decision processes (CPOMDPs) have been used to model various real-world phenomena. However, they are notoriously difficult to solve to optimality, and there exist only a few approximation methods for obtaining high-quality solutions. In this study, grid-based approximations are used in combination with linear programming (LP) models to generate approximate policies for CPOMDPs. A detailed numerical study is conducted with six CPOMDP problem instances considering both their finite and infinite horizon formulations. The quality of approximation algorithms for solving unconstrained POMDP problems is established through a comparative analysis with exact solution methods. Then, the performance of the LP-based CPOMDP solution approaches for varying budget levels is evaluated. Finally, the flexibility of LP-based approaches is demonstrated by applying deterministic policy constraints, and a detailed investigation into their impact on rewards and CPU run time is provided. For most of the finite horizon problems, deterministic policy constraints are found to have little impact on expected reward, but they introduce a significant increase to CPU run time. For infinite horizon problems, the reverse is observed: deterministic policies tend to yield lower expected total rewards than their stochastic counterparts, but the impact of deterministic constraints on CPU run time is negligible in this case. Overall, these results demonstrate that LP models can effectively generate approximate policies for both finite and infinite horizon problems while providing the flexibility to incorporate various additional constraints into the underlying model. [ABSTRACT FROM AUTHOR]

Copyright of Przegląd Elektrotechniczny is the property of Przeglad Elektrotechniczny and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs. We also provide a short overview of the recent literature in this area of research. [ABSTRACT FROM AUTHOR]

We propose a new variational model in Sobolev–Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to image processing. The characteristic feature of the proposed model is that the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. So, we deal with a constrained minimization problem that lives in variable Sobolev–Orlicz spaces. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations. [ABSTRACT FROM AUTHOR]

We present a new methodology to approximate a trapezoidal fuzzy numbers set by using smoothing radial basis functions (R B F s). The methodology uses different error and similarity indices to determine and compare the accuracy of the approximation of the given trapezoidal fuzzy data. For the proposed approximation method a fuzzy radial basis functions type are defined, called fuzzy smoothing radial basis functions under tension. The computation of one of these approximation functions from a given trapezoidal fuzzy data set is described and some convergence results are proved. Finally, some examples in two-dimensions are given to compare the behavior of the presented method by using the proposed error and similarity indices for different configurations of the fuzzy smoothing radial basis functions under tension. [ABSTRACT FROM AUTHOR]

The complete multiple-input–multiple-output (MIMO) channel simulation in deterministic techniques can be a computationally intensive task, due to the inherent scenario's complexity and number of antennas. The spatial coherence among MIMO elements is a reasonable assumption to approximate the channel from a single point simulation. In this letter, a novel method to incorporate a geometrical approximation of the MIMO channel into a three-dimensional ray launching (3D-RL) algorithm is presented. The method is antenna type independent and the orientation of the array is embedded in the antenna representation. Relevant information of the MIMO channel characteristics like the root mean square (rms) delay spread, the maximum delay spread, phase and channel capacity are obtained and compared with the full 3D-RL simulation of the entire MIMO array, achieving 93.4% reduction in computational time. [ABSTRACT FROM AUTHOR]

The paper presents research on the approximation of variable-order fractional operators by recurrent neural networks. The research focuses on two basic variable-order fractional operators, i.e., integrator and differentiator. The study includes variations of the order of each fractional operator. The recurrent neural network architecture based on GRU (Gated Recurrent Unit) cells functioned as a neural approximation for selected fractional operators. The paper investigates the impact of the number of neurons in the hidden layer, treated as a hyperparameter, on the quality of modeling error. Training of the established recurrent neural network was performed on synthetic data sets. Data for training was prepared based on the modified Grünwald-Letnikov definition of variable-order fractional operators suitable for convenient numerical computing without memory effects. The research presented in this paper showed that recurrent network architecture based on GRU-type cells can satisfactorily approximate targeted simple yet functional variable-order fractional operators with minor modeling errors. In addition, the research also compares the presented solution with basic and recurrent neural networks that utilize Tapped Delay Lines (TDL) in their structure. The presented solution is a novel approach to the approximation of VO-FC operators. It has the advantage of automatic selection of neural approximator parameters by optimization based on data customized for specific requirements.

Radio telescopes produce large volumes of data that need to be processed to obtain high-resolution sky images. This is a complex task that requires computing systems that provide both high performance and high energy efficiency. Hardware accelerators such as GPUs (Graphics Processing Units) and FPGAs (Field Programmable Gate Arrays) can provide these two features and are thus an appealing option for this application. Most HPC (High-Performance Computing) systems operate in double precision (64-bit) or in single precision (32-bit), and radio-astronomical imaging is no exception. With reduced precision computing, smaller data types (e.g., 16-bit) are used to improve energy efficiency and throughput performance in noise-tolerant applications. We demonstrate that reduced precision can also be used to produce high-quality sky images. To this end, we analyze the gridding component (Image-Domain Gridding) of the widely-used WSClean imaging application. Gridding is typically one of the most time-consuming steps in the imaging process and, therefore, an excellent candidate for acceleration. We identify the minimum required exponent and mantissa bits for a custom floating-point data type. Then, we propose the first custom floating-point accelerator on a Xilinx Alveo U50 FPGA using High-Level Synthesis. Our reduced-precision implementation improves the throughput and energy efficiency of respectively $1.84\times$ and $2.03\times$ compared to the single-precision floating-point baseline on the same FPGA. Our solution is also $2.12\times$ faster and $3.46\times$ more energy-efficient than an Intel i9 9900k CPU (Central Processing Unit) and manages to keep up in throughput with an AMD RX 550 GPU.

A scheme often used for error tolerance of arithmetic circuits is the so-called Reduced Precision Redundancy (RPR). Rather than replicating multiple times the entire module, RPR uses reduced precision (inexact) copies to significantly reduce the redundancy overhead, while still being able to correct the largest errors. This paper focuses on the low-power operation for RPR; a new scheme is proposed. At circuit level, power gating is initially utilized in the arithmetic modules to power off one of the modules (i.e., the exact module) when the inexact modules’ error is smaller than the threshold. The proposed design is applicable to (unsigned integer) addition, multiplication, and MAC (multiply and add) by proposing RPR implementations that reduce the power consumption with a limited impact on its error correction capability. The proposed schemes have been implemented and tested for various applications (image and DCT processing). The results show that they can significantly reduce power consumption; moreover, the simulation results show that the Mean Square Error (MSE) at the proposed schemes’ output is low.

In this paper, the Wentzel–Kramers–Brillouin (WKB) method is extended to be applicable for conformable Hamiltonian systems, where the concept of conformable operator with fractional order α is involved. The WKB approximation for the α -wave function is derived for potentials which slowly vary in space. Some illustrative examples to demonstrate the method are presented. The quantities of the conformable form are found to be in exact agreement with the corresponding traditional quantities when α = 1. [ABSTRACT FROM AUTHOR]

Thomson’s multitaper method estimates the power spectrum of a signal from $N$ equally spaced samples by averaging $K$ tapered periodograms. Discrete prolate spheroidal sequences (DPSS) are used as tapers since they provide excellent protection against spectral leakage. Thomson’s multitaper method is widely used in applications, but most of the existing theory is qualitative or asymptotic. Furthermore, many practitioners use a DPSS bandwidth $W$ and number of tapers that are smaller than what the theory suggests is optimal because the computational requirements increase with the number of tapers. We revisit Thomson’s multitaper method from a linear algebra perspective involving subspace projections. This provides additional insight and helps us establish nonasymptotic bounds on some statistical properties of the multitaper spectral estimate, which are similar to existing asymptotic results. We show using $K=2NW-O(\log (NW))$ tapers instead of the traditional $2NW-O(1)$ tapers better protects against spectral leakage, especially when the power spectrum has a high dynamic range. Our perspective also allows us to derive an $\epsilon $ -approximation to the multitaper spectral estimate which can be evaluated on a grid of frequencies using $O\left({\log (NW)\log \tfrac {1}{ \epsilon }}\right)$ FFTs instead of $K=O(NW)$ FFTs. This is useful in problems where many samples are taken, and thus, using many tapers is desirable. [ABSTRACT FROM AUTHOR]

Time series approximation techniques can provide approximate results for the data in another new space by dimensionality reduction or feature extraction. In this study, we propose a new time series approximation strategy based on the Fuzzy C-Means (FCM) clustering and elaborate on a comprehensive analysis of relationships between reconstruction error and classification performance when dealing with various representation (approximation) mechanisms of time series. Typically, time series approximation leads to the representation of original time series in the space of lower dimensionality compared to the dimensionality of the original input space. We reveal, quantify, and visualize the relationships between the reconstruction error and classification error (classification rate) for several commonly encountered representation methods. Through carefully structured experiments completed for sixteen publicly available datasets, we demonstrate experimentally and analytically that the classification error obtained for time series in the developed representation space becomes smaller than when dealing with original time series. It has been also observed that the reconstruction error decreases when increasing the dimensionality of the representation space. In addition, when compared with the state-of-the-art algorithms reported in the literature, experimental results show the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

This article extends the previous research on the notion of a standardized call function and how to obtain an approximate model of the Black-Scholes formula via the hyperbolic tangent. Although the Black-Scholes approach is outdated and suffers from many limitations, it is still widely used to derive the implied volatility of options. This is particularly important for traders because it represents the risk of the underlying, and is the main factor in the option price. The approximation error of the suggested solution was estimated and the results compared with the most common methods available in the literature. A new formula was provided to correct some cases of underestimation of implied volatility. Graphic evidence, stress tests and Monte Carlo analysis confirm the quality of the results obtained. Finally, further literature is provided as to why implied volatility is used in decision making. [ABSTRACT FROM AUTHOR]

In this article, we extend the adaptive tracking control to more general nonlinear systems with multiple uncertainties, including output constraint, input delay, unknown parameter, and external disturbances for the first time. Without any growth assumptions, the adaptive backstepping technique is combined with the parameter separation technique to solve the parametric nonlinearities while the related results need to apply restrictive assumptions or use an approximation-based scheme to deal with them. To alleviate the serious uncertainties caused by output constraint and input delay, the barrier Lyapunov function (BLF) and the Pade approximation method are employed in a unified framework when the control coefficient is known. Under the case of the unknown control coefficient, the Nussbaum gain technique is further combined to compensate it. Then, under both cases, universal adaptive state-feedback control strategies merged with rigorous stability analysis are proposed, respectively, which guarantees all the signals are uniformly ultimately bounded. In addition, the reference signal tracks the system output into a compact set of the origin and the constraint of output is not violated. Finally, the proposed controllers are applied to an inverted pendulum system, which demonstrates the designed controller is effective. [ABSTRACT FROM AUTHOR]

In This Research paper, described different linearization methods in terms of analytical and numerical simulation (Computer added linearized -FE) method and Digital image correlation method for prediction of loading analysis. Article can be helping for linearization in nonlinearity of systems, that what is using different methods likes Numerical (approximation), Analytical (Exact) and Experimental. Where, analytical is difficult to calculate in computers so it uses numerical methods. That is nearer to exact values so we can determine behaviour of dynamical systems. Numerical simulation is static with different time, so analysis is easy for its. Numerical methods work on algorithms, so it is easy to evaluate for calculating in computers. For future aspects image correlation method is more helpful to determine behaviour of nonlinear dynamic system which will discretize system with different displacement and FEM analyses to dynamically nonlinear system. [ABSTRACT FROM AUTHOR]

Copyright of Computer Measurement & Control is the property of Magazine Agency of Computer Measurement & Control and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Graph matching, the problem of aligning a pair of graphs so as to minimize their edge disagreements, has received widespread attention owing to its broad spectrum of applications in data science. As the problem is NP–hard in the worst-case, a variety of approximation algorithms have been proposed for obtaining high quality, suboptimal solutions. In this article, we approach the task of designing an efficient polynomial-time approximation algorithm for graph matching from a previously unconsidered perspective. Our key result is that graph matching can be formulated as maximizing a monotone, supermodular set function subject to matroid intersection constraints. We leverage this fact to apply a discrete optimization variant of the minorization-maximization algorithm which exploits supermodularity of the objective function to iteratively construct and maximize a sequence of global lower bounds on the objective. At each step, we solve a maximum weight matching problem in a bipartite graph. Differing from prior approaches, the algorithm exploits the combinatorial structure inherent in the problem to generate a sequence of iterates featuring monotonically non-decreasing objective value while always adhering to the combinatorial matching constraints. Experiments on real-world data demonstrate the empirical effectiveness of the algorithm relative to the prevailing state-of-the-art. [ABSTRACT FROM AUTHOR]

The next generation massive access has attracted considerable recent attention due to its potential in smart meters, industrial internet of things (IIoT), and smart traffics, etc. However, how to achieve the minimum queuing delay in massive access still remains open, thereby making quality-of-service (QoS) assurance a challenging issue in practice. In this paper, we aim at minimizing the average queuing delay by applying cross-layer scheduling with joint channel and buffer awareness, the complexity of which increases exponentially with the number of users. Fortunately, with massive users or devices, mean-field approximation can be adopted to substantially simplify the design and analysis of the delay-optimal scheduling. More specifically, we present a cocktail filling policy and a queue-aware multiuser diversity protocol, in which all backlogged packets of a user will be served by either NOMA or TDMA mode respectively, if the user’s channel gain is beyond a certain threshold. The average queuing delay and queue-length-violation probability are derived based on a Markov model. Numerical results will also demonstrate that the mean-field approximation based joint physical and network layer scheduling is capable of improving the QoS in massive access. [ABSTRACT FROM AUTHOR]

Outpatient clinics typically face a critical issue in patient adherence to their clinic appointments and suffer a huge backlog of patients waiting to be seen. The goal of this study is to enhance care provider productivity and patient access through effective clinic appointment scheduling. To achieve this goal, we introduce discrete-time bulk-service queues to model the backlog dynamics accounting for the no-show behavior of patients and consider different overbooking strategies to reduce backlogs at a minimum risk of working overtime. Numerically stable solution procedures and approximation methods are introduced to significantly reduce the computational burden, making the queueing models a practical tool for outpatient appointment template design. The insights obtained from this work will support capacity planning decision-making and are critical to improving the operational efficiency of outpatient clinics. Note to Practitioners—The operation and management of appointment systems is becoming much more sophisticated these years; however, worldwide, outpatient clinics still suffer from a common conundrum, which is the extensive backlog accumulation. Patients not showing up to their clinic appointments are one of the major contributors to this issue. The overbooking approach, which redesigns the appointment system by releasing more appointment slots than budgeted, is typically employed to address this no-show issue. However, the practice of overbooking should be carried out carefully, and the performance of system backlog as well as the risk of overworking should be considered simultaneously. This work provides a practical tool to assist the management of appointment capacity in outpatient clinics. [ABSTRACT FROM AUTHOR]

Solving mixed-integer nonlinear programs (MINLPs) is hard from both a theoretical and practical perspective. Decomposing the nonlinear and the integer part is promising from a computational point of view. In general, however, no bounds on the objective value gap can be established and iterative procedures with potentially many subproblems are necessary. The situation is different for mixed-integer optimal control problems with binary variables that switch over time. Here, a priori bounds were derived for a decomposition into one continuous nonlinear control problem and one mixed-integer linear program, the combinatorial integral approximation (CIA) problem. In this article, we generalize and extend the decomposition idea. First, we derive different decompositions and analyze the implied a priori bounds. Second, we propose several strategies to recombine promising candidate solutions for the binary control functions in the original problem. We present the extensions for ordinary differential equations-constrained problems. These extensions are transferable in a straightforward way, though, to recently suggested variants for certain partial differential equations, for algebraic equations, for additional combinatorial constraints, and for discrete time problems. We implemented all algorithms and subproblems in AMPL for a proof-of-concept study. Numerical results show the improvement compared to the standard CIA decomposition with respect to objective function value and compared to general-purpose MINLP solvers with respect to runtime. [ABSTRACT FROM AUTHOR]

The envelope theory is a method to easily obtain approximate, but reliable, solutions for some quantum many-body problems. Quite general Hamiltonians can be considered for systems composed of an arbitrary number of different particles in D dimensions. In the case of identical particles, a compact set of three equations can be written to find the eigensolutions. This set provides also a nice interpretation and a starting point to improve the method. It is shown here that a similar set of seven equations can be determined for a system containing an arbitrary number of two different particles. [ABSTRACT FROM AUTHOR]

The purpose of this work is to produce a regularity theory for a class of parabolic Isaacs equations. Our techniques are based on approximation methods which allow us to connect our problem with a Bellman parabolic model. An approximation regime for the coefficients, combined with a smallness condition on the source term unlocks new regularity results in Sobolev and Hölder spaces. [ABSTRACT FROM AUTHOR]

Spectral clustering techniques are valuable tools in signal processing and machine learning for partitioning complex data sets. The effectiveness of spectral clustering stems from constructing a non-linear embedding based on creating a similarity graph and computing the spectral decomposition of the Laplacian matrix. However, spectral clustering methods fail to scale to large data sets because of high computational cost and memory usage. A popular approach for addressing these problems utilizes the Nyström method, an efficient sampling-based algorithm for computing low-rank approximations to large positive semi-definite matrices. This paper demonstrates how the previously popular approach of Nyström-based spectral clustering has severe limitations. Existing time-efficient methods ignore critical information by prematurely reducing the rank of the similarity matrix associated with sampled points. Also, current understanding is limited regarding how utilizing the Nyström approximation will affect the quality of spectral embedding approximations. To address the limitations, this work presents a principled spectral clustering algorithm that exploits spectral properties of the similarity matrix associated with sampled points to regulate accuracy-efficiency trade-offs. We provide theoretical results to reduce the current gap and present numerical experiments with real and synthetic data. Empirical results demonstrate the efficacy and efficiency of the proposed method compared to existing spectral clustering techniques based on the Nyström method and other efficient methods. The overarching goal of this work is to provide an improved baseline for future research directions to accelerate spectral clustering.

This research paper aims to analyze the minimax approach used in the semivariogram fitting process that forms one stage of the kriging operation performed for interpolation. The conventional method uses the weighted least squares fit for various theoretical functions such as stable, exponential, spherical. However, several recent approaches have been developed using machine learning regression techniques. This research employs the ordinary kriging technique where the proposed minimax approach is expected to increase the accuracy of the interpolation resulted by reducing the error of the final result. Kriging, which is based on the stochastic method, is widely used for spatial values and has been proven to be a better predicting process than deterministic methods. The novel approach to ordinary kriging discussed here, the minimax approach, is able to increase result accuracy based on the experiments performed. Minimax can predict the weights of the semivariogram values better than the weighted least-squares method and performs faster than machine learning approaches.

This paper is concerned with performance analysis for bounded persistent disturbances of continuous-time linear time-invariant (LTI) systems. Such an analysis can be done by computing the L∞-induced norm of continuous-time LTI systems since it corresponds to the worst maximum magnitude of the output for the worst persistent external input with a unit magnitude. In our preceding study, piecewise constant and linear approximation schemes for analyzing this norm have been developed through two alternative approximation approaches, one for the input and the other for the relevant kernel function, via the fast-lifting technique. The approximation errors in these approximation schemes have been shown to converge to 0 at the rates of 1/N and 1/N2, respectively, as the fast-lifting parameter N is increased. Along this line, this paper aims at developing generalized techniques that offer improved accuracy named the piecewise quadratic and cubic approximation schemes. The generalization and the associated accuracy improvement discussed in this paper are not limited to the increased orders of approximation but are extended further to taking advantage of the freedom in the point around which relevant functions are expanded to Taylor series. The approximation errors in the piecewise quadratic and cubic approximation schemes are shown to converge to 0 at the rates of 1/N3 and 1/N4, respectively, regardless of the point at which the Taylor expansion is applied. Finally, effectiveness of the developed computation methods is confirmed through a numerical example.

In this paper, the perturbation theory is extended to be applicable for systems containing conformable derivative of fractional order α. This is needed as an essential and powerful approximation method for describing systems with conformable differential equations that are difficult to solve analytically. The work here is derived and discussed for the conformable Hamiltonian systems that appears in the conformable quantum mechanics. The required α -corrections for the energy eigenvalues and eigenfunctions are derived. To demonstrate this extension, three illustrative examples are given, and the standard values obtained by the traditional theory are recovered when α = 1. [ABSTRACT FROM AUTHOR]

The combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discrete-valued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discrete-valued approximants that are defined on given decompositions of the domain. It is known that the resulting discrete controls can be constructed such that they converge to a relaxed control in the weak ∗ topology of L ∞ if the grid constant of this decomposition is driven to zero. We exploit this insight to formulate a general approximation result for optimization problems, which feature discrete and distributed optimization variables, and which are governed by a compact control-to-state operator. We analyze the topology induced by the grid refinements and prove convergence rates of the control vectors for two problem classes. We use a reconstruction problem from signal processing to demonstrate both the applicability of the method outside the scope of differential equations, the predominant case in the literature, and the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

Multibang regularization and combinatorial integral approximation decompositions are two actively researched techniques for integer optimal control. We consider a class of polyhedral functions that arise particularly as convex lower envelopes of multibang regularizers and show that they have beneficial properties with respect to regularization of relaxations of integer optimal control problems. We extend the algorithmic framework of the combinatorial integral approximation such that a subsequence of the computed discrete-valued controls converges to the infimum of the regularized integer control problem. [ABSTRACT FROM AUTHOR]

I develop inverse scattering methods for velocity reconstruction in the subsurface, based on multiple scattering theory, Gaussian beams and nonlinear Born approximation. This method is based on a new version of the distorted Born iterative inverse scattering method. It directly uses an explicit representation of the data sensitivity function in terms of Green functions, rather than the indirect optimization approach based on the adjoint state method. I propose three direct scattering methods for forward modeling, namely, Gaussian beam integral equation propagators, nonlinear Born approximation, and multiple scattering. First, I apply a sensitivity kernel incorporating the Gaussian beam summation-based integral equation method and compute the background medium Green’s function in an inhomogeneous medium. Then I extend an approximate solution of the Lippmann–Schwinger equation, referred to as nonlinear Born approximation to inverse scattering problem. I apply the nonlinear Born approximation to forward modeling at each iteration. Furthermore, I combine the Gaussian beams with the multiple scattering theory for nonlinear scattering problems. I obtain the inversion results using the proposed approaches, which shows the capability for inverse scattering imaging. I have carried out several numerical experiments that involve reconstructing the resampled Marmousi and Society of Exploration Geophysicists (SEG)/European Association of Geoscientists and Engineers (EAGE) salt models from its smoothed version. [ABSTRACT FROM AUTHOR]