Recently in Charalambopoulos et al. (2020), we presented a methodology aiming at reconstructing bounded total variation ( $TV$ ) conductivities via a technique simulating the so-called half-quadratic minimization approach, encountered in Aubert & Kornprobst (2002, Mathematical Problems in Image Processing. New York, NY: Springer). The method belongs to a duality framework, in which the auxiliary function $omega (x)$ was introduced, offering a tool for smoothing the members of the admissible set of conductivity profiles. The dual variable $omega (x)$ , in that approach, after every external update, served in the formation of an intermediate optimization scheme, concerning exclusively the sought conductivity $alpha (x)$ . In this work, we develop a novel investigation stemming from the previous approach, having though two different fundamental components. First, we do not detour herein the $BV$ -assumption on the conductivity profile, which means that the functional under optimization contains the $TV$ of $alpha (x)$ itself. Secondly, the auxiliary dual variable $omega (x)$ and the conductivity $alpha (x)$ acquire an equivalent role and concurrently, a parallel pacing in the minimization process. A common characteristic between these two approaches is that the function $omega (x)$ is an indicator of the conductivity's ‘jump’ set. A fortiori, this crucial property has been ameliorated herein, since the reciprocal role of the elements of the pair $(alpha ,omega )$ offers a self-monitoring structure very efficient to the minimization descent.
{"title":"A dual self-monitored reconstruction scheme on the TV-regularized inverse conductivity problem","authors":"Vanessa Markaki;Drosos Kourounis;Antonios Charalambopoulos","doi":"10.1093/imamat/hxab011","DOIUrl":"10.1093/imamat/hxab011","url":null,"abstract":"Recently in Charalambopoulos et al. (2020), we presented a methodology aiming at reconstructing bounded total variation (\u0000<tex>$TV$</tex>\u0000) conductivities via a technique simulating the so-called half-quadratic minimization approach, encountered in Aubert & Kornprobst (2002, Mathematical Problems in Image Processing. New York, NY: Springer). The method belongs to a duality framework, in which the auxiliary function \u0000<tex>$omega (x)$</tex>\u0000 was introduced, offering a tool for smoothing the members of the admissible set of conductivity profiles. The dual variable \u0000<tex>$omega (x)$</tex>\u0000, in that approach, after every external update, served in the formation of an intermediate optimization scheme, concerning exclusively the sought conductivity \u0000<tex>$alpha (x)$</tex>\u0000. In this work, we develop a novel investigation stemming from the previous approach, having though two different fundamental components. First, we do not detour herein the \u0000<tex>$BV$</tex>\u0000-assumption on the conductivity profile, which means that the functional under optimization contains the \u0000<tex>$TV$</tex>\u0000 of \u0000<tex>$alpha (x)$</tex>\u0000 itself. Secondly, the auxiliary dual variable \u0000<tex>$omega (x)$</tex>\u0000 and the conductivity \u0000<tex>$alpha (x)$</tex>\u0000 acquire an equivalent role and concurrently, a parallel pacing in the minimization process. A common characteristic between these two approaches is that the function \u0000<tex>$omega (x)$</tex>\u0000 is an indicator of the conductivity's ‘jump’ set. A fortiori, this crucial property has been ameliorated herein, since the reciprocal role of the elements of the pair \u0000<tex>$(alpha ,omega )$</tex>\u0000 offers a self-monitoring structure very efficient to the minimization descent.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"604-630"},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46877833","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 considers the stationary problem of density-suppressed motility models proposed in Fu et al. (2012) and Liu et al. (2011) in one dimension with Neumman boundary conditions. The models consist of parabolic equations with cross-diffusion and degeneracy. We employ the global bifurcation theory and Helly compactness theorem to explore the conditions under which non-constant stationary (pattern) solutions exist and asymptotic profiles of solutions as some parameter value is small. When the cell growth is not considered, we are able to show the monotonicity of solutions and hence achieve a global bifurcation diagram by treating the chemical diffusion rate as a bifurcation parameter. Furthermore, we show that the solutions have boundary spikes as the chemical diffusion rate tends to zero and identify the conditions for the non-existence of non-constant solutions. When transformed to specific motility functions, our results indeed give sharp conditions on the existence of non-constant stationary solutions. While with the cell growth, the structure of global bifurcation diagram is much more complicated and in particular the solution loses the monotonicity property. By treating the growth rate as a bifurcation parameter, we identify a minimum range of growth rate in which non-constant stationary solutions are warranted, while a global bifurcation diagram can still be attained in a special situation. We use numerical simulations to test our analytical results and illustrate that patterns can be very intricate and stable stationary solutions may not exist when the parameter value is outside the minimal range identified in our paper.
本文考虑Fu et al.(2012)和Liu et al.(2011)提出的密度抑制运动模型在一维具有Neumman边界条件的平稳性问题。该模型由具有交叉扩散和简并的抛物型方程组成。利用全局分岔理论和Helly紧性定理，探讨了非常平稳(模式)解存在的条件和当某参数值较小时解的渐近轮廓。当不考虑细胞生长时，我们可以将化学扩散速率作为分岔参数来显示解的单调性，从而得到全局分岔图。进一步，我们证明了当化学扩散速率趋于零时，解具有边界尖峰，并确定了非常数解不存在的条件。当转化为具体的运动函数时，我们的结果确实给出了非常平稳解存在的尖锐条件。但随着细胞的增长，全局分岔图的结构变得复杂，特别是解的单调性逐渐丧失。通过将增长率作为分岔参数，我们确定了增长率的最小范围，在此范围内保证了非常平稳解，而在特殊情况下仍然可以得到全局分岔图。我们使用数值模拟来测试我们的分析结果，并说明模式可能非常复杂，当参数值超出我们论文中确定的最小范围时，稳定的固定解可能不存在。
{"title":"Steady states and pattern formation of the density-suppressed motility model","authors":"Zhi-An Wang;Xin Xu","doi":"10.1093/imamat/hxab006","DOIUrl":"10.1093/imamat/hxab006","url":null,"abstract":"This paper considers the stationary problem of density-suppressed motility models proposed in Fu et al. (2012) and Liu et al. (2011) in one dimension with Neumman boundary conditions. The models consist of parabolic equations with cross-diffusion and degeneracy. We employ the global bifurcation theory and Helly compactness theorem to explore the conditions under which non-constant stationary (pattern) solutions exist and asymptotic profiles of solutions as some parameter value is small. When the cell growth is not considered, we are able to show the monotonicity of solutions and hence achieve a global bifurcation diagram by treating the chemical diffusion rate as a bifurcation parameter. Furthermore, we show that the solutions have boundary spikes as the chemical diffusion rate tends to zero and identify the conditions for the non-existence of non-constant solutions. When transformed to specific motility functions, our results indeed give sharp conditions on the existence of non-constant stationary solutions. While with the cell growth, the structure of global bifurcation diagram is much more complicated and in particular the solution loses the monotonicity property. By treating the growth rate as a bifurcation parameter, we identify a minimum range of growth rate in which non-constant stationary solutions are warranted, while a global bifurcation diagram can still be attained in a special situation. We use numerical simulations to test our analytical results and illustrate that patterns can be very intricate and stable stationary solutions may not exist when the parameter value is outside the minimal range identified in our paper.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"577-603"},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41883152","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}
Reactive settling denotes the process of sedimentation of small solid particles dispersed in a viscous fluid with simultaneous reactions between the components that constitute the solid and liquid phases. This process is of particular importance for the simulation and control of secondary settling tanks (SSTs) in water resource recovery facilities (WRRFs), formerly known as wastewater treatment plants. A spatially 1D model of reactive settling in an SST is formulated by combining a mechanistic model of sedimentation with compression with a model of biokinetic reactions. In addition, the cross-sectional area of the tank is allowed to vary as a function of height. The final model is a system of strongly degenerate parabolic, nonlinear partial differential equations that include discontinuous coefficients to describe the feed, underflow and overflow mechanisms, as well as singular source terms that model the feed mechanism. A finite difference scheme for the final model is developed by first deriving a method-of-lines formulation (discrete in space, continuous in time) and then passing to a fully discrete scheme by a time discretization. The advantage of this formulation is its compatibility with common practice in development of software for WRRFs. The main mathematical result is an invariant-region property, which implies that physically relevant numerical solutions are produced. Simulations of denitrification in SSTs in WRRFs illustrate the model and its discretization.
{"title":"A method-of-lines formulation for a model of reactive settling in tanks with varying cross-sectional area","authors":"Raimund Bürger;Julio Careaga;Stefan Diehl","doi":"10.1093/imamat/hxab012","DOIUrl":"https://doi.org/10.1093/imamat/hxab012","url":null,"abstract":"Reactive settling denotes the process of sedimentation of small solid particles dispersed in a viscous fluid with simultaneous reactions between the components that constitute the solid and liquid phases. This process is of particular importance for the simulation and control of secondary settling tanks (SSTs) in water resource recovery facilities (WRRFs), formerly known as wastewater treatment plants. A spatially 1D model of reactive settling in an SST is formulated by combining a mechanistic model of sedimentation with compression with a model of biokinetic reactions. In addition, the cross-sectional area of the tank is allowed to vary as a function of height. The final model is a system of strongly degenerate parabolic, nonlinear partial differential equations that include discontinuous coefficients to describe the feed, underflow and overflow mechanisms, as well as singular source terms that model the feed mechanism. A finite difference scheme for the final model is developed by first deriving a method-of-lines formulation (discrete in space, continuous in time) and then passing to a fully discrete scheme by a time discretization. The advantage of this formulation is its compatibility with common practice in development of software for WRRFs. The main mathematical result is an invariant-region property, which implies that physically relevant numerical solutions are produced. Simulations of denitrification in SSTs in WRRFs illustrate the model and its discretization.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"514-546"},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50350066","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}
A common realization of superhydrophobic surfaces comprises of a periodic array of cylindrical bubbles which are trapped in a periodically grooved solid substrate. We consider the thermocapillary animation of liquid motion by a macroscopic temperature gradient which is longitudinally applied over such a bubble mattress. Assuming a linear variation of the interfacial tension with the temperature, at slope $sigma _T$ , we seek the effective velocity slip attained by the liquid at large distances away from the mattress. We focus upon the dilute limit, where the groove width $2c$ is small compared with the array period $2l$ . The requisite velocity slip in the applied-gradient direction, determined by a local analysis about a single bubble, is provided by the approximation $$begin{align*}& pi frac{Gsigma_T c^2}{mu l} I(alpha), end{align*}$$ wherein $G$ is the applied-gradient magnitude, $mu $ is the liquid viscosity and $I(alpha )$ , a non-monotonic function of the protrusion angle $alpha $ , is provided by the quadrature, $$begin{align*}& I(alpha) = frac{2}{sinalpha} int_0^inftyfrac{sinh salpha}{ cosh s(pi-alpha) sinh s pi} , textrm{d} s. end{align*}$$
超疏水表面的一种常见实现包括捕获在周期性槽状固体衬底中的圆柱形气泡的周期性阵列。我们考虑了液体运动的热毛细动画的宏观温度梯度，这是纵向施加在这种气泡床垫。假设界面张力随温度呈线性变化，斜率为$sigma _T$，我们寻求液体在远离床垫很远的地方获得的有效速度滑移。我们将重点放在稀释极限上，其中槽宽$2c$与阵列周期$2l$相比较小。在施加梯度方向上所需的速度滑移，由对单个气泡的局部分析确定，由近似$$begin{align*}& pi frac{Gsigma_T c^2}{mu l} I(alpha), end{align*}$$提供，其中$G$是施加梯度幅度，$mu $是液体粘度，$I(alpha )$是突出角$alpha $的非单调函数，由正交函数提供。 $$begin{align*}& I(alpha) = frac{2}{sinalpha} int_0^inftyfrac{sinh salpha}{ cosh s(pi-alpha) sinh s pi} , textrm{d} s. end{align*}$$
{"title":"Longitudinal thermocapillary slip about a dilute periodic mattress of protruding bubbles","authors":"Ehud Yariv;Toby L Kirk","doi":"10.1093/imamat/hxab004","DOIUrl":"10.1093/imamat/hxab004","url":null,"abstract":"A common realization of superhydrophobic surfaces comprises of a periodic array of cylindrical bubbles which are trapped in a periodically grooved solid substrate. We consider the thermocapillary animation of liquid motion by a macroscopic temperature gradient which is longitudinally applied over such a bubble mattress. Assuming a linear variation of the interfacial tension with the temperature, at slope \u0000<tex>$sigma _T$</tex>\u0000, we seek the effective velocity slip attained by the liquid at large distances away from the mattress. We focus upon the dilute limit, where the groove width \u0000<tex>$2c$</tex>\u0000 is small compared with the array period \u0000<tex>$2l$</tex>\u0000. The requisite velocity slip in the applied-gradient direction, determined by a local analysis about a single bubble, is provided by the approximation \u0000<tex>$$begin{align*}& pi frac{Gsigma_T c^2}{mu l} I(alpha), end{align*}$$</tex>\u0000 wherein \u0000<tex>$G$</tex>\u0000 is the applied-gradient magnitude, \u0000<tex>$mu $</tex>\u0000 is the liquid viscosity and \u0000<tex>$I(alpha )$</tex>\u0000, a non-monotonic function of the protrusion angle \u0000<tex>$alpha $</tex>\u0000, is provided by the quadrature, \u0000<tex>$$begin{align*}& I(alpha) = frac{2}{sinalpha} int_0^inftyfrac{sinh salpha}{ cosh s(pi-alpha) sinh s pi} , textrm{d} s. end{align*}$$</tex>","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"490-501"},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48907885","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}
In this work, we propose mathematical formulations that detail the effect of the dielectric strength of dielectric material on the spatial distribution of electric field in an infinite space with a conducting plate. Using the dielectric strength of air as the maximum limit for the magnitude of electric field intensity and the equivalence of stored charge between two different zones, we determine the size of the dielectric breakdown region (the extended region with ionized material) for the conducting strip and the conducting disk charged to an electric voltage. The size of dielectric breakdown is proportional to the square of the applied voltage, and decreases with the increase of the width/radius of the conducting strip/disk.
{"title":"Dielectric breakdown sizes of conducting plates","authors":"Mimi X Yang;Fuqian Yang;Sanboh Lee","doi":"10.1093/imamat/hxab013","DOIUrl":"10.1093/imamat/hxab013","url":null,"abstract":"In this work, we propose mathematical formulations that detail the effect of the dielectric strength of dielectric material on the spatial distribution of electric field in an infinite space with a conducting plate. Using the dielectric strength of air as the maximum limit for the magnitude of electric field intensity and the equivalence of stored charge between two different zones, we determine the size of the dielectric breakdown region (the extended region with ionized material) for the conducting strip and the conducting disk charged to an electric voltage. The size of dielectric breakdown is proportional to the square of the applied voltage, and decreases with the increase of the width/radius of the conducting strip/disk.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"502-513"},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43821516","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}
An inequality on torsional rigidity is established. For tangential polygons, this inequality is stronger than an inequality of Polya and Szego for convex domains.
建立了扭转刚度的不等式。对于切向多边形，这个不等式强于Polya和Szego对于凸域的不等式。
{"title":"Torsional rigidity for tangential polygons","authors":"G. Keady","doi":"10.1093/IMAMAT/HXAB022","DOIUrl":"https://doi.org/10.1093/IMAMAT/HXAB022","url":null,"abstract":"\u0000 An inequality on torsional rigidity is established. For tangential polygons, this inequality is stronger than an inequality of Polya and Szego for convex domains.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43312312","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 investigate the reciprocity gap functional method, which has been developed in the inverse scattering theory, in the context of electrical impedance tomography. In particular, we aim to reconstruct an inclusion contained in a body, whose conductivity is different from the conductivity of the surrounding material. Numerical examples are given, showing the performance of our algorithm.
{"title":"Reconstruction of inclusions in electrical conductors","authors":"M. Cristo, Giacomo Milan","doi":"10.1093/imamat/hxaa030","DOIUrl":"https://doi.org/10.1093/imamat/hxaa030","url":null,"abstract":"\u0000 We investigate the reciprocity gap functional method, which has been developed in the inverse scattering theory, in the context of electrical impedance tomography. In particular, we aim to reconstruct an inclusion contained in a body, whose conductivity is different from the conductivity of the surrounding material. Numerical examples are given, showing the performance of our algorithm.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"85 1","pages":"933-950"},"PeriodicalIF":1.2,"publicationDate":"2020-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxaa030","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43740457","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}
Results are presented for nonlinear equilibrium solutions of the Navier–Stokes equations in the boundary layer set up by a flat plate started impulsively from rest. The solutions take the form of a wave–roll–streak interaction, which takes place in a layer located at the edge of the boundary layer. This extends previous results for similar nonlinear equilibrium solutions in steady 2D boundary layers. The results are derived asymptotically and then compared to numerical results obtained by marching the reduced boundary-region disturbance equations forward in time. It is concluded that the previously found canonical free-stream coherent structures in steady boundary layers can be embedded in unbounded, unsteady shear flows.
{"title":"Free-stream coherent structures in the unsteady Rayleigh boundary layer","authors":"Eleanor C. Johnstone, P. Hall","doi":"10.1093/imamat/hxaa038","DOIUrl":"https://doi.org/10.1093/imamat/hxaa038","url":null,"abstract":"\u0000 Results are presented for nonlinear equilibrium solutions of the Navier–Stokes equations in the boundary layer set up by a flat plate started impulsively from rest. The solutions take the form of a wave–roll–streak interaction, which takes place in a layer located at the edge of the boundary layer. This extends previous results for similar nonlinear equilibrium solutions in steady 2D boundary layers. The results are derived asymptotically and then compared to numerical results obtained by marching the reduced boundary-region disturbance equations forward in time. It is concluded that the previously found canonical free-stream coherent structures in steady boundary layers can be embedded in unbounded, unsteady shear flows.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"85 1","pages":"1021-1040"},"PeriodicalIF":1.2,"publicationDate":"2020-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxaa038","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44181833","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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