Marvin Fritz;Christina Kuttler;Mabel L Rajendran;Barbara Wohlmuth;Laura Scarabosio
In this work, we present and analyse a system of coupled partial differential equations, which models tumour growth under the influence of subdiffusion, mechanical effects, nutrient supply and chemotherapy. The subdiffusion of the system is modelled by a time fractional derivative in the equation governing the volume fraction of the tumour cells. The mass densities of the nutrients and the chemotherapeutic agents are modelled by reaction diffusion equations. We prove the existence and uniqueness of a weak solution to the model via the Faedo–Galerkin method and the application of appropriate compactness theorems. Lastly, we propose a fully discretized system and illustrate the effects of the fractional derivative and the influence of the fractional parameter in numerical examples.
{"title":"On a subdiffusive tumour growth model with fractional time derivative","authors":"Marvin Fritz;Christina Kuttler;Mabel L Rajendran;Barbara Wohlmuth;Laura Scarabosio","doi":"10.1093/imamat/hxab009","DOIUrl":"https://doi.org/10.1093/imamat/hxab009","url":null,"abstract":"In this work, we present and analyse a system of coupled partial differential equations, which models tumour growth under the influence of subdiffusion, mechanical effects, nutrient supply and chemotherapy. The subdiffusion of the system is modelled by a time fractional derivative in the equation governing the volume fraction of the tumour cells. The mass densities of the nutrients and the chemotherapeutic agents are modelled by reaction diffusion equations. We prove the existence and uniqueness of a weak solution to the model via the Faedo–Galerkin method and the application of appropriate compactness theorems. Lastly, we propose a fully discretized system and illustrate the effects of the fractional derivative and the influence of the fractional parameter in numerical examples.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50350067","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}
The numerical solution of partial differential equations that govern fluid dynamics with turbulence and combustion is challenging due to the multiscale nature of the dynamical system and the need to resolve small-scale physical features. In addition, the uncertainties in the dynamical system, including those in the physical models and parameters, initial and boundary conditions and numerical methods, impact the computational fluid dynamics (CFD) prediction of turbulence and chemical reactions. To improve the CFD prediction, this study focuses on the development and application of a maximum likelihood ensemble filter (MLEF), an ensemble-based data assimilation (DA), for flows featuring combustion and/or turbulence. MLEF finds the optimal analysis and its uncertainty by maximizing the posterior probability density function. The novelty of the study lies in the combination of advanced DA and CFD methods for a new comprehensive application to predict engineering fluid dynamics. The study combines important aspects, including an ensemble-based DA with analysis and uncertainty estimation, an augmented control vector that simultaneously adjusts initial conditions and model empirical parameters and an application of DA to CFD modeling of combustion and flows with complex geometry. The DA performance is validated by a turbulent Couette flow. The new CFD–DA system is then applied to solve the time-evolving shear-layer mixing with methane-air combustion and the turbulent flow over a bluff-body geometry. Results demonstrate the improvement of estimates of model parameters and the uncertainty reduction in initial conditions (ICs) for CFD modeling of flames and flows by the MLEF method.
{"title":"The maximum likelihood ensemble filter for computational flame and fluid dynamics","authors":"Yijun Wang;Stephen Guzik;Milija Zupanski;Xinfeng Gao","doi":"10.1093/imamat/hxab010","DOIUrl":"10.1093/imamat/hxab010","url":null,"abstract":"The numerical solution of partial differential equations that govern fluid dynamics with turbulence and combustion is challenging due to the multiscale nature of the dynamical system and the need to resolve small-scale physical features. In addition, the uncertainties in the dynamical system, including those in the physical models and parameters, initial and boundary conditions and numerical methods, impact the computational fluid dynamics (CFD) prediction of turbulence and chemical reactions. To improve the CFD prediction, this study focuses on the development and application of a maximum likelihood ensemble filter (MLEF), an ensemble-based data assimilation (DA), for flows featuring combustion and/or turbulence. MLEF finds the optimal analysis and its uncertainty by maximizing the posterior probability density function. The novelty of the study lies in the combination of advanced DA and CFD methods for a new comprehensive application to predict engineering fluid dynamics. The study combines important aspects, including an ensemble-based DA with analysis and uncertainty estimation, an augmented control vector that simultaneously adjusts initial conditions and model empirical parameters and an application of DA to CFD modeling of combustion and flows with complex geometry. The DA performance is validated by a turbulent Couette flow. The new CFD–DA system is then applied to solve the time-evolving shear-layer mixing with methane-air combustion and the turbulent flow over a bluff-body geometry. Results demonstrate the improvement of estimates of model parameters and the uncertainty reduction in initial conditions (ICs) for CFD modeling of flames and flows by the MLEF method.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44645899","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}
Pub Date : 2021-05-15DOI: 10.21203/RS.3.RS-563354/V1
Zofia Wr'oblewska, P. Kowalczyk, Lukasz Plociniczak
We consider a classical spring-mass model of human running which is built upon an inverted elastic pendulum. Based on our previous results concerning asymptotic solutions for large spring constant (or small angle of attack), we construct analytical approximations of solutions in the considered model. The model itself consists of two sets of differential equations - one set describes the motion of the centre of mass of a runner in contact with the ground (support phase), and the second set describes the phase of no contact with the ground (flight phase). By appropriately concatenating asymptotic solutions for the two phases we are able to reduce the dynamics to a one-dimensional apex to apex return map. We find sufficient conditions for this map to have a unique stable fixed point. By numerical continuation of fixed points with respect to energy, we find a transcritical bifurcation in our model system.
{"title":"Stability of Fixed Points in an Approximate Solution of the Spring-mass Running Model","authors":"Zofia Wr'oblewska, P. Kowalczyk, Lukasz Plociniczak","doi":"10.21203/RS.3.RS-563354/V1","DOIUrl":"https://doi.org/10.21203/RS.3.RS-563354/V1","url":null,"abstract":"\u0000 We consider a classical spring-mass model of human running which is built upon an inverted elastic pendulum. Based on our previous results concerning asymptotic solutions for large spring constant (or small angle of attack), we construct analytical approximations of solutions in the considered model. The model itself consists of two sets of differential equations - one set describes the motion of the centre of mass of a runner in contact with the ground (support phase), and the second set describes the phase of no contact with the ground (flight phase). By appropriately concatenating asymptotic solutions for the two phases we are able to reduce the dynamics to a one-dimensional apex to apex return map. We find sufficient conditions for this map to have a unique stable fixed point. By numerical continuation of fixed points with respect to energy, we find a transcritical bifurcation in our model system.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41588949","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 note we consider 1D problems within the context of a new class of elastic bodies. Under suitable conditions on the constitutive equations we prove instability and nonexistence of solutions similar to those in place for the linearized theory. The last section is devoted to describing the spatial behavior of the solutions.
{"title":"On the instability, nonexistence and spatial behaviour of the one-dimensional response of a new class of elastic bodies","authors":"R Quintanilla;K R Rajagopal","doi":"10.1093/imamat/hxab014","DOIUrl":"10.1093/imamat/hxab014","url":null,"abstract":"In this note we consider 1D problems within the context of a new class of elastic bodies. Under suitable conditions on the constitutive equations we prove instability and nonexistence of solutions similar to those in place for the linearized theory. The last section is devoted to describing the spatial behavior of the solutions.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41910739","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}
The general three-component nonlinear Schrödinger (gtc-NLS) equations are completely integrable and contain the self-focusing, defocusing and mixed cases, which are applied in many physical fields. In this paper, we would like to use the Fokas method to explore the initial-boundary value (IBV) problem for the gtc-NLS equations with a $4times 4$ matrix Lax pair on a finite interval based on the inverse scattering transform. The solutions of the gtc-NLS equations can be expressed using the solution of a $4times 4$ matrix Riemann–Hilbert (RH) problem constructed in the complex $k$ -plane. The jump matrices of the RH problem can be explicitly found in terms of three spectral functions related to the initial data, and the Dirichlet–Neumann boundary data, respectively. The global relation between the distinct spectral functions is also proposed to derive two distinct but equivalent types of representations of the Dirichlet–Neumann boundary value problems. Particularly, the relevant formulae for the boundary value problems on the finite interval can generate ones on the half-line as the length of the interval closes to infinity. Finally, we also analyse the linearizable boundary conditions for the Gel'fand–Levitan–Marchenko representation. These results will be useful to further study the solution properties of the IBV problem of the gtc-NLS system by using the Deift–Zhou's nonlinear steepest descent method and some numerical methods.
{"title":"An initial-boundary value problem for the general three-component nonlinear Schrödinger equations on a finite interval","authors":"Zhenya Yan","doi":"10.1093/imamat/hxab007","DOIUrl":"10.1093/imamat/hxab007","url":null,"abstract":"The general three-component nonlinear Schrödinger (gtc-NLS) equations are completely integrable and contain the self-focusing, defocusing and mixed cases, which are applied in many physical fields. In this paper, we would like to use the Fokas method to explore the initial-boundary value (IBV) problem for the gtc-NLS equations with a \u0000<tex>$4times 4$</tex>\u0000 matrix Lax pair on a finite interval based on the inverse scattering transform. The solutions of the gtc-NLS equations can be expressed using the solution of a \u0000<tex>$4times 4$</tex>\u0000 matrix Riemann–Hilbert (RH) problem constructed in the complex \u0000<tex>$k$</tex>\u0000-plane. The jump matrices of the RH problem can be explicitly found in terms of three spectral functions related to the initial data, and the Dirichlet–Neumann boundary data, respectively. The global relation between the distinct spectral functions is also proposed to derive two distinct but equivalent types of representations of the Dirichlet–Neumann boundary value problems. Particularly, the relevant formulae for the boundary value problems on the finite interval can generate ones on the half-line as the length of the interval closes to infinity. Finally, we also analyse the linearizable boundary conditions for the Gel'fand–Levitan–Marchenko representation. These results will be useful to further study the solution properties of the IBV problem of the gtc-NLS system by using the Deift–Zhou's nonlinear steepest descent method and some numerical methods.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45172944","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}
The concern in this paper is the problem of finding—or, at least, approximating—functions, defined within and on the boundary of a tangential polygon, functions whose Laplacian is $-1$ and which satisfy a homogeneous Robin boundary condition on the boundary. The parameter in the Robin condition is denoted by $beta $ . The integral of the solution over the interior, denoted by $Q$ , is, in the context of flows in a microchannel, the volume flow rate. A variational estimate of the dependence of $Q$ on $beta $ and the polygon's geometry is studied. Classes of tangential polygons treated include regular polygons and triangles, especially isosceles: the variational estimate $R(beta)$ is a rational function which approximates $Q(beta)$ closely.
{"title":"Steady slip flow of Newtonian fluids through tangential polygonal microchannels","authors":"Grant Keady","doi":"10.1093/imamat/hxab008","DOIUrl":"10.1093/imamat/hxab008","url":null,"abstract":"The concern in this paper is the problem of finding—or, at least, approximating—functions, defined within and on the boundary of a tangential polygon, functions whose Laplacian is \u0000<tex>$-1$</tex>\u0000 and which satisfy a homogeneous Robin boundary condition on the boundary. The parameter in the Robin condition is denoted by \u0000<tex>$beta $</tex>\u0000. The integral of the solution over the interior, denoted by \u0000<tex>$Q$</tex>\u0000, is, in the context of flows in a microchannel, the volume flow rate. A variational estimate of the dependence of \u0000<tex>$Q$</tex>\u0000 on \u0000<tex>$beta $</tex>\u0000 and the polygon's geometry is studied. Classes of tangential polygons treated include regular polygons and triangles, especially isosceles: the variational estimate \u0000<tex>$R(beta)$</tex>\u0000 is a rational function which approximates \u0000<tex>$Q(beta)$</tex>\u0000 closely.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47756270","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":null,"pages":null},"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":null,"pages":null},"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}
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":null,"pages":null},"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}
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":null,"pages":null},"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}
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