Approximate Answers Research Articles

Numerical Solution of Differential Equations Using the Wavelet-Based Galerkin Method with Fibonacci Wavelets

Differential equations form the foundation of scientific theories that address numerous real-world physical challenges. Numerical methods enable the resolution of complex problems through relatively simple operations. A significant advantage of numerical methods, compared to analytical methods, is their ease of implementation on modern computers, allowing for quicker solutions. Galerkin's method belongs to a broader category of numerical techniques. Additionally, wavelet analysis represents a promising domain within applied and computational research. This paper establishes a wavelet-based Galerkin method for numerically solving differential equations, utilizing Fibonacci wavelets as trial functions. The proposed method yields results comparable to existing techniques and provides solutions that closely approximate exact answers for certain problems, thereby demonstrating its effectiveness and accuracy.

ActiveReach: an active learning framework for approximate reachability query answering in large-scale graphs.

With graph reachability query, one can answer whether there exists a path between two query vertices in a given graph. The existing reachability query processing solutions use traditional reachability index structures and can only compute exact answers, which may take a long time to resolve in large graphs. In contrast, with an approximate reachability query, one can offer a compromise by enabling users to strike a trade-off between query time and the accuracy of the query result. In this study, we propose a framework, dubbed ActiveReach, for learning index structures to answer approximate reachability query. ActiveReach is a two-phase framework that focuses on embedding nodes in a reachability space. In the first phase, we leverage node attributes and positional information to create reachability-aware embeddings for each node. These embeddings are then used as nodes' attributes in the second phase. In the second phase, we incorporate the new attributes and include reachability information as labels in the training data to generate embeddings in a reachability space. In addition, computing reachability for all training data may not be practical. Therefore, selecting a subset of data to compute reachability effectively and enhance reachability prediction performance is challenging. ActiveReach addresses this challenge by employing an active learning approach in the second phase to selectively compute reachability for a subset of node pairs, thus learning the approximate reachability for the entire graph. Our extensive experimental study with various real attributed large-scale graphs demonstrates the effectiveness of each component of our framework.

Numerical Computation and Statistical Interpretations of Heat Transfer of Tween-20/ethyl Acetate Nanofluid Flow with Melting Rheological Quality and Activation Energy

Tween-20 plays a significant role in the biological, food, and pharmaceutical industries. Additionally, it plays a vital role in improving the quality of reverse mechanisms of multi-drug resistance. This paper uses artificial neural computing to examine Tween-20 nanoparticles’ behaviors in a base fluid called ethyl acetate to enhance the applications in nanomedicine, drug delivery and biotechnology. The study focuses on the nonlinear model of a viscous fluid at the stagnation point, where mixed convection processes and activation energy are present. The study incorporates slip velocity and melting boundary conditions to examine heat and mass transfer, taking into account thermal and solutal stratification. Various fields of research and technology, specially in industrial engineering that include hydrodynamics, panto-graph systems, and biomedical mathematics, extensively utilize artificial computing. The datasets are based on velocity, temperature, and concentration outlines. The fourth-order Runge-Kutta method is used to generate the datasets. To validate the LMM-ABNNs, we have compared them with a numerical solution, showing a high level of agreement. We utilize the error histogram and mean square error results to validate the performance, scrutinize the training and testing methods, and explore the validity of the approximate answer. Furthermore, the quality characteristics such as skin friction, rate of heat, and mass movement (Nusselt and Sherwood numbers) are statistically analyzed to forecast the model’s durability. This article is the best example of investigating different fluid parameters with the latest artificial neural computing.

Analysis of Cauchy reaction-diffusion equations involving Atangana-Baleanu fractional operator

This study investigates the Cauchy reaction-diffusion equation (CRDE) with the Atangana-Baleanu differential operator. The existence and uniqueness of solutions to fractional starting value issues are begun using the fixed-point theorem and contraction principle, respectively. The proposed study uses the natural variation iteration technique (NVIM) to get an approximate solution for nonlinear fractional reaction-diffusion equations. This study's approximate answers are compared to other solutions found using known methodologies, and the results are discussed. The devised technique has benefits in terms of accuracy and computational cost efficiency, which may be used to solve nonlinear fractional reaction-diffusion equations.

Backbone Index and GNN Models for Skyline Path Query Evaluation over Multi-cost Road Networks

Skyline path queries (SPQs) extend skyline queries to multi-dimensional networks, such as multi-cost road networks (MCRNs). Such queries return a set of non-dominated paths between two given network nodes. Despite the existence of extensive works on evaluating different SPQ variants, SPQ evaluation is still very inefficient due to the nonexistence of efficient index structures to support such queries. Existing index building approaches for supporting shortest-path query execution, when directly extended to support SPQs, use an unreasonable amount of space and time to build, making them impractical for processing large graphs. In this article, we propose a novel index structure, backbone index , and a corresponding index construction method that condenses an initial MCRN to multiple smaller summarized graphs with different granularity. We present efficient approaches to find approximate solutions to SPQs by utilizing the backbone index structure. Furthermore, considering making good use of historical query and query results, we propose two models, S kyline P ath G raph N eural N etwork (SP-GNN) and T ransfer SP-GNN (TSP-GNN), to support effective SPQ processing. Our extensive experiments on real-world large road networks show that the backbone index can support finding meaningful approximate SPQ solutions efficiently. The backbone index can be constructed in a reasonable time, which dramatically outperforms the construction of other types of indexes for road networks. As far as we know, this is the first compact index structure that can support efficient approximate SPQ evaluation on large MCRNs. The results on the SP-GNN and TSP-GNN models also show that both models can help get approximate SPQ answers efficiently.

Heuristic determination of parent β matrix orientation from Burgers oriented α precipitates in Ti alloys

A heuristic stereographic tool has been developed supporting identification of un-measurable parent β grain orientations in Ti alloys from electron back scattered diffraction (EBSD) without computation. The method involves the use of inverse pole figure (IPF) Euler angle color patterns of α laths in Burgers oriented α/β crystals. The approach employs the use of composite stereographic projections of two fundamental α/β variants of the Burgers orientation relationship (BOR), overlaid with a commonly used IPF Euler angle color scheme for α lath orientations. Use of the heuristic tool associates many α lath orientations via EBSD-IPF derived Euler angle color patterns with the required, but usually un-measurable parent β grain orientation. Results from use of the heuristic tool are compared with observations from EBSD-IPF scans that directly measured α lath and parent β grain orientations. A reference look-up table associating characteristic α lath color patterns with orientation of their underlying parent β grain is provided. It is shown that the use of the heuristic tool to determine orientations of interest enables identification of un-measurable parent β grain orientations rather than obtaining an approximate answer using complicated computational methods. The new tool yields fairly accurate results avoiding post-processing, and often does not require the presence of more than two or three variants.

Approximate and exact solution of Korteweg de Vries problem using Aboodh Adomian polynomial method

This study introduce Aboodh Adomian polynomial Method (AAPM) to solve nonlinear third order KdV problems providing it approximate and exact solution. To get the approximate analytical answers to the issues, the Aboodh transform approach was used. Given that the Aboodh transform cannot handle the nonlinear elements in the equation, the Adomian polynomial was thought to be a crucial tool for linearizing the associated nonlinearities. All of the issues examined demonstrated the strength and effectiveness of the Adomian polynomial and Aboodh transforms in solving various nonlinear equations when compared to other well-known methods. To show how this strategy may be applied and is beneficial, three cases were examined.

Numerical Solution of Volterra's Integral Equations by Collocation and Taylor Method

In this article, two numerical methods (collocation method and Taylor method) are introduced to solve Volterra integral equation system. In the collocation method, using Bessel polynomials and collocation points, we convert the device into a matrix form and solve the device using the matrix form and obtain an approximate solution for the device. This answer is such that the larger the N becomes, the approximate answer is closer to the exact answer of the device. In Taylor method, With the use of the Taylor series, the system of integral equations is transformed into a matrix equation, and by integrating the output of this new system, a system of algebraic equations is created. By working through this device, we can obtain a rough solution for the Integral Equations device. Using these two methods when the answers are polynomial, we can get real answers.

Ballistic to diffusive crossover in a weakly interacting Fermi gas

In the absence of disorder and interactions, fermions move coherently and their associated charge and energy exhibit ballistic spreading, even at finite energy density. In the presence of weak interactions and a finite energy density, fermion-fermion scattering leads to a crossover between early-time ballistic and late-time diffusive transport. The relevant crossover timescales and the transport coefficients are both functions of interaction strength, but the question of determining the precise functional dependence is likely impossible to answer exactly. In this work we develop a numerical method (fDAOE) which is powerful enough to provide an approximate answer to this question, and which is consistent with perturbative arguments in the limit of very weak interactions. Our algorithm, which adapts the existing dissipation-assisted operator evolution (DAOE) to fermions, is applicable to systems of interacting fermions at high temperatures. The algorithm approximates the exact dynamics by systematically discarding information from high n-point functions, and is tailored to capture noninteracting dynamics exactly. Applying our method to a microscopic model of interacting fermions, we numerically determine crossover timescales and diffusion constants for a wide range of interaction strengths. In the limit of weak interaction strength (Δ), we demonstrate that the crossover from ballistic to diffusive transport happens at a time tD∼1/Δ2 and that the diffusion constant similarly scales as D∼1/Δ2. We confirm that these scalings are consistent with a perturbative Fermi's golden rule calculation, and we provide a heuristic operator-spreading picture for the crossover between ballistic and diffusive transport. Published by the American Physical Society 2024

DIDS: Double Indices and Double Summarizations for Fast Similarity Search

Data series has been one of the significant data forms in various applications. It becomes imperative to devise a data series index that supports both approximate and exact similarity searches for large data series collections in high-dimensional metric spaces. The state-of-the-art works employ summarizations and indices to reduce the accesses to the data series. However, we discover two significant flaws that severely limit performance enhancement. Firstly, the state-of-the-art works often employ segment-based summarizations, whose lower bound distances decrease significantly when representing a data series collection, resulting in numerous invalid accesses. Secondly, the disk-based indices for the exact search mainly rely on tree-based indices, which results in low-quality approximate answers, consequently impacting the exact search. To address these problems, we propose a novel solution, Double Indices and Double Summarizations (DIDS). Besides segment-based summarizations, DIDS introduces reference-point-based summarizations to improve the pruning rate by the sorted-based representation strategy. Moreover, DIDS employs reference points and a cost model to cluster similar data series, and uses a graph-based approach to interconnect various regions, enhancing approximate search capabilities. We conduct experiments on extensive datasets, validating the superior search performance of DIDS.

GAN-Based Tabular Data Generator for Constructing Synopsis in Approximate Query Processing: Challenges and Solutions

In data-driven systems, data exploration is imperative for making real-time decisions. However, big data are stored in massive databases that are difficult to retrieve. Approximate Query Processing (AQP) is a technique for providing approximate answers to aggregate queries based on a summary of the data (synopsis) that closely replicates the behavior of the actual data; this can be useful when an approximate answer to queries is acceptable in a fraction of the real execution time. This study explores the novel utilization of a Generative Adversarial Network (GAN) for the generation of tabular data that can be employed in AQP for synopsis construction. We thoroughly investigate the unique challenges posed by the synopsis construction process, including maintaining data distribution characteristics, handling bounded continuous and categorical data, and preserving semantic relationships, and we then introduce the advancement of tabular GAN architectures that overcome these challenges. Furthermore, we propose and validate a suite of statistical metrics tailored for assessing the reliability of GAN-generated synopses. Our findings demonstrate that advanced GAN variations exhibit a promising capacity to generate high-fidelity synopses, potentially transforming the efficiency and effectiveness of AQP in data-driven systems.

Finding Minimum Connected Subgraphs With Ontology Exploration on Large RDF Data

In this paper, we study the following problem: given a knowledge graph (KG) and a set of input vertices (representing concepts or entities) and edge labels, we aim to find the smallest connected subgraphs containing all of the inputs. This problem plays a key role in KG-based search engines and natural language question answering systems, and it is a natural extension of the Steiner tree problem, which is known to be NP-hard. We present RECON, a system for finding approximate answers. RECON aims at achieving high accuracy with instantaneous response (i.e., sub-second/millisecond delay) over KGs with hundreds of millions edges without resorting to expensive computational resources. Furthermore, when no answer exists due to disconnection between concepts and entities, RECON refines the input to a semantically similar one based on the ontology, and attempts to find answers with respect to the refined input. We conduct a comprehensive experimental evaluation of RECON. In particular we compare it with five existing approaches for finding approximate Steiner trees. Our experiments on four large real and synthetic KGs show that RECON significantly outperforms its competitors and incurs a much smaller memory footprint.

A Numerical Approximation of the Stochastic Ito-Volterra Integral Equation

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms and the solution are stochastic processes. Numerous studies have employed orthogonal polynomials, however most of them focus on deterministic rather than stochastic systems. This is the reason why in this study, we looked into a numerical solution for the stochastic Ito-Volterra integral equation using the explicit finite difference scheme and Bernstein polynomials as trial functions. The equidistant collocation procedure was used to calculate the unknown constant parameters in between and reach the desired approximation. The method was evaluated and contrasted with the Block Pulse method for approximate answers based on the aforementioned method, which were obtained and compared with others in the literature.

ShadowAQP: Efficient Approximate Group-by and Join Query via Attribute-Oriented Sample Size Allocation and Data Generation

Approximate query processing (AQP) is one of the key techniques to cope with big data querying problem on account that it obtains approximate answers efficiently. To address non-trivial sample selection and heavy sampling cost issues in AQP, we propose ShadowAQP, an efficient and accurate approach based on attribute-oriented sample size allocation and data generation. We select samples according to group-by and join attributes, and determine the sample size for each group of unique value combinations to improve query accuracy. We design a conditional variational autoencoder model with automatic table data encoding and model update strategies. To further improve accuracy and efficiency, we propose a set of extensions, including parallel multi-round sampling aggregation, data outlier-aware sampling, and dimension reduction optimization. Evaluation results on diversified datasets show that, compared with SOTA approaches, ShadowAQP achieves 5.8× query speed performance improvement on average (up to 12.8×), while reducing query error by 74% on average (up to 95%) at the same time.

Querying Data Exchange Settings Beyond Positive Queries

Abstract Data exchange, the problem of transferring data from a source schema to a target schema, has been studied for several years. The semantics of answering positive queries over the target schema has been defined in early work, but little attention has been paid to more general queries. A few proposals of semantics for more general queries exist but they either do not properly extend the standard semantics under positive queries, giving rise to counterintuitive answers, or they make query answering undecidable even for the most important data exchange settings, for example, with weakly-acyclic dependencies. The goal of this paper is to provide a new semantics for data exchange that is able to deal with general queries. At the same time, we want our semantics to coincide with the classical one when focusing on positive queries, and to not trade-off too much in terms of complexity of query answering. We show that query answering is undecidable in general under the new semantics, but it is $\text{co}\text{NP}\text{-complete}$ when the dependencies are weakly-acyclic. Moreover, in the latter case, we show that exact answers under our semantics can be computed by means of logic programs with choice, thus exploiting existing efficient systems. For more efficient computations, we also show that our semantics allows for the construction of a representative target instance, similar in spirit to a universal solution, that can be exploited for computing approximate answers in polynomial time.

Accelerating Aggregation Queries on Unstructured Streams of Data

Analysts and scientists are interested in querying streams of video, audio, and text to extract quantitative insights. For example, an urban planner may wish to measure congestion by querying the live feed from a traffic camera. Prior work has used deep neural networks (DNNs) to answer such queries in the batch setting. However, much of this work is not suited for the streaming setting because it requires access to the entire dataset before a query can be submitted or is specific to video. Thus, to the best of our knowledge, no prior work addresses the problem of efficiently answering queries over multiple modalities of streams. In this work we propose InQuest, a system for accelerating aggregation queries on unstructured streams of data with statistical guarantees on query accuracy. InQuest leverages inexpensive approximation models ("proxies") and sampling techniques to limit the execution of an expensive high-precision model (an "oracle") to a subset of the stream. It then uses the oracle predictions to compute an approximate query answer in real-time. We theoretically analyzed InQuest and show that the expected error of its query estimates converges on stationary streams at a rate inversely proportional to the oracle budget. We evaluated our algorithm on six real-world video and text datasets and show that InQuest achieves the same root mean squared error (RMSE) as two streaming baselines with up to 5.0x fewer oracle invocations. We further show that InQuest can achieve up to 1.9x lower RMSE at a fixed number of oracle invocations than a state-of-the-art batch setting algorithm.

NeuroSketch: Fast and Approximate Evaluation of Range Aggregate Queries with Neural Networks.

Range aggregate queries (RAQs) are an integral part of many real-world applications, where, often, fast and approximate answers for the queries are desired. Recent work has studied answering RAQs using machine learning (ML) models, where a model of the data is learned to answer the queries. However, there is no theoretical understanding of why and when the ML based approaches perform well. Furthermore, since the ML approaches model the data, they fail to capitalize on any query specific information to improve performance in practice. In this paper, we focus on modeling "queries" rather than data and train neural networks to learn the query answers. This change of focus allows us to theoretically study our ML approach to provide a distribution and query dependent error bound for neural networks when answering RAQs. We confirm our theoretical results by developing NeuroSketch, a neural network framework to answer RAQs in practice. Extensive experimental study on real-world, TPC-benchmark and synthetic datasets show that NeuroSketch answers RAQs multiple orders of magnitude faster than state-of-the-art and with better accuracy.

A comparative study of Sumudu HPM and Elzaki HPM for coupled Burgers’ equation

The two hybrid algorithms Sumudu HPM and Elzaki HPM are used in the current study to tackle coupled Burgers' equations and produce accurate results. To demonstrate the validity of the given approaches, three instances are used. Applying Sumudu HPM and Elzaki HPM yields the same approximate and exact answers in all of the examples taken into consideration, which is proved with the help of the accompanying figures. It attests to the entire acceptance and accuracy of the solutions produced by these methods. The proposed regimes also have error and convergence analyses available. The current analytical regimes offer a more effective method of handling partial differential equations than the intricate numerical systems. It is also asserted that exact and approximation solutions are compatible. Also announced is the planned regime's numerical convergence.

On Efficient Approximate Queries over Machine Learning Models

The question of answering queries over ML predictions has been gaining attention in the database community. This question is challenging because finding high quality answers by invoking an oracle such as a human expert or an expensive deep neural network model on every single item in the DB and then applying the query, can be prohibitive. We develop a novel unified framework for approximate query answering by leveraging a proxy to minimize the oracle usage of finding high quality answers for both Precision-Target (PT) and Recall-Target (RT) queries. Our framework uses a judicious combination of invoking the expensive oracle on data samples and applying the cheap proxy on the DB objects. It relies on two assumptions. Under the P roxy Q uality assumption, we develop two algorithms: PQA that efficiently finds high quality answers with high probability and no oracle calls, and PQE, a heuristic extension that achieves empirically good performance with a small number of oracle calls. Alternatively, under the C ore S et C losure assumption, we develop two algorithms: CSC that efficiently returns high quality answers with high probability and minimal oracle usage, and CSE, which extends it to more general settings. Our extensive experiments on five real-world datasets on both query types, PT and RT, demonstrate that our algorithms outperform the state-of-the-art and achieve high result quality with provable statistical guarantees.

Numerical study of Casson-Nanofluid flow past an exponentially stretching sheet filled by porous medium in presence of velocity and thermal slip effects

An inquiry of the non-Newtonian Casson flow of nanofluid through a non-linear exponentially expanding sheet is one of the things that will be covered in the course of this work. When a magnetic field, chemical reaction, thermophoresis, and Brownian motion phenomena are present, the influence of coupled velocity and thermal slip conditions on porous embedded fluid flow is investigated. These phenomena include Brownian motion. In particular, the flow of fluid through porous embedded structures is studied in the presence of a magnetic field. Because the mechanics of fluid motion are extremely sequential and non-linear, fluid dynamics may be described as having these characteristics. In order to acquire an approximate answer, what has been done here is to combine the Runge-Kutta method with a numerical approach that makes use of the shooting technique and a shooting scheme. This has been done in order to get an approximation. The degree to which the thermo-fluid parameters are susceptible to change is shown by both the tabular and graphical representations of the data. When the new findings are contrasted with the findings of the earlier study, there may be found to be a high degree of congruence between the two sets of findings.