Mathematical and Computational Applications

20 pages, 748 KiB

Open AccessArticle

Causal Analysis to Explain the Performance of Algorithms: A Case Study for the Bin Packing Problem

by Jenny Betsabé Vázquez-Aguirre, Guadalupe Carmona-Arroyo, Marcela Quiroz-Castellanos and Nicandro Cruz-Ramírez

Math. Comput. Appl. 2024, 29(5), 73; https://doi.org/10.3390/mca29050073 - 28 Aug 2024

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Abstract

This work presents a knowledge discovery approach through Causal Bayesian Networks for understanding the conditions under which the performance of an optimization algorithm can be affected by the characteristics of the instances of a combinatorial optimization problem (COP). We introduce a case study for the causal analysis of the performance of two state-of-the-art algorithms for the one-dimensional Bin Packing Problem (BPP). We meticulously selected the set of features associated with the parameters that define the instances of the problem. Subsequently, we evaluated the algorithmic performance on instances with distinct features. Our analysis scrutinizes both instance features and algorithm performance, aiming to identify causes influencing the performance of the algorithms. The proposed study successfully identifies specific values affecting algorithmic effectiveness and efficiency, revealing shared causes within some value ranges across both algorithms. The knowledge generated establishes a robust foundation for future research, enabling predictions of algorithmic performance, as well as the selection and design of heuristic strategies for improving the performance in the most difficult instances. The causal analysis employed in this study did not require specific configurations, making it an invaluable tool for analyzing the performance of different algorithms in other COPs. Full article

(This article belongs to the Special Issue New Trends in Computational Intelligence and Applications 2023)

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10 pages, 669 KiB

Open AccessArticle

Analysis of a First-Order Delay Model under a History Function with Discontinuity

by Weam G. Alharbi

Math. Comput. Appl. 2024, 29(5), 72; https://doi.org/10.3390/mca29050072 - 24 Aug 2024

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Abstract

This paper analyzes the first-order delay equation

y^{'} (t) = α y (t) + β y (t - τ)

subject to a history function in addition to an initial condition that assumes discontinuity at [...] Read more.

This paper analyzes the first-order delay equation

y^{'} (t) = α y (t) + β y (t - τ)

subject to a history function in addition to an initial condition that assumes discontinuity at

t = 0

. The method of steps is successfully applied to derive the exact solution in an explicit form. In addition, a unified formula is provided to describe the solution in any finite sub-interval of the problem’s domain. The characteristics and properties of the solution are theoretically investigated and then confirmed through several plots. The behavior of the solution and its derivative are examined and interpreted. The results show that the method of steps is an effective method of solution to treat the current delay model. The present successful analysis can be used to investigate other delay models with complex initial conditions. Furthermore, the present approach can be generalized to include the inhomogeneous version of the current model without using numerical methods. Full article

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17 pages, 4513 KiB

Open AccessArticle

Machine Learning Based Extraction of Boundary Conditions from Doppler Echo Images for Patient Specific Coarctation of the Aorta: Computational Fluid Dynamics Study

by Vincent Milimo Masilokwa Punabantu, Malebogo Ngoepe, Amit Kumar Mishra, Thomas Aldersley, John Lawrenson and Liesl Zühlke

Math. Comput. Appl. 2024, 29(5), 71; https://doi.org/10.3390/mca29050071 - 23 Aug 2024

Viewed by 369

Abstract

Patient-specific computational fluid dynamics (CFD) studies on coarctation of the aorta (CoA) in resource-constrained settings are limited by the available imaging modalities for geometry and velocity data acquisition. Doppler echocardiography is considered a suitable velocity acquisition modality due to its low cost and safety. This study aims to investigate the application of classical machine learning (ML) methods to create an adequate and robust approach to obtain boundary conditions (BCs) from Doppler echocardiography images for haemodynamic modelling using CFD. Our proposed approach combines ML and CFD to model haemodynamic flow within the region of interest. The key feature of the approach is the use of ML models to calibrate the inlet and outlet BCs of the CFD model. In the ML model, patient heart rate served as the crucial input variable due to its temporal variation across the measured vessels. ANSYS Fluent was used for the CFD component of the study, whilst the Scikit-learn Python library was used for the ML component. We validated our approach against a real clinical case of severe CoA before intervention. The maximum coarctation velocity of our simulations was compared to the measured maximum coarctation velocity obtained from the patient whose geometry was used within the study. Of the 5 mL models used to obtain BCs, the top model was within 5% of the maximum measured coarctation velocity. The framework demonstrated that it was capable of taking into account variations in the patient’s heart rate between measurements. Therefore, it allowed for the calculation of BCs that were physiologically realistic when the measurements across each vessel were scaled to the same heart rate while providing a reasonably accurate solution. Full article

(This article belongs to the Special Issue Current Problems and Advances in Computational and Applied Mechanics (AfriComp6))

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Math. Comput. Appl., Volume 29, Issue 5 (October 2024) – 3 articles

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