Larkspur Brudvik-Lindner, Dimitrios Mitsotakis, Athanasios E Tzavaras
Abstract We consider a dissipative, dispersive system of the Boussinesq type, which describes wave phenomena in scenarios where dissipation plays a significant role. Examples include undular bores in rivers or oceans, where turbulence-induced dissipation significantly influences their behavior. In this study, we demonstrate that the proposed system admits traveling wave solutions known as diffusive-dispersive shock waves. These solutions can be categorized as oscillatory and regularized shock waves, depending on the interplay between dispersion and dissipation effects. By comparing numerically computed solutions with laboratory data, we observe that the proposed model accurately captures the behavior of undular bores over a broad range of phase speeds. Traditionally, undular bores have been approximated using the original Peregrine system, which, even though it doesn’t possess these as traveling wave solutions, tends to offer accurate approximations within suitable time scales. To shed light on this phenomenon, we demonstrate that the discrepancy between the solutions of the dissipative Peregrine system and the non-dissipative counterpart is proportional to the product of the dissipation coefficient and the observation time.
{"title":"Oscillatory and regularized shock waves for a dissipative Peregrine-Boussinesq system","authors":"Larkspur Brudvik-Lindner, Dimitrios Mitsotakis, Athanasios E Tzavaras","doi":"10.1093/imamat/hxad030","DOIUrl":"https://doi.org/10.1093/imamat/hxad030","url":null,"abstract":"Abstract We consider a dissipative, dispersive system of the Boussinesq type, which describes wave phenomena in scenarios where dissipation plays a significant role. Examples include undular bores in rivers or oceans, where turbulence-induced dissipation significantly influences their behavior. In this study, we demonstrate that the proposed system admits traveling wave solutions known as diffusive-dispersive shock waves. These solutions can be categorized as oscillatory and regularized shock waves, depending on the interplay between dispersion and dissipation effects. By comparing numerically computed solutions with laboratory data, we observe that the proposed model accurately captures the behavior of undular bores over a broad range of phase speeds. Traditionally, undular bores have been approximated using the original Peregrine system, which, even though it doesn’t possess these as traveling wave solutions, tends to offer accurate approximations within suitable time scales. To shed light on this phenomenon, we demonstrate that the discrepancy between the solutions of the dissipative Peregrine system and the non-dissipative counterpart is proportional to the product of the dissipation coefficient and the observation time.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135728896","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}
{"title":"Correction to: On the use of asymptotically motivated gauge functions to obtain convergent series solutions to nonlinear ODEs","authors":"","doi":"10.1093/imamat/hxad026","DOIUrl":"https://doi.org/10.1093/imamat/hxad026","url":null,"abstract":"","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136378025","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}
Abstract The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic non-local delayed reaction–diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition at the finite end. This equation models the spatial–temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions. The theoretical results are illustrated by application to the stochastic non-local delayed Nicholson’s blowfly equation.
{"title":"Random attractors for a stochastic nonlocal delayed reaction-diffusion equation on a semi-infinite interval","authors":"Wenjie Hu, Quanxin Zhu, Tomás Caraballo","doi":"10.1093/imamat/hxad025","DOIUrl":"https://doi.org/10.1093/imamat/hxad025","url":null,"abstract":"Abstract The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic non-local delayed reaction–diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition at the finite end. This equation models the spatial–temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions. The theoretical results are illustrated by application to the stochastic non-local delayed Nicholson’s blowfly equation.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135148643","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}
Abstract We investigate the problem of learning a water quality model (BOD-DO model) from given data. Assuming that all parameters in the model are constants, we reformulate the problem as a system of linear equations for the unknown terms. Since in practice the system is often under-determined or over-determined and the observed data are noisy, we use an $l^{1}$-weighted regularization method to find a stable approximate solution. Then, Nesterov’s algorithm is used to solve the regularized problem. Learning models with variable coefficients are also discussed. Numerical examples show that our approach works well with noisy data and has the ability to learn the BOD-DO model.
{"title":"Learning river water quality models by <i>l1</i>-weighted regularization","authors":"Dinh Nho Hào, Duong Xuan Hiep, Pham Quy Muoi","doi":"10.1093/imamat/hxad023","DOIUrl":"https://doi.org/10.1093/imamat/hxad023","url":null,"abstract":"Abstract We investigate the problem of learning a water quality model (BOD-DO model) from given data. Assuming that all parameters in the model are constants, we reformulate the problem as a system of linear equations for the unknown terms. Since in practice the system is often under-determined or over-determined and the observed data are noisy, we use an $l^{1}$-weighted regularization method to find a stable approximate solution. Then, Nesterov’s algorithm is used to solve the regularized problem. Learning models with variable coefficients are also discussed. Numerical examples show that our approach works well with noisy data and has the ability to learn the BOD-DO model.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134950164","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}
We are concerned with the inverse scattering problem of recovering the refractive indices and transmission coefficients by the corresponding acoustic far-field measurement encoded into the scattering amplitude. Our first uniqueness result is to determine a constant refractive index by the fixed incident direction scattering amplitude, the proof of which is mainly based on the discreteness of the corresponding interior transmission eigenvalues. Then motivated by the previous work Xiang & Yan (2021), the second uniqueness result is established to recover a piecewise constant refractive index from the far-field pattern at a fixed frequency.
{"title":"Uniqueness of refractive indices and transmission coefficients by an inhomogeneous medium in acoustic scattering","authors":"Jianlin Xiang, Guozheng Yan","doi":"10.1093/imamat/hxad022","DOIUrl":"https://doi.org/10.1093/imamat/hxad022","url":null,"abstract":"\u0000 We are concerned with the inverse scattering problem of recovering the refractive indices and transmission coefficients by the corresponding acoustic far-field measurement encoded into the scattering amplitude. Our first uniqueness result is to determine a constant refractive index by the fixed incident direction scattering amplitude, the proof of which is mainly based on the discreteness of the corresponding interior transmission eigenvalues. Then motivated by the previous work Xiang & Yan (2021), the second uniqueness result is established to recover a piecewise constant refractive index from the far-field pattern at a fixed frequency.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41592336","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}
Abstract In this paper, we investigate the use of bilevel optimization for model learning in variational imaging problems. Bilevel learning is an alternative approach to deep learning methods, which leads to fully interpretable models. However, it requires a detailed analytical insight into the underlying mathematical model. We focus on the bilevel learning problem for total variation models with spatially- and patch-dependent parameters. Our study encompasses the directional differentiability of the solution mapping, the derivation of optimality conditions, and the characterization of the Bouligand subdifferential of the solution operator. We also propose a two-phase trust-region algorithm for solving the problem and present numerical tests using the CelebA dataset.
{"title":"Interpretable Model Learning in Variational Imaging: A Bilevel Optimization Approach","authors":"Juan Carlos De los Reyes, David Villacís","doi":"10.1093/imamat/hxad024","DOIUrl":"https://doi.org/10.1093/imamat/hxad024","url":null,"abstract":"Abstract In this paper, we investigate the use of bilevel optimization for model learning in variational imaging problems. Bilevel learning is an alternative approach to deep learning methods, which leads to fully interpretable models. However, it requires a detailed analytical insight into the underlying mathematical model. We focus on the bilevel learning problem for total variation models with spatially- and patch-dependent parameters. Our study encompasses the directional differentiability of the solution mapping, the derivation of optimality conditions, and the characterization of the Bouligand subdifferential of the solution operator. We also propose a two-phase trust-region algorithm for solving the problem and present numerical tests using the CelebA dataset.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134951867","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}
This paper concerns the use of asymptotic expansions for the efficient solving of forward and inverse problems involving a nonlinear singularly perturbed time-dependent reaction–diffusion–advection equation. By using an asymptotic expansion with the local coordinates in the transition-layer region, we prove the existence and uniqueness of a smooth solution with a sharp transition layer for a three-dimensional partial differential equation. Moreover, with the help of asymptotic expansion, a simplified model is derived for the corresponding inverse source problem, which is close to the original inverse problem over the entire region except for a narrow transition layer. We show that such simplification does not reduce the accuracy of the inversion results when the measurement data contain noise. Based on this simpler inversion model, an asymptotic-expansion regularization algorithm is proposed for efficiently solving the inverse source problem in the three-dimensional case. A model problem shows the feasibility of the proposed numerical approach.
{"title":"Solving forward and inverse problems involving a nonlinear three-dimensional partial differential equation via asymptotic expansions","authors":"D. Chaikovskii, Ye Zhang","doi":"10.1093/imamat/hxad021","DOIUrl":"https://doi.org/10.1093/imamat/hxad021","url":null,"abstract":"\u0000 This paper concerns the use of asymptotic expansions for the efficient solving of forward and inverse problems involving a nonlinear singularly perturbed time-dependent reaction–diffusion–advection equation. By using an asymptotic expansion with the local coordinates in the transition-layer region, we prove the existence and uniqueness of a smooth solution with a sharp transition layer for a three-dimensional partial differential equation. Moreover, with the help of asymptotic expansion, a simplified model is derived for the corresponding inverse source problem, which is close to the original inverse problem over the entire region except for a narrow transition layer. We show that such simplification does not reduce the accuracy of the inversion results when the measurement data contain noise. Based on this simpler inversion model, an asymptotic-expansion regularization algorithm is proposed for efficiently solving the inverse source problem in the three-dimensional case. A model problem shows the feasibility of the proposed numerical approach.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47356608","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}
Abstract Deep neural networks are capable of state-of-the-art performance in many classification tasks. However, they are known to be vulnerable to adversarial attacks—small perturbations to the input that lead to a change in classification. We address this issue from the perspective of backward error and condition number, concepts that have proved useful in numerical analysis. To do this, we build on the work of Beuzeville, T., Boudier, P., Buttari, A., Gratton, S., Mary, T. and Pralet S. (2021) Adversarial attacks via backward error analysis. hal-03296180, version 3. In particular, we develop a new class of attack algorithms that use componentwise relative perturbations. Such attacks are highly relevant in the case of handwritten documents or printed texts where, for example, the classification of signatures, postcodes, dates or numerical quantities may be altered by changing only the ink consistency and not the background. This makes the perturbed images look natural to the naked eye. Such ‘adversarial ink’ attacks therefore reveal a weakness that can have a serious impact on safety and security. We illustrate the new attacks on real data and contrast them with existing algorithms. We also study the use of a componentwise condition number to quantify vulnerability.
{"title":"Adversarial ink: componentwise backward error attacks on deep learning","authors":"Lucas Beerens, Desmond J Higham","doi":"10.1093/imamat/hxad017","DOIUrl":"https://doi.org/10.1093/imamat/hxad017","url":null,"abstract":"Abstract Deep neural networks are capable of state-of-the-art performance in many classification tasks. However, they are known to be vulnerable to adversarial attacks—small perturbations to the input that lead to a change in classification. We address this issue from the perspective of backward error and condition number, concepts that have proved useful in numerical analysis. To do this, we build on the work of Beuzeville, T., Boudier, P., Buttari, A., Gratton, S., Mary, T. and Pralet S. (2021) Adversarial attacks via backward error analysis. hal-03296180, version 3. In particular, we develop a new class of attack algorithms that use componentwise relative perturbations. Such attacks are highly relevant in the case of handwritten documents or printed texts where, for example, the classification of signatures, postcodes, dates or numerical quantities may be altered by changing only the ink consistency and not the background. This makes the perturbed images look natural to the naked eye. Such ‘adversarial ink’ attacks therefore reveal a weakness that can have a serious impact on safety and security. We illustrate the new attacks on real data and contrast them with existing algorithms. We also study the use of a componentwise condition number to quantify vulnerability.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135210757","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}
Conformal capacity is a mathematical quantity relevant to a wide range of physical and mathematical problems and recently there has been a resurgence of interest in devising new methods for its computation. In this paper we show how ideas from matched asymptotics can be used to derive estimates for conformal capacity. The formulas derived here are explicit, and there is evidence that they provide excellent approximations to the exact capacity values even well outside the expected range of validity.
{"title":"Estimating conformal capacity using asymptotic matching","authors":"Hiroyuki Miyoshi, D. Crowdy","doi":"10.1093/imamat/hxad018","DOIUrl":"https://doi.org/10.1093/imamat/hxad018","url":null,"abstract":"\u0000 Conformal capacity is a mathematical quantity relevant to a wide range of physical and mathematical problems and recently there has been a resurgence of interest in devising new methods for its computation. In this paper we show how ideas from matched asymptotics can be used to derive estimates for conformal capacity. The formulas derived here are explicit, and there is evidence that they provide excellent approximations to the exact capacity values even well outside the expected range of validity.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43277795","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}
We consider a large cloud of vectors formed at each layer of a standard neural network, corresponding to a large number of separate inputs which were presented independently to the classifier. Although the embedding dimension (the total possible degrees of freedom) reduces as we pass through successive layers, from input to output, the actual dimensionality of the point clouds that the layers contain does not necessarily reduce. We argue that this phenomenon may result in a vulnerability to (universal) adversarial attacks (which are small specific perturbations). This analysis requires us to estimate the intrinsic dimension of point clouds (with values between 20 and 200) within embedding spaces of dimension 1000 up to 800,000. This needs some care. If the cloud dimension actually increases from one layer to the next it implies there is some ‘volume filling’ over-folding, and thus there exist possible small directional perturbations in the latter space that are equivalent to shifting large distances within the former space, thus inviting possibility of universal and imperceptible attacks.
{"title":"Mappings, dimensionality and reversing out of deep neural networks","authors":"Zhaofang Cui, P. Grindrod","doi":"10.1093/imamat/hxad019","DOIUrl":"https://doi.org/10.1093/imamat/hxad019","url":null,"abstract":"\u0000 We consider a large cloud of vectors formed at each layer of a standard neural network, corresponding to a large number of separate inputs which were presented independently to the classifier. Although the embedding dimension (the total possible degrees of freedom) reduces as we pass through successive layers, from input to output, the actual dimensionality of the point clouds that the layers contain does not necessarily reduce. We argue that this phenomenon may result in a vulnerability to (universal) adversarial attacks (which are small specific perturbations). This analysis requires us to estimate the intrinsic dimension of point clouds (with values between 20 and 200) within embedding spaces of dimension 1000 up to 800,000. This needs some care. If the cloud dimension actually increases from one layer to the next it implies there is some ‘volume filling’ over-folding, and thus there exist possible small directional perturbations in the latter space that are equivalent to shifting large distances within the former space, thus inviting possibility of universal and imperceptible attacks.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42770755","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}
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