Numerical study of an adaptive domain decomposition algorithm based on Chebyshev tau method for solving singular perturbed problems

Domain decomposition methods are robust and parallel scalable, preconditioned iterative algorithms for the solution of the large linear systems arising in the discretization of elliptic partial dif...

SummaryBalancing domain decomposition by constraints algorithms with adaptive primal constraints are developed in a concise variational framework for the weighted plane wave least‐squares discretization of Helmholtz equations with high and various wave numbers. The unknowns to be solved in this preconditioned system are defined on elements rather than vertices or edges, which are different from the well‐known discretizations such as the classical finite element method. Through choosing suitable “interface” and appropriate primal constraints with complex coefficients and introducing some local techniques, we developed a two‐level adaptive balancing domain decomposition by constraints algorithm for the plane wave least‐squares discretization, and the condition number of the preconditioned system is proved to be bounded above by a user‐defined tolerance and a constant that is only dependent on the maximum number of interfaces per subdomain. A multilevel algorithm is also attempted to resolve the bottleneck in large‐scale coarse problem. Numerical results are carried out to confirm the theoretical results and illustrate the efficiency of the proposed algorithms.

Scientific machine learning (SciML), an area of research where techniques from machine learning and scientific computing are combined, has become of increasing importance and receives growing attention. Here, our focus is on a very specific area within SciML given by the combination of domain decomposition methods (DDMs) with machine learning techniques for the solution of partial differential equations. The aim of the present work is to make an attempt of providing a review of existing and also new approaches within this field as well as to present some known results in a unified framework; no claim of completeness is made. As a concrete example of machine learning enhanced DDMs, an approach is presented which uses neural networks to reduce the computational effort in adaptive DDMs while retaining their robustness. More precisely, deep neural networks are used to predict the geometric location of constraints which are needed to define a robust coarse space. Additionally, two recently published deep domain decomposition approaches are presented in a unified framework. Both approaches use physics‐constrained neural networks to replace the discretization and solution of the subdomain problems of a given decomposition of the computational domain. Finally, a brief overview is given of several further approaches which combine machine learning with ideas from DDMs to either increase the performance of already existing algorithms or to create completely new methods.

In this article, we are concerned with the numerical treatment of nonlinear elliptic boundary value problems. Our method of choice is a domain decomposition strategy. Partially following the lines from (Cohen, Dahmen and deVore, SIAM J Numer Anal 41 (2003), 1785–1823; Kappei, Appl Anal J Sci 90 (2011), 1323–1353; Lui, SIAM J Sci Comput 21 (2000), 1506–1523; Stevenson and Werner, Math Comp 78 (2009), 619–644), we develop an adaptive additive Schwarz method using wavelet frames. We show that the method converges with an asymptotically optimal rate and support our theoretical results with numerical tests in one and two space dimensions. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Emphasis has been on combustion and flame propagation. The research program was on modeling, analysis and computation of combustion phenomena, with emphasis on transition from laminar to turbulent combustion. Nonlinear dynamics and pattern formation were investigated in the transition. Stability of combustion waves, and transitions to complex waves are described. Combustion waves possess large activation energies, so that chemical reactions are significant only in thin layers, or reaction zones. In limit of infinite activation energy, the zones shrink to moving surfaces, (fronts) which must be found during the analysis, so that (moving free boundary problems). The studies are carried out for limiting case with fronts, while the numerical studies are carried out for finite, though large, activation energy. Accurate resolution of the solution in the reaction zones is essential, otherwise false predictions of dynamics are possible. Since the the reaction zones move, adaptive pseudo-spectral methods were developed. The approach is based on a synergism of analytical and computational methods. The numerical computations build on and extend the analytical information. Furthermore, analytical solutions serve as benchmarks for testing the accuracy of the computation. Finally, ideas from analysis (singular perturbation theory) have induced new approaches to computations. The computational results suggest new analysis to be considered. Among the recent interesting results, was spatio-temporal chaos in combustion. One goal is extension of the adaptive pseudo-spectral methods to adaptive domain decomposition methods. Efforts have begun to develop such methods for problems with multiple reaction zones, corresponding to problems with more complex, and more realistic chemistry. Other topics included stochastics, oscillators, Rysteretic Josephson junctions, DC SQUID, Markov jumps, laser with saturable absorber, chemical physics, Brownian movement, combustion synthesis, etc.

Submarine optical fiber communication cables (subsequently referred to as submarine cables) form the backbone of the Internet’s infrastructure. Damage to these cables can precipitate Internet outages with far-reaching socio-economic impacts. The prevailing practice of manual cable routing is laborious, considering the thousands of kilometers these cables span. It also fails to strike an optimal balance between cost and risk due to its lack of scalability and precision. The Fast Marching Method (FMM), a non-iterative, precise numerical approach capable of solving the Eikonal equation, offers a viable alternative by optimizing the required path between source and destination, considering a summary objective function of costs and risk factors. An interpretation of its solution signifies the optimum value of the objective function between a starting point and all other points. However, the sequential nature of the FMM algorithm suffers from computational limitations and impedes direct parallelization. In this study, we introduce an Adaptive Parallel FMM (APFMM), an innovative approach utilizing adaptive domain decomposition and dynamic multi-resolution analysis. This scalable and widely applicable method can overcome the limitations of existing methods and achieve planning of high-precision, ultra-long-distance (over 14,000 km) submarine cable routes over the Earth’s surface. Simulated experiment results corroborate that APFMM effectively overcomes the computational challenges posed by the sequential FMM when dealing with large datasets. Additionally, it reduces the running time by more than 81% compared to the traditional parallel FMM. This marks a substantial advancement in facilitating efficient, automated, high-precision planning for long-distance submarine cable paths. Note to Practitioners—This paper introduces APFMM, a novel technique based on adaptive domain decomposition and multi-resolution analysis, facilitating high-precision, ultra-long-distance (over 14,000 km) submarine cable path planning. Results from simulated experiments show that APFMM not only overcomes the computational constraints associated with the sequential FMM for large datasets but also cuts the running time by more than 81% relative to the conventional parallel FMM. This breakthrough improvement holds significant implications for practitioners in submarine cable design and construction. With the use of APFMM, designers can plan and optimize cable paths more efficiently, thereby lowering cabling costs and enhancing network resilience. Furthermore, the application of APFMM is not limited to submarine cable path planning and can be employed in other domains involving large-scale data processing and complex path planning, such as electricity cables, gas pipelines, and transportation route planning. While our focus in this paper is primarily on submarine cable path planning, we anticipate practitioners extending the application of APFMM to other use cases, realizing broader utility and benefits.

Adaptive domain decomposition for unstructured meshes applied to the direct simulation monte carlo method

Performing direct simulation Monte Carlo (DSMC) calculations on parallel processors may result in a strongly unbalanced distribution of work. This situation affects the performances leading to very low speedups. We present a method which makes use of adaptive domain decomposition to dynamically distribute the computational work load. The major advantage of this method is the possibility to redistribute the fraction of work which depends on the domain decomposition only. The method has been implemented on a transputer system with up to 8 processors (20 MHz Inmos T800), using the FORTRAN version of Parasoft Express Library for parallel compilers. The problems studied were hypersonic flows of a chemically reacting gas in a two-dimensional domain. The adaptive decomposition method worked well in uniformly distributing the computational work over an arbitrary number of processors and reduced the average execution time by over the 38% in certain cases. >

Urban waste management poses significant challenges due to the growing volume and complexity of waste generated in cities. Effective classification of waste into degradable and non-degradable categories is critical for optimizing waste disposal, promoting recycling, and mitigating environmental impacts. This report presents a comprehensive exploration of an adaptive machine learning algorithm for image-based waste classification, designed to enhance municipal waste management systems. The proposed approach integrates advanced image processing techniques with machine learning models, such as Convolutional Neural Networks (CNNs) and their efficient variants, to classify waste with high accuracy. A citizen reporting interface allows users to capture and upload waste images, which are processed in real-time by a centralized system. The adaptive algorithm incorporates preprocessing steps, feature extraction, and model inference to deliver actionable insights, enabling municipal authorities to deploy appropriate waste collection mechanisms promptly. Through a detailed literature review, the report examines existing methodologies, benchmarks their performance, and identifies key challenges, including data quality, scalability, and integration with urban infrastructure. Case studies from global implementations illustrate the transformative potential of AI-driven waste management systems. Furthermore, the report outlines future directions, such as leveraging advanced deep learning architectures, real-time deployment using edge computing, and expanding classification to include hazardous and recyclable waste categories. This work contributes to the growing field of smart waste management by demonstrating how machine learning can address critical urban challenges. By fostering collaboration between technology and sustainability, the proposed framework lays the foundation for cleaner and more sustainable cities.

An adaptive algorithm based on wavelets is proposed for the fast numerical solution of control problems governed by elliptic boundary value problems with Dirichlet boundary control. A quadratic cost functional representing Sobolev norms of the state and a regularization in terms of the control is to be minimized subject to linear constraints in weak form. In particular, the constraints are formulated as a saddle point problem that allows to handle the varying boundary conditions explicitly. In the framework of (biorthogonal) wavelets, a representer for the functional is derived in terms of l2-norms of wavelet expansion coefficients and the constraints are written in form of an l2 automorphism. Standard techniques from optimization are then used to deduce the resulting first order necessary conditions as a (still infinite) system in l2. Applying the machinery developed in [8,9] which has been extended to control problems in [14], an adaptive method is proposed which can be interpreted as an inexact gradient method for the control. In each iteration step, in turn the primal and the adjoint saddle point system are solved up to a prescribed accuracy by an adaptive iterative Uzawa algorithm for saddle point problems which has been proposed in [10]. Under these premises, it can be shown that the adaptive algorithm containing now three layers of iterations is asymptotically optimal. This means that the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet-best N-term approximation of the solution, and the total computational work is proportional to the number of computational unknowns.

In this paper we present the derivation and the application of an adaptive nonintrusive spectral technique for uncertainty quantification. Spectral techniques can be used to reconstruct stochastic quantities of interest by means of a Fourier-like expansion. Their application to uncertainty propagation problems can be performed in a nonintrusive fashion by evaluating a set of projection integrals that is used to reconstruct the spectral expansion. We present the derivation of a new adaptive quadrature algorithm, based on the definition of a sparse grid, which can be used to evaluate these spectral coefficients. This new adaptive algorithm is applied to a reference uncertainty quantification problem consisting of a coupled time-dependent model. The benefits of using such an adaptive method are analyzed and discussed from the uncertainty propagation and computational points of view.

To accurately evaluate the remaining life (RUL) of rolling bearings under small sample conditions and strong noise interference, a RUL prediction scheme using adaptive variational mode decomposition (VMD) and double-discriminator conditional CycleGAN (DD-cCycleGAN) is put forward. Combining chimp optimization algorithm (ChOA) with VMD, an adaptive VMD algorithm based on ChOA is presented, which selects effective mode components for reconstruction and reduces interference from strong background noise. A DD-cCycleGAN is developed to generate new samples which not only retain sample information of source domain, but also resemble samples of target one. A LSTM network after training is utilized to predict the bearing RUL in test samples. The performance of this scheme was validated by using the XJTU-SY bearing test dataset. The comparison analyses demonstrate this scheme has strong noise resistance and high accuracy.

A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem

A domain decomposition algorithm for viscous/inviscid coupling

Aspects for a block version of the interval Cholesky algorithm