This paper is concerned with the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary in dimension 3. For arbitrary dimension $ Ngeq 2 $, in [18], we have shown that its long-time dynamics is characterised by a spreading-vanishing dichotomy; moreover, we have found a threshold condition on the kernel function that governs the onset of accelerated spreading, and determined the spreading speed when it is finite. In a more recent work [19], we have obtained sharp estimates of the spreading rate when the kernel function $ J(|x|) $ behaves like $ |x|^{-beta} $ as $ |x|toinfty $ in $ {mathbb R}^N $ ($ Ngeq 2 $). In this paper, we obtain more accurate estimates for the spreading rate when $ N = 3 $, which employs the fact that the formulas relating the involved kernel functions in the proofs of [19] become particularly simple in dimension $ 3 $.
{"title":"The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in $ {mathbb R}^3 $","authors":"Yihong Du, W. Ni","doi":"10.3934/mine.2023041","DOIUrl":"https://doi.org/10.3934/mine.2023041","url":null,"abstract":"<abstract><p>This paper is concerned with the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary in dimension 3. For arbitrary dimension $ Ngeq 2 $, in <sup>[<xref ref-type=\"bibr\" rid=\"b18\">18</xref>]</sup>, we have shown that its long-time dynamics is characterised by a spreading-vanishing dichotomy; moreover, we have found a threshold condition on the kernel function that governs the onset of accelerated spreading, and determined the spreading speed when it is finite. In a more recent work <sup>[<xref ref-type=\"bibr\" rid=\"b19\">19</xref>]</sup>, we have obtained sharp estimates of the spreading rate when the kernel function $ J(|x|) $ behaves like $ |x|^{-beta} $ as $ |x|toinfty $ in $ {mathbb R}^N $ ($ Ngeq 2 $). In this paper, we obtain more accurate estimates for the spreading rate when $ N = 3 $, which employs the fact that the formulas relating the involved kernel functions in the proofs of <sup>[<xref ref-type=\"bibr\" rid=\"b19\">19</xref>]</sup> become particularly simple in dimension $ 3 $.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ludovica Cicci, S. Fresca, S. Pagani, A. Manzoni, A. Quarteroni
The numerical simulation of several virtual scenarios arising in cardiac mechanics poses a computational challenge that can be alleviated if traditional full-order models (FOMs) are replaced by reduced order models (ROMs). For example, in the case of problems involving a vector of input parameters related, e.g., to material coefficients, projection-based ROMs provide mathematically rigorous physics-driven surrogate ROMs. In this work we demonstrate how, once trained, ROMs yield extremely accurate predictions (according to a prescribed tolerance) – yet cheaper than the ones provided by FOMs – of the structural deformation of the left ventricular tissue over an entire heartbeat, and of related output quantities of interest, such as the pressure-volume loop, for any desired input parameter values within a prescribed parameter range. However, the construction of ROM approximations for time-dependent cardiac mechanics is not straightforward, because of the highly nonlinear and multiscale nature of the problem, and almost never addressed. Our approach relies on the reduced basis method for parameterized partial differential equations. This technique performs a Galerkin projection onto a low-dimensional space for the displacement variable; the reduced space is built from a set of solution snapshots – obtained for different input parameter values and time instances – of the high-fidelity FOM, through the proper orthogonal decomposition technique. Then, suitable hyper-reduction techniques, such as the Discrete Empirical Interpolation Method, are exploited to efficiently handle nonlinear and parameter-dependent terms. In this work we show how a fast and reliable approximation of the time-dependent cardiac mechanical model can be achieved by a projection-based ROM, taking into account both passive and active mechanics for the left ventricle providing all the building blocks of the methodology, and highlighting those challenging aspects that are still open.
{"title":"Projection-based reduced order models for parameterized nonlinear time-dependent problems arising in cardiac mechanics","authors":"Ludovica Cicci, S. Fresca, S. Pagani, A. Manzoni, A. Quarteroni","doi":"10.3934/mine.2023026","DOIUrl":"https://doi.org/10.3934/mine.2023026","url":null,"abstract":"The numerical simulation of several virtual scenarios arising in cardiac mechanics poses a computational challenge that can be alleviated if traditional full-order models (FOMs) are replaced by reduced order models (ROMs). For example, in the case of problems involving a vector of input parameters related, e.g., to material coefficients, projection-based ROMs provide mathematically rigorous physics-driven surrogate ROMs. In this work we demonstrate how, once trained, ROMs yield extremely accurate predictions (according to a prescribed tolerance) – yet cheaper than the ones provided by FOMs – of the structural deformation of the left ventricular tissue over an entire heartbeat, and of related output quantities of interest, such as the pressure-volume loop, for any desired input parameter values within a prescribed parameter range. However, the construction of ROM approximations for time-dependent cardiac mechanics is not straightforward, because of the highly nonlinear and multiscale nature of the problem, and almost never addressed. Our approach relies on the reduced basis method for parameterized partial differential equations. This technique performs a Galerkin projection onto a low-dimensional space for the displacement variable; the reduced space is built from a set of solution snapshots – obtained for different input parameter values and time instances – of the high-fidelity FOM, through the proper orthogonal decomposition technique. Then, suitable hyper-reduction techniques, such as the Discrete Empirical Interpolation Method, are exploited to efficiently handle nonlinear and parameter-dependent terms. In this work we show how a fast and reliable approximation of the time-dependent cardiac mechanical model can be achieved by a projection-based ROM, taking into account both passive and active mechanics for the left ventricle providing all the building blocks of the methodology, and highlighting those challenging aspects that are still open.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this manuscript, we consider well-known multi-task learning (MTL) models from the literature for linear regression problems, such as clustered MTL or weakly constrained MTL. We propose novel reformulations of the training problem for these models, based on mixed-integer quadratic programming (MIQP) techniques. We show that our approach allows to drive the optimization process up to certified global optimality, exploiting popular off-the-shelf software solvers. By computational experiments on both synthetic and real-world datasets, we show that this strategy generally leads to improvements in terms of the predictive performance of the models, if compared to the classical local optimization techniques, based on alternating minimization strategies, that are usually employed. We also suggest a number of possible extensions of our model that should further improve the quality of the obtained regressors, introducing, for example, sparsity and features selection elements.
{"title":"Mixed-integer quadratic programming reformulations of multi-task learning models","authors":"M. Lapucci, Davide Pucci","doi":"10.3934/mine.2023020","DOIUrl":"https://doi.org/10.3934/mine.2023020","url":null,"abstract":"In this manuscript, we consider well-known multi-task learning (MTL) models from the literature for linear regression problems, such as clustered MTL or weakly constrained MTL. We propose novel reformulations of the training problem for these models, based on mixed-integer quadratic programming (MIQP) techniques. We show that our approach allows to drive the optimization process up to certified global optimality, exploiting popular off-the-shelf software solvers. By computational experiments on both synthetic and real-world datasets, we show that this strategy generally leads to improvements in terms of the predictive performance of the models, if compared to the classical local optimization techniques, based on alternating minimization strategies, that are usually employed. We also suggest a number of possible extensions of our model that should further improve the quality of the obtained regressors, introducing, for example, sparsity and features selection elements.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a new full explanation of the Tacoma Narrows Bridge collapse, occurred on November 7, 1940. Our explanation involves both structural phenomena, such as parametric resonances, and sophisticated mathematical tools, such as the Floquet theory. Contrary to all previous attempts, our explanation perfectly fits, both qualitatively and quantitatively, with what was observed that day. With this explanation at hand, we set up and partially solve some optimal control and shape optimization problems (both analytically and numerically) aiming to improve the stability of bridges. The control parameter to be optimized is the strength of a partial damping term whose role is to decrease the energy within the deck. Shape optimization intends to give suggestions for the design of future bridges.
{"title":"A new detailed explanation of the Tacoma collapse and some optimization problems to improve the stability of suspension bridges","authors":"F. Gazzola, M. Jleli, B. Samet","doi":"10.3934/mine.2023045","DOIUrl":"https://doi.org/10.3934/mine.2023045","url":null,"abstract":"We give a new full explanation of the Tacoma Narrows Bridge collapse, occurred on November 7, 1940. Our explanation involves both structural phenomena, such as parametric resonances, and sophisticated mathematical tools, such as the Floquet theory. Contrary to all previous attempts, our explanation perfectly fits, both qualitatively and quantitatively, with what was observed that day. With this explanation at hand, we set up and partially solve some optimal control and shape optimization problems (both analytically and numerically) aiming to improve the stability of bridges. The control parameter to be optimized is the strength of a partial damping term whose role is to decrease the energy within the deck. Shape optimization intends to give suggestions for the design of future bridges.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In these notes we review some recent results on the homoenergetic solutions for the Boltzmann equation obtained in [4,20,21,22]. These solutions are a particular class of non-equilibrium solutions of the Boltzmann equation which are useful to describe the dynamics of Boltzmann gases under shear, expansion or compression. Therefore, they do not behave asymptotically for long times as Maxwellian distributions, at least for all the choices of the collision kernels, and their behavior strongly depends on the homogeneity of the collision kernel and on the particular form of the hyperbolic terms which describe the deformation taking plance in the gas. We consider here the case of simple shear deformation and present different possible long-time asymptotics of these solutions. We discuss the current knowledge about the long-time behaviour of the homoenergetic solutions as well as some conjectures and open problems.
{"title":"Homoenergetic solutions of the Boltzmann equation: the case of simple-shear deformations","authors":"A. Nota, J. Velázquez","doi":"10.3934/mine.2023019","DOIUrl":"https://doi.org/10.3934/mine.2023019","url":null,"abstract":"<abstract><p>In these notes we review some recent results on the homoenergetic solutions for the Boltzmann equation obtained in <sup>[<xref ref-type=\"bibr\" rid=\"b4\">4</xref>,<xref ref-type=\"bibr\" rid=\"b20\">20</xref>,<xref ref-type=\"bibr\" rid=\"b21\">21</xref>,<xref ref-type=\"bibr\" rid=\"b22\">22</xref>]</sup>. These solutions are a particular class of non-equilibrium solutions of the Boltzmann equation which are useful to describe the dynamics of Boltzmann gases under shear, expansion or compression. Therefore, they do not behave asymptotically for long times as Maxwellian distributions, at least for all the choices of the collision kernels, and their behavior strongly depends on the homogeneity of the collision kernel and on the particular form of the hyperbolic terms which describe the deformation taking plance in the gas. We consider here the case of simple shear deformation and present different possible long-time asymptotics of these solutions. We discuss the current knowledge about the long-time behaviour of the homoenergetic solutions as well as some conjectures and open problems.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study an overdetermined problem that arises as the Euler-Lagrange equation of a weighted variational problem in elasticity. Based on a detailed linear analysis by spherical harmonics, we prove the existence and local uniqueness as well as an optimal stability estimate for the shape of a domain allowing the solvability of the overdetermined problem. Our linear analysis reveals that the solution structure is strongly related to the choice of parameters in the problem. In particular, the global uniqueness holds for the pair of the parameters lying in a triangular region.
{"title":"Linear stability analysis of overdetermined problems with non-constant data","authors":"Michiaki Onodera","doi":"10.3934/mine.2023048","DOIUrl":"https://doi.org/10.3934/mine.2023048","url":null,"abstract":"We study an overdetermined problem that arises as the Euler-Lagrange equation of a weighted variational problem in elasticity. Based on a detailed linear analysis by spherical harmonics, we prove the existence and local uniqueness as well as an optimal stability estimate for the shape of a domain allowing the solvability of the overdetermined problem. Our linear analysis reveals that the solution structure is strongly related to the choice of parameters in the problem. In particular, the global uniqueness holds for the pair of the parameters lying in a triangular region.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The focus of the article is the electro-magneto-hydrodynamics of an Oldroyd 6-constants fluid flow through parallel micro-plates with heat transfer. The medium between the micro-plates is porous and we use the Darcy-Brinkman-Forchheimer model for it. Numerical calculations, using the shooting method, were performed to solve the non-linear equations that emanate from the modeling. The results for the velocity mechanism, the Nusselt number and the temperature distribution are graphically shown. The analysis of the problem focuses on the effects of several fluid and heat transfer parameters, such as the Hartmann number, the Brinkmann number, the Darcy-Brinkman-Forchheimer parameter, the Darcy parameter, the viscous dissipation, and the Joule heating coefficient.
{"title":"Oldroyd 6-constant Electro-magneto-hydrodynamic fluid flow through parallel micro-plates with heat transfer using Darcy-Brinkman-Forchheimer model: A parametric investigation","authors":"M. M. Bhatti, E. Michaelides","doi":"10.3934/mine.2023051","DOIUrl":"https://doi.org/10.3934/mine.2023051","url":null,"abstract":"The focus of the article is the electro-magneto-hydrodynamics of an Oldroyd 6-constants fluid flow through parallel micro-plates with heat transfer. The medium between the micro-plates is porous and we use the Darcy-Brinkman-Forchheimer model for it. Numerical calculations, using the shooting method, were performed to solve the non-linear equations that emanate from the modeling. The results for the velocity mechanism, the Nusselt number and the temperature distribution are graphically shown. The analysis of the problem focuses on the effects of several fluid and heat transfer parameters, such as the Hartmann number, the Brinkmann number, the Darcy-Brinkman-Forchheimer parameter, the Darcy parameter, the viscous dissipation, and the Joule heating coefficient.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, dedicated to Ireneo Peral, we study the regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy potentials in the right hand side.
{"title":"Hardy potential versus lower order terms in Dirichlet problems: regularizing effects","authors":"D. Arcoya, L. Boccardo, L. Orsina","doi":"10.3934/mine.2023004","DOIUrl":"https://doi.org/10.3934/mine.2023004","url":null,"abstract":"<abstract><p>In this paper, dedicated to Ireneo Peral, we study the regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy potentials in the right hand side.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present several results of embedding type for parabolic Morrey and $ L_{p} $ spaces with or without mixed norms. Some other interpolation results for parabolic Morrey spaces are also given. The main object of investigation is the term $ b^{i}D_{i}u $ and the ways to estimate it in various Morrey and $ L_{p} $ spaces in order to be able to treat it as a perturbation term in the parabolic equations.
{"title":"On parabolic Adams's, the Chiarenza-Frasca theorems, and some other results related to parabolic Morrey spaces","authors":"N. Krylov","doi":"10.3934/mine.2023038","DOIUrl":"https://doi.org/10.3934/mine.2023038","url":null,"abstract":"We present several results of embedding type for parabolic Morrey and $ L_{p} $ spaces with or without mixed norms. Some other interpolation results for parabolic Morrey spaces are also given. The main object of investigation is the term $ b^{i}D_{i}u $ and the ways to estimate it in various Morrey and $ L_{p} $ spaces in order to be able to treat it as a perturbation term in the parabolic equations.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive interior curvature bounds for admissible solutions of a class of mixed Hessian curvature equations subject to affine Dirichlet data. As an application, we study a Plateau type problem for locally convex Weingarten hypersurfaces.
{"title":"Interior curvature bounds for a type of mixed Hessian quotient equations","authors":"Weimin Sheng, Shucan Xia","doi":"10.3934/mine.2023040","DOIUrl":"https://doi.org/10.3934/mine.2023040","url":null,"abstract":"<abstract><p>We derive interior curvature bounds for admissible solutions of a class of mixed Hessian curvature equations subject to affine Dirichlet data. As an application, we study a Plateau type problem for locally convex Weingarten hypersurfaces.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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