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A homogenised model for the motion of evaporating fronts in porous media 多孔介质中蒸发锋运动的均匀化模型
IF 1.9 4区 数学
European Journal of Applied Mathematics Pub Date : 2023-01-23 DOI: 10.1017/s0956792522000419
E. Luckins, C. Breward, I. Griffiths, C. Please
{"title":"A homogenised model for the motion of evaporating fronts in porous media","authors":"E. Luckins, C. Breward, I. Griffiths, C. Please","doi":"10.1017/s0956792522000419","DOIUrl":"https://doi.org/10.1017/s0956792522000419","url":null,"abstract":"\u0000 Evaporation within porous media is both a multiscale and interface-driven process, since the phase change at the evaporating interfaces within the pores generates a vapour flow and depends on the transport of vapour through the porous medium. While homogenised models of flow and chemical transport in porous media allow multiscale processes to be modelled efficiently, it is not clear how the multiscale effects impact the interface conditions required for these homogenised models. In this paper, we derive a homogenised model, including effective interface conditions, for the motion of an evaporation front through a porous medium, using a combined homogenisation and boundary layer analysis. This analysis extends previous work for a purely diffusive problem to include both gas flow and the advective–diffusive transport of material. We investigate the effect that different microscale models describing the chemistry of the evaporation have on the homogenised interface conditions. In particular, we identify a new effective parameter, \u0000 \u0000 \u0000 \u0000$mathcal{L}$\u0000\u0000 \u0000 , the average microscale interface length, which modifies the effective evaporation rate in the homogenised model. Like the effective diffusivity and permeability of a porous medium, \u0000 \u0000 \u0000 \u0000$mathcal{L}$\u0000\u0000 \u0000 may be found by solving a periodic cell problem on the microscale. We also show that the different microscale models of the interface chemistry result in fundamentally different fine-scale behaviour at, and near, the interface.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47679907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Battleship, tomography and quantum annealing 战舰、层析成像和量子退火
IF 1.9 4区 数学
European Journal of Applied Mathematics Pub Date : 2023-01-16 DOI: 10.1017/s0956792522000377
W. Casper, Taylor Grimes
{"title":"Battleship, tomography and quantum annealing","authors":"W. Casper, Taylor Grimes","doi":"10.1017/s0956792522000377","DOIUrl":"https://doi.org/10.1017/s0956792522000377","url":null,"abstract":"\u0000 The classic game of Battleship involves two players taking turns attempting to guess the positions of a fleet of vertically or horizontally positioned enemy ships hidden on a \u0000 \u0000 \u0000 \u0000$10times 10$\u0000\u0000 \u0000 grid. One variant of this game, also referred to as Battleship Solitaire, Bimaru or Yubotu, considers the game with the inclusion of X-ray data, represented by knowledge of how many spots are occupied in each row and column in the enemy board. This paper considers the Battleship puzzle problem: the problem of reconstructing an enemy fleet from its X-ray data. We generate non-unique solutions to Battleship puzzles via certain reflection transformations akin to Ryser interchanges. Furthermore, we demonstrate that solutions of Battleship puzzles may be reliably obtained by searching for solutions of the associated classical binary discrete tomography problem which minimise the discrete Laplacian. We reformulate this optimisation problem as a quadratic unconstrained binary optimisation problem and approximate solutions via a simulated annealer, emphasising the future practical applicability of quantum annealers to solving discrete tomography problems with predefined structure.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43618966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Partial Euler operators and the efficient inversion of Div 偏欧拉算子及Div的有效反演
IF 1.9 4区 数学
European Journal of Applied Mathematics Pub Date : 2022-12-16 DOI: 10.1017/S0956792523000037
P. Hydon
{"title":"Partial Euler operators and the efficient inversion of Div","authors":"P. Hydon","doi":"10.1017/S0956792523000037","DOIUrl":"https://doi.org/10.1017/S0956792523000037","url":null,"abstract":"Abstract The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems have conservation laws of arbitrarily high order that must be found with the aid of computer algebra. Even low-order conservation laws of complex systems can be hard to find and invert. This paper describes a new, efficient approach to the inversion problem. Two main tools are developed: partial Euler operators and partial scalings. These lead to a line integral formula for the inversion of a total derivative and a procedure for inverting a given total divergence concisely.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48745043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Two approximate symmetry frameworks for nonlinear partial differential equations with a small parameter: Comparisons, relations, approximate solutions 小参数非线性偏微分方程的两个近似对称框架:比较，关系，近似解
IF 1.9 4区 数学
European Journal of Applied Mathematics Pub Date : 2022-12-16 DOI: 10.1017/S0956792522000407
Mahmood R. Tarayrah, Brian Pitzel, A. Cheviakov
{"title":"Two approximate symmetry frameworks for nonlinear partial differential equations with a small parameter: Comparisons, relations, approximate solutions","authors":"Mahmood R. Tarayrah, Brian Pitzel, A. Cheviakov","doi":"10.1017/S0956792522000407","DOIUrl":"https://doi.org/10.1017/S0956792522000407","url":null,"abstract":"Abstract The frameworks of Baikov–Gazizov–Ibragimov (BGI) and Fushchich–Shtelen (FS) approximate symmetries are used to study symmetry properties of partial differential equations with a small parameter. In general, it is shown that unlike the case of ordinary differential equations (ODEs), unstable BGI point symmetries of unperturbed partial differential equations (PDEs) do not necessarily yield local approximate symmetries for the perturbed model. While some relations between the BGI and FS approaches can be established, the two methods yield different approximate symmetry classifications. Detailed classifications are presented for two nonlinear PDE families. The second family includes a one-dimensional wave equation describing the wave motion in a hyperelastic material with a single family of fibers. For this model, approximate symmetries can be used to compute approximate closed-form solutions. Wave breaking times are found numerically and using the approximate solutions, which yield comparable results.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42174109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Constant rank factorisations of smooth maps, with applications to sonar 光滑地图的常秩因子分解及其在声纳中的应用
IF 1.9 4区 数学
European Journal of Applied Mathematics Pub Date : 2022-12-01 DOI: 10.1017/s0956792522000365
Michael Robinson
{"title":"Constant rank factorisations of smooth maps, with applications to sonar","authors":"Michael Robinson","doi":"10.1017/s0956792522000365","DOIUrl":"https://doi.org/10.1017/s0956792522000365","url":null,"abstract":"\u0000 Sonar systems are frequently used to classify objects at a distance by using the structure of the echoes of acoustic waves as a proxy for the object’s shape and composition. Traditional synthetic aperture processing is highly effective in solving classification problems when the conditions are favourable but relies on accurate knowledge of the sensor’s trajectory relative to the object being measured. This article provides several new theoretical tools that decouple object classification performance from trajectory estimation in synthetic aperture sonar processing. The key insight is that decoupling the trajectory from classification-relevant information involves factoring a function into the composition of two functions. The article presents several new general topological invariants for smooth functions based on their factorisations over function composition. These invariants specialise to the case when a sonar platform trajectory is deformed by a non-small perturbation. The mathematical results exhibited in this article apply well beyond sonar classification problems. This article is written in a way that supports full mathematical generality.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43371338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A modelling framework for efficient reduced order simulations of parametrised lithium-ion battery cells 参数化锂离子电池单元高效降阶模拟的建模框架
IF 1.9 4区 数学
European Journal of Applied Mathematics Pub Date : 2022-11-29 DOI: 10.1017/S0956792522000353
M. Landstorfer, Mario Ohlberger, S. Rave, M. Tacke
{"title":"A modelling framework for efficient reduced order simulations of parametrised lithium-ion battery cells","authors":"M. Landstorfer, Mario Ohlberger, S. Rave, M. Tacke","doi":"10.1017/S0956792522000353","DOIUrl":"https://doi.org/10.1017/S0956792522000353","url":null,"abstract":"Abstract In this contribution, we present a modelling and simulation framework for parametrised lithium-ion battery cells. We first derive a continuum model for a rather general intercalation battery cell on the basis of non-equilibrium thermodynamics. In order to efficiently evaluate the resulting parameterised non-linear system of partial differential equations, the reduced basis method is employed. The reduced basis method is a model order reduction technique on the basis of an incremental hierarchical approximate proper orthogonal decomposition approach and empirical operator interpolation. The modelling framework is particularly well suited to investigate and quantify degradation effects of battery cells. Several numerical experiments are given to demonstrate the scope and efficiency of the modelling framework.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49274262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Exact nonclassical symmetry solutions of Lotka–Volterra-type population systems lotka - volterra型种群系统的精确非经典对称解
IF 1.9 4区 数学
European Journal of Applied Mathematics Pub Date : 2022-11-25 DOI: 10.1017/S095679252200033X
P. Broadbridge, R. Cherniha, J. Goard
{"title":"Exact nonclassical symmetry solutions of Lotka–Volterra-type population systems","authors":"P. Broadbridge, R. Cherniha, J. Goard","doi":"10.1017/S095679252200033X","DOIUrl":"https://doi.org/10.1017/S095679252200033X","url":null,"abstract":"Abstract New classes of conditionally integrable systems of nonlinear reaction–diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator–prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka–Volterra system, but they have additional features.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42441098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Existence and stability of singular patterns in a fractional Ginzburg–Landau equation with a mean field 具有平均场的分数阶Ginzburg-Landau方程奇异模式的存在性和稳定性
IF 1.9 4区 数学
European Journal of Applied Mathematics Pub Date : 2022-11-11 DOI: 10.1017/s0956792522000286
Mingchen Gao, M. Winter, Wen Yang
{"title":"Existence and stability of singular patterns in a fractional Ginzburg–Landau equation with a mean field","authors":"Mingchen Gao, M. Winter, Wen Yang","doi":"10.1017/s0956792522000286","DOIUrl":"https://doi.org/10.1017/s0956792522000286","url":null,"abstract":"\u0000 In this paper, we consider the existence and stability of singular patterns in a fractional Ginzburg–Landau equation with a mean field. We prove the existence of three types of singular steady-state patterns (double fronts, single spikes, and double spikes) by solving their respective consistency conditions. In the case of single spikes, we prove the stability of single small spike solution for sufficiently large spatial period by studying an explicit non-local eigenvalue problem which is equivalent to the original eigenvalue problem. For the other solutions, we prove the instability by using the variational characterisation of eigenvalues. Finally, we present the results of some numerical computations of spike solutions based on the finite difference methods of Crank–Nicolson and Adams–Bashforth.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45218613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A deterministic gradient-based approach to avoid saddle points 基于确定性梯度的避鞍点方法
IF 1.9 4区 数学
European Journal of Applied Mathematics Pub Date : 2022-11-09 DOI: 10.1017/s0956792522000316
L. M. Kreusser, S. J. Osher, B. Wang
{"title":"A deterministic gradient-based approach to avoid saddle points","authors":"L. M. Kreusser, S. J. Osher, B. Wang","doi":"10.1017/s0956792522000316","DOIUrl":"https://doi.org/10.1017/s0956792522000316","url":null,"abstract":"<p>Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning (ML) models efficiently. First-order methods such as gradient descent (GD) are usually the methods of choice for training ML models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher et al., arXiv:1806.06317], called modified LSGD (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region’s dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230705121909977-0335:S0956792522000316:S0956792522000316_inline1.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$lfloor (n-1)/2rfloor$\u0000</span></span>\u0000</span>\u0000</span>, and hence it is significantly smaller than that of GD whose dimension is <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230705121909977-0335:S0956792522000316:S0956792522000316_inline2.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$n-1$\u0000</span></span>\u0000</span>\u0000</span>.</p>","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138539587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A deterministic gradient-based approach to avoid saddle points 基于确定性梯度的避鞍点方法
IF 1.9 4区 数学
European Journal of Applied Mathematics Pub Date : 2022-11-09 DOI: 10.1017/s0956792522000316
L. M. Kreusser, S. J. Osher, B. Wang
{"title":"A deterministic gradient-based approach to avoid saddle points","authors":"L. M. Kreusser, S. J. Osher, B. Wang","doi":"10.1017/s0956792522000316","DOIUrl":"https://doi.org/10.1017/s0956792522000316","url":null,"abstract":"<p>Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning (ML) models efficiently. First-order methods such as gradient descent (GD) are usually the methods of choice for training ML models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher et al., arXiv:1806.06317], called modified LSGD (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region’s dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230705121909977-0335:S0956792522000316:S0956792522000316_inline1.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$lfloor (n-1)/2rfloor$\u0000</span></span>\u0000</span>\u0000</span>, and hence it is significantly smaller than that of GD whose dimension is <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230705121909977-0335:S0956792522000316:S0956792522000316_inline2.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$n-1$\u0000</span></span>\u0000</span>\u0000</span>.</p>","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138539663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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