Approximation Operators Research Articles

AxOTreeS : A Tree S earch Approach to Synthesizing FPGA-based A ppro x imate O perators

Approximate computing (AxC) provides the scope for achieving disproportionate gains in a system’s power, performance, and area (PPA) metrics by leveraging an application’s inherent error-resilient behavior (BEHAV). Trading computational accuracy for performance gains makes AxC an attractive proposition for implementing computationally complex AI/ML-based applications on resource-constrained embedded systems. The growing diversity of application domains using AI/ML has also led to the increasing usage of FPGA-based embedded systems. However, implementing AxC for FPGAs has primarily been limited to the post-processing of ASIC-optimized approximate operators (AxOs). This approach usually involves selecting from a set of AxOs that have been optimized for a gate-based implementation in an ASIC. While such an approach does allow leveraging existing knowledge of ASIC-based AxO design, it limits the scope for considering the challenges and opportunities associated with FPGA’s LUT-based computation structures. Similarly, the few works considering the LUT-based computing for AxO design use generic optimization approaches that do not allow integrating problem-specific prior knowledge—empirical and/or statistical. To this end, we propose a novel tree search-based approach to AxO synthesis for FPGAs. Specifically, we present a design methodology using Monte Carlo Tree Search (MCTS)-based search tree traversal that allows the designer to integrate statistical data, such as correlation, into the AxOs optimization. With the proposed methods, we report improvements over standard MCTS algorithm-based results as well as improved hypervolume for both operator-level and application-specific DSE, compared to state-of-the-art design methodologies.

Unveiling the Potential of Sheffer Polynomials: Exploring Approximation Features with Jakimovski–Leviatan Operators

In this article, we explore the construction of Jakimovski–Leviatan operators for Durrmeyer-type approximation using Sheffer polynomials. Constructing positive linear operators for Sheffer polynomials enables us to analyze their approximation properties, including weighted approximations and convergence rates. The application of approximation theory has earned significant attention from scholars globally, particularly in the fields of engineering and mathematics. The investigation of these approximation properties and their characteristics holds considerable importance in these disciplines.

Fundamental properties of fuzzy rough sets based on triangular norms and fuzzy implications: the properties characterized by fuzzy neighborhood and fuzzy topology

Fuzzy rough set models are useful tools for dealing with fuzzy and real-valued data. They have been used in many real-world applications. In this paper, we investigate the fuzzy rough set model based on triangular norms and fuzzy implications. First, we extend some results in the published literature by removing the condition that is the continuity of triangular norms, and obtain more general conclusion about fuzzy upper approximation operators. Then, for the fuzzy neighborhood and the fuzzy lower approximation operator based on fuzzy implications, we investigate their characterization with each other. Finally, we establish the relationships between fuzzy rough sets and fuzzy topology. In this work, researches on the properties of fuzzy rough sets based on triangular norms which need not be continuous provide generalization results for fuzzy rough set theory from viewpoint of mathematics.

Co-Variational Inequality Problem Involving Two Generalized Yosida Approximation Operators

We focus our study on a co-variational inequality problem involving two generalized Yosida approximation operators in real uniformly smooth Banach space. We show some characteristics of a generalized Yosida approximation operator, which are used in our main proof. We apply the concept of nonexpansive sunny retraction to obtain a solution to our problem. Convergence analysis is also discussed.

β-fuzzy equivalence relations, β-fuzzy partitions and the rough set model

This paper selects a residuated lattice L as the truth value table and selects a fixed element β∈L as the degree criterion, we introduce concepts of β-fuzzy equivalence relations and β-fuzzy partitions. The β-fuzzy partition is a natural generalization of fuzzy β-coverings. It is shown that they are one-to-one corresponding if and only if β=1. In order to check the reasonableness of β-fuzzy partitions, we define a pair of fuzzy upper/lower rough approximation operators and show that each of them has two equivalent descriptions, which is not possessed by fuzzy β-coverings.

Hemimetric-based λ-valued fuzzy rough sets

A λ-subset, or a [0,λ]-valued fuzzy subset, is a mapping from a nonempty set to the interval [0,λ]. In this paper, we use the notion of hemimetrics, a kind of distance functions, as the basic structure to define and study fuzzy rough set model of λ-subsets by using the usual addition and subtraction of real numbers. We define a pair of fuzzy upper/lower approximation operators and investigate their properties and interrelations. These two operators have nice logical descriptions by using the related Lukasiewicz logical systems. We show that upper definable sets, lower definable sets and definable sets are equivalent, and they form an Alexandrov fuzzy topology. A processing of a λ-subset via fuzzy upper/lower approximation operators can actually considered as a processing of the related image, and thus has potential applications in image processing.

Toward Optimal Softcore Carry-aware Approximate Multipliers on Xilinx FPGAs

Domain-specific accelerators for signal processing, image processing, and machine learning are increasingly being implemented on SRAM-based field-programmable gate arrays (FPGAs). Owing to the inherent error tolerance of such applications, approximate arithmetic operations, in particular, the design of approximate multipliers, have become an important research problem. Truncation of lower bits is a widely used approximation approach; however, analyzing and limiting the effects of carry-propagation due to this approximation has not been explored in detail yet. In this article, an optimized carry-aware approximate radix-4 Booth multiplier design is presented that leverages the built-in slice look-up tables (LUTs) and carry-chain resources in a novel configuration. The proposed multiplier simplifies the computation of the upper and lower bits and provides significant benefits in terms of FPGA resource usage (LUTs saving 38.5%–42.9%), Power Delay Product (PDP saving 49.4%–53%), performance metric (LUTs × critical path delay (CPD) × PDP saving 68.9%–73.1%) and errors (70% improvement in mean relative error distance) compared to the latest state-of-the-art designs. Therefore, the proposed designs are an attractive choice to implement multiplication on FPGA-based accelerators.

Fuzzy Product Rough Sets as a Special Case of Fuzzy T-Rough Sets

Fuzzy T-rough set consists of a set ???? and a T-similarity relation on ????, where T is a lower semicontinuous triangular norm. In this paper, axiomatic definition for fuzzy ????-rough sets and its upper approximation operator were proposed. The method employed was by relaxing the arbitrary T and adopting its special case ???????? (product triangular norm). The results obtained suggests an easier way of being specific to the product case of fuzzy rough sets and computations regarding its upper approximation operators. Some important propositions and examples were also provided.

Rough set model based on axiomatic fuzzy set

Both fuzzy set and rough set are important mathematical tools to describe incomplete and uncertain information, and they are highly complementary to each other. What is more, most fuzzy rough sets are obtained by combining Zadeh fuzzy sets and Pawlak rough sets. There are few reports about the combination of axiomatic fuzzy sets and Pawlak rough sets. For this reason, we propose the axiomatic fuzzy rough sets (namely rough set model with respect to the axiomatic fuzzy set) establishing on fuzzy membership space. In this paper, we first present a similarity description method based on vague partitions. Then the concept of similarity operator is proposed to describe uncertainty in the fuzzy approximation space. Finally, some characterizations concerning upper and lower approximation operators are shown, including basic properties. Furthermore, we give a algorithm to verify the effectiveness and efficiency of the model.

A bi-variable precision rough set model and its application to attribute reduction

Significant indicators for measuring algorithms for attribute reduction include their fault tolerance. The majority of existing algorithms rely on the incorporation of fault tolerance at the upper and lower approximation operators. However, information granules partitioned by these algorithms are not inherently fault-tolerant. To enhance the fault tolerance of attribute reduction algorithms, this study introduces a new model based on variable precision rough sets, comprising of three parameters: the inclusion of fault tolerance in the comparison of attribute values between two objects, the partition of information granules, and the calculation of upper and lower approximation operators. As a result, the model tolerates errors, enhances the fault-tolerance ability of information granules, and improves its inclusivity. Using this new model, we design a novel algorithm that deletes both redundant and noise attributes to improve the classification accuracy. Through analysis of classification accuracy and robustness, we confirm that the novel algorithm's fault tolerance and robustness are maintained even after adding random Gaussian noise. The parameters included as part of the model are also found to be mostly unchanged after the addition of random Gaussian noise.

Synthesis of sparse planar arrays with multiple patterns: An automatic coordinate pairing approach

An innovative algorithm to synthesize a non-uniform sparse planar array with as few elements as possible is proposed in this paper. The algorithm is aimed at multiple different beampatterns, and can directly estimate the exact positions of the planar array elements, avoiding the problem of coordinate pairing between X-axis and Y-axis. To achieve this, the proposed method utilizes the rotational invariance caused by the translation invariant structure of the steering vector. It derives the array element position from the data matrix based on unitary ESPRIT and employs a low-rank approximation operation to construct a matrix containing the position information of the reconstructed sparse array elements. Automatic pairing is then realized by assigning the real and imaginary parts of the matrix eigenvalues to the abscissa and ordinate of elements one-to-one. The paper presents two different simulation scenarios to demonstrate the performance of the proposed algorithm, with the results showing a reduction in the number of array elements by 20%-30% while ensuring low reconstruction error, low time, and space complexity. Overall, the proposed algorithm provides an effective solution for synthesizing non-uniform sparse planar arrays with fewer elements, which can significantly reduce costs and complexity in various applications.

A robust algorithmic cum integrated approach of interval-valued fuzzy hypersoft set and OOPCS for real estate pursuit.

Due to the vast variety of aspects that must be made-many of which are in opposition to one another-choosing a home can be difficult for those without much experience. Individuals need to spend more time making decisions because they are difficult, which results in making poor choices. To overcome residence selection issues, a computational approach is necessary. Unaccustomed people can use decision support systems to help them make decisions of expert quality. The current article explains the empirical procedure in that field in order to construct decision-support system for selecting a residence. The main goal of this study is to build a weighted product mechanism-based decision-support system for residential preference. The said house short-listing estimation is based on several key requirements derived from the interaction between the researchers and experts. The results of the information processing show that the normalized product strategy can rank the available alternatives to help individuals choose the best option. The interval valued fuzzy hypersoft set (IVFHS-set) is a broader variant of the fuzzy soft set that resolves the constraints of the fuzzy soft set from the perspective of the utilization of the multi-argument approximation operator. This operator maps sub-parametric tuples into a power set of universe. It emphasizes the segmentation of every attribute into a disjoint attribute valued set. These characteristics make it a whole new mathematical tool for handling problems involving uncertainties. This makes the decision-making process more effective and efficient. Furthermore, the traditional TOPSIS technique as a multi-criteria decision-making strategy is discussed in a concise manner. A new decision-making strategy, "OOPCS" is constructed with modifications in TOPSIS for fuzzy hypersoft set in interval settings. The proposed strategy is applied to a real-world multi-criteria decision-making scenario for ranking the alternatives to check and demonstrate their efficiency and effectiveness.

A consistent multi-resolution particle method for fluid-driven granular dynamics

Granular dynamics driven by fluid flow is ubiquitous in many industrial and natural processes, such as fluvial and coastal sediment transport. Yet, their complex multiphysics and multi-scale nature challenge numerical models’ accuracy and efficiency. Here, we develop a new multi-resolution mesh-free particle method, based on an enhanced weakly-compressible Moving Particle Semi-implicit (MPS) method, to study the dynamics of rapid fluid-driven granular erosion. We propose and validate a novel multi-resolution multiphase MPS formulation for the consistent and conservative form of the governing equations, including particle stabilization techniques. First, we discuss the numerical accuracy and convergence of the proposed approximation operators through two numerical benchmark cases: the multi-viscosity Poiseuille flow and the multi-density hydrostatic pressure. Then, coupling the developed model with a generalized rheology equation, we investigate the water dam-break waves over movable beds. The particle convergence study confirms that the proposed multi-resolution formulation predicts the analytical solutions with acceptable accuracy and order of convergence. Validating the multiphase granular flow reveals that the mechanical behavior of this fluid-driven problem is highly sensitive to the water-sediment density ratio; the bed with lighter grains experiences extreme erosion and interface deformations. For the bed with a heavier material but different geometrical setups, the surge speed, and the transport layer thickness remain almost identical (away from the gate). Furthermore, while the multi-resolution model accurately estimates the global sediment dynamics, the single-resolution model underestimates the flow evolution. Overall, the qualitative and quantitative analysis of results emphasizes the importance of multi-scale multi-density interactions in fluid-driven modeling.

Representations of L-fuzzy rough approximation operators

In recent years, there has been tremendous interest in developing lattice-valued rough set theory. In this framework, we primarily take into account the following two problems. Firstly, we include the well-known intuitionistic, interval-valued, neutrosophic and Pythagorean fuzzy rough sets into the framework of lattice-valued rough sets. Specifically, the four kinds of rough sets can be considered as special lattice-valued rough sets. Secondly, based on a completely distributive lattice L, we provide representations of the upper and lower L-fuzzy rough approximation operators by using four kinds of cut sets of an L-fuzzy set and an L-fuzzy relation, which generalizes the existing results in the case that L=[0,1]. In particular, we show that representations of intuitionistic and interval-valued fuzzy rough approximation operators provided by Zhou and Sun are special examples of our proposed representations.

Composition operators on Dirichlet spaces over the half-plane

ABSTRACT As a continuation of the study of polynomial approximation and composition operators on Dirichlet spaces of unit disk, which has settled a problem posed by Cima in 1976, the present paper aims to consider the case of the unbounded domains, such as the half-plane. Specifically, we may obtain the rational approximations in the Dirichlet spaces and characterize the composition operators which have a dense range on the Dirichlet spaces over the half-plane. Moreover, this paper also considers the relationship between the Dirichlet spaces and Hardy spaces on half-plane.

Approximations of semiring-valued fuzzy sets with applications in new fuzzy structures

Many of the new types of fuzzy sets, such as intuitionistic, neutrosophic or fuzzy soft sets, can be transformed into one common type of fuzzy sets, called (R,R⁎)-fuzzy sets, with values in a set R that is a common underlying set of complete commutative idempotent semirings R and R⁎. For (R,R⁎)-fuzzy sets, the theory of lower and upper approximations by (R,R⁎)-relations is defined and properties of these approximations are investigated. The transformation of new types of fuzzy sets into (R,R⁎)-fuzzy sets is used to show the possibility of introducing approximation operators in new types of fuzzy sets without having to define these operators separately for each new type of fuzzy set.

Novel results on G -lower L -fuzzy rough approximation operator in ( O , G ) -fuzzy rough set model1

Lately, Jiang and Hu (H.B. Jiang, B.Q. Hu, On ( O , G ) -fuzzy rough sets based on overlap and grouping functions over complete lattices, Int. J. Approx. Reason. 144 (2022) 18-50.) put forward ( O , G ) -fuzzy rough sets via overlap and grouping functions over complete lattices. Meanwhile, they showed the characterizations of O -upper and G -lower L -fuzzy rough approximation operators in ( O , G ) -fuzzy rough set model based on some of specific L -fuzzy relations and studied the topological properties of the proposed model. Nevertheless, we discover that the partial results given by Jiang and Hu could be further optimized. So, as a replenish of the above article, in this paper, based on G -lower L -fuzzy rough approximation operator in ( O , G ) -fuzzy rough set model, we further explore several new conclusions on the relationship between G -lower L -fuzzy rough approximation operator and different L -fuzzy relations. In particular, the equivalent descriptions of relationship between G -lower L -fuzzy rough approximation operator and O -transitive ( O -Euclidean) L -fuzzy relations are investigated, which are not involved in above literature and can make the theoretical results of this newly fuzzy rough set model more perfect.

Sliding-Mode-Based Attitude Tracking Control of Spacecraft Under Reaction Wheel Uncertainties

The attitude tracking operations of an on-orbit spacecraft with degraded performance exhibited by potential actuator uncertainties (including failures and misalignments) can be extraordinarily challenging. Thus, the control law development for the attitude tracking task of spacecraft subject to actuator (namely reaction wheel) uncertainties is addressed in this paper. More specially, the attitude dynamics model of the spacecraft is firstly established under actuator failures and misalignment (without a small angle approximation operation). Then, a new non-singular sliding manifold with fixed time convergence and anti-unwinding properties is proposed, and an adaptive sliding mode control (SMC) strategy is introduced to handle actuator uncertainties, model uncertainties and external disturbances simultaneously. Among this, an explicit misalignment angles range that could be treated herein is offered. Lyapunov-based stability analyses are employed to verify that the reaching phase of the sliding manifold is completed in finite time, and the attitude tracking errors are ensured to converge to a small region of the closest equilibrium point in fixed time once the sliding manifold enters the reaching phase. Finally, the beneficial features of the designed controller are manifested via detailed numerical simulation tests.

On a generalization of the extended best polynomial approximation operator in Orlicz–Lorentz spaces

AbstractIn this paper, we consider the best polynomial approximation operator defined on an Orlicz–Lorentz space , and its extension to , where w is a non‐negative continuous weight function and is the derivative of ϕ, which is not required to be an Orlicz function. Our work generalizes a recent result in this field on an Orlicz–Lorentz space generated by an Orlicz function. In addition, we establish some properties and estimates for any extended best polynomial approximation.

The axiomatic characterization on fuzzy variable precision rough sets based on residuated lattice

Axiomatization is a lively research direction in fuzzy rough set theory. Fuzzy variable precision rough set (FVPRS) incorporates fault-tolerant factors to fuzzy rough set, so its axiomatic description becomes more complicated and difficult to realize. In this paper, we present an axiomatic approach to FVPRSs based on residuated lattice (L-fuzzy variable precision rough set (LFVPRS)). First, a pair of mappings with three axioms is utilized to characterize the upper (resp., lower) approximation operator of LFVPRS. This is distinct from the characterization on upper (resp., lower) approximation operator of fuzzy rough set, which consists of one mapping with two axioms. Second, utilizing the notion of correlation degree (resp., subset degree) of fuzzy sets, three characteristic axioms are grouped into a single axiom. At last, various special LFVPRS generated by reflexive, symmetric and transitive L-fuzzy relation and their composition are also characterized by axiomatic set and single axiom, respectively.