Siqing Li, Leevan Ling, Steven J. Ruuth, Xuemeng Wang
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1163-1185, June 2024. Abstract. We are interested in generating surfaces with arbitrary roughness and forming patterns on the surfaces. Two methods are applied to construct rough surfaces. In the first method, some superposition of wave functions with random frequencies and angles of propagation are used to get periodic rough surfaces with analytic parametric equations. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace–Beltrami operator and in the generation of pattern formation. Numerical experiments show that the patterns become irregular as the amplitude and frequency of the rough surface increase. For the sake of easy generalization to closed manifolds, we propose a second construction method for rough surfaces, which uses random nodal values and discretized heat filters. We provide numerical evidence that both surface construction methods yield comparable patterns to those observed in real-life animals.
{"title":"Realistic Pattern Formations on Surfaces by Adding Arbitrary Roughness","authors":"Siqing Li, Leevan Ling, Steven J. Ruuth, Xuemeng Wang","doi":"10.1137/22m1518001","DOIUrl":"https://doi.org/10.1137/22m1518001","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1163-1185, June 2024. <br/> Abstract. We are interested in generating surfaces with arbitrary roughness and forming patterns on the surfaces. Two methods are applied to construct rough surfaces. In the first method, some superposition of wave functions with random frequencies and angles of propagation are used to get periodic rough surfaces with analytic parametric equations. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace–Beltrami operator and in the generation of pattern formation. Numerical experiments show that the patterns become irregular as the amplitude and frequency of the rough surface increase. For the sake of easy generalization to closed manifolds, we propose a second construction method for rough surfaces, which uses random nodal values and discretized heat filters. We provide numerical evidence that both surface construction methods yield comparable patterns to those observed in real-life animals.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"418 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259022","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}
Agustín Gabriel Yabo, Jean-Baptiste Caillau, Jean-Luc Gouzé
SIAM Journal on Applied Mathematics, Ahead of Print. Abstract. The study of living microorganisms using resource allocation models has been key in elucidating natural behaviors of bacteria, by allowing allocation of microbial resources to be represented through optimal control strategies. The approach can also be applied to research in microbial cell factories, to investigate the optimal production of value-added compounds regulated by an external control. The latter is the subject of this paper, in which we study batch bioprocessing from a resource allocation perspective. Based on previous works, we propose a simple bacterial growth model accounting for the dynamics of the bioreactor and intracellular composition, and we analyze its asymptotic behavior and stability. Using optimization and optimal control theory, we study the production of biomass and metabolites of interest for infinite- and finite-time horizons. The resulting optimal control problems are studied using Pontryagin’s maximum principle and numerical methods, and the solutions found are characterized by the presence of the Fuller phenomenon (producing an infinite set of switching points occurring in a finite-time window) at the junctions with a second-order singular arc. The approach, inspired by biotechnological engineering, aims to shed light upon the role of cellular composition and resource allocation during batch processing and, at the same time, poses very interesting and challenging mathematical problems.
{"title":"Optimal Bacterial Resource Allocation Strategies in Batch Processing","authors":"Agustín Gabriel Yabo, Jean-Baptiste Caillau, Jean-Luc Gouzé","doi":"10.1137/22m1506328","DOIUrl":"https://doi.org/10.1137/22m1506328","url":null,"abstract":"SIAM Journal on Applied Mathematics, Ahead of Print. <br/> Abstract. The study of living microorganisms using resource allocation models has been key in elucidating natural behaviors of bacteria, by allowing allocation of microbial resources to be represented through optimal control strategies. The approach can also be applied to research in microbial cell factories, to investigate the optimal production of value-added compounds regulated by an external control. The latter is the subject of this paper, in which we study batch bioprocessing from a resource allocation perspective. Based on previous works, we propose a simple bacterial growth model accounting for the dynamics of the bioreactor and intracellular composition, and we analyze its asymptotic behavior and stability. Using optimization and optimal control theory, we study the production of biomass and metabolites of interest for infinite- and finite-time horizons. The resulting optimal control problems are studied using Pontryagin’s maximum principle and numerical methods, and the solutions found are characterized by the presence of the Fuller phenomenon (producing an infinite set of switching points occurring in a finite-time window) at the junctions with a second-order singular arc. The approach, inspired by biotechnological engineering, aims to shed light upon the role of cellular composition and resource allocation during batch processing and, at the same time, poses very interesting and challenging mathematical problems.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"34 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141196806","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1140-1162, June 2024. Abstract. First hitting times (FHTs) describe the time it takes a random “searcher” to find a “target” and are used to study timescales in many applications. FHTs have been well-studied for diffusive search, especially for small targets, which is called the narrow capture or narrow escape problem. In this paper, we study the FHT to small targets for a one-dimensional superdiffusive search described by a Lévy flight. By applying the method of matched asymptotic expansions to a fractional differential equation we obtain an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order [math] (describing a [math]-stable Lévy flight whose squared displacement scales as [math] in time [math]) and targets of radius [math], we show that the MFHT is order one for [math] and diverges as [math] for [math] and [math] for [math]. We then use our asymptotic results to identify the value of [math] which minimizes the average MFHT and find that (a) this optimal value of [math] vanishes for sparse targets and (b) the value [math] (corresponding to an inverse square Lévy search) is optimal in only very specific circumstances. We confirm our results by comparison to both deterministic numerical solutions of the associated fractional differential equation and stochastic simulations.
{"title":"First Hitting Time of a One-Dimensional Lévy Flight to Small Targets","authors":"Daniel Gomez, Sean D. Lawley","doi":"10.1137/23m1586239","DOIUrl":"https://doi.org/10.1137/23m1586239","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1140-1162, June 2024. <br/> Abstract. First hitting times (FHTs) describe the time it takes a random “searcher” to find a “target” and are used to study timescales in many applications. FHTs have been well-studied for diffusive search, especially for small targets, which is called the narrow capture or narrow escape problem. In this paper, we study the FHT to small targets for a one-dimensional superdiffusive search described by a Lévy flight. By applying the method of matched asymptotic expansions to a fractional differential equation we obtain an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order [math] (describing a [math]-stable Lévy flight whose squared displacement scales as [math] in time [math]) and targets of radius [math], we show that the MFHT is order one for [math] and diverges as [math] for [math] and [math] for [math]. We then use our asymptotic results to identify the value of [math] which minimizes the average MFHT and find that (a) this optimal value of [math] vanishes for sparse targets and (b) the value [math] (corresponding to an inverse square Lévy search) is optimal in only very specific circumstances. We confirm our results by comparison to both deterministic numerical solutions of the associated fractional differential equation and stochastic simulations.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"72 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259062","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1079-1095, June 2024. Abstract. Active Brownian particles (ABPs) are a model for nonequilibrium systems in which the constituent particles are self-propelled in addition to their Brownian motion. Compared to the well-studied mean first passage time (MFPT) of passive Brownian particles, the MFPT of ABPs is much less developed. In this paper, we study the MFPT for ABPs in a 1-D domain with absorbing boundary conditions at both ends of the domain. To reveal the effect of swimming on the MFPT, we consider an asymptotic analysis in the weak-swimming or small Péclet ([math]) number limit. In particular, analytical expressions for the survival probability and the MFPT are developed up to [math]. We explore the effects of the starting positions and starting orientations on the MFPT. Our analysis shows that if the starting orientations are biased towards one side of the domain, the MFPT as a function of the starting position becomes asymmetric about the center of the domain. The analytical results were confirmed by the numerical solutions of the full PDE model.
{"title":"Asymptotic Analysis and Simulation of Mean First Passage Time for Active Brownian Particles in 1-D","authors":"Sarafa A. Iyaniwura, Zhiwei Peng","doi":"10.1137/23m1593917","DOIUrl":"https://doi.org/10.1137/23m1593917","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1079-1095, June 2024. <br/> Abstract. Active Brownian particles (ABPs) are a model for nonequilibrium systems in which the constituent particles are self-propelled in addition to their Brownian motion. Compared to the well-studied mean first passage time (MFPT) of passive Brownian particles, the MFPT of ABPs is much less developed. In this paper, we study the MFPT for ABPs in a 1-D domain with absorbing boundary conditions at both ends of the domain. To reveal the effect of swimming on the MFPT, we consider an asymptotic analysis in the weak-swimming or small Péclet ([math]) number limit. In particular, analytical expressions for the survival probability and the MFPT are developed up to [math]. We explore the effects of the starting positions and starting orientations on the MFPT. Our analysis shows that if the starting orientations are biased towards one side of the domain, the MFPT as a function of the starting position becomes asymmetric about the center of the domain. The analytical results were confirmed by the numerical solutions of the full PDE model.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"24 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061644","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1039-1059, June 2024. Abstract. Consider a system of two parallel solid cylinders of equal radius made of a homogeneous material. We study the stability of a liquid bridge of circular cylinder shape between both solid cylinders. It is proved that if the circular cylinder liquid is concave, then it is stable. If the circular cylinder liquid is convex, we establish conditions on the radius of the cylinder liquid and the contact angle that ensure that long convex circular cylinders are not stable. Estimates for the length of these convex cylinders are given.
{"title":"Stability of a Capillary Circular Cylinder between Two Parallel Cylinders","authors":"Rafael López","doi":"10.1137/23m1602139","DOIUrl":"https://doi.org/10.1137/23m1602139","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1039-1059, June 2024. <br/> Abstract. Consider a system of two parallel solid cylinders of equal radius made of a homogeneous material. We study the stability of a liquid bridge of circular cylinder shape between both solid cylinders. It is proved that if the circular cylinder liquid is concave, then it is stable. If the circular cylinder liquid is convex, we establish conditions on the radius of the cylinder liquid and the contact angle that ensure that long convex circular cylinders are not stable. Estimates for the length of these convex cylinders are given.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"36 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925364","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}
Tianyu Kong, Diyi Liu, Mitchell Luskin, Alexander B. Watson
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1011-1038, June 2024. Abstract. We consider the problem of numerically computing the quantum dynamics of an electron in twisted bilayer graphene. The challenge is that atomic-scale models of the dynamics are aperiodic for generic twist angles because of the incommensurability of the layers. The Bistritzer–MacDonald PDE model, which is periodic with respect to the bilayer’s moiré pattern, has recently been shown to rigorously describe these dynamics in a parameter regime. In this work, we first prove that the dynamics of the tight-binding model of incommensurate twisted bilayer graphene can be approximated by computations on finite domains. The main ingredient of this proof is a speed of propagation estimate proved using Combes–Thomas estimates. We then provide extensive numerical computations, which clarify the range of validity of the Bistritzer–MacDonald model.
{"title":"Modeling of Electronic Dynamics in Twisted Bilayer Graphene","authors":"Tianyu Kong, Diyi Liu, Mitchell Luskin, Alexander B. Watson","doi":"10.1137/23m1595941","DOIUrl":"https://doi.org/10.1137/23m1595941","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1011-1038, June 2024. <br/> Abstract. We consider the problem of numerically computing the quantum dynamics of an electron in twisted bilayer graphene. The challenge is that atomic-scale models of the dynamics are aperiodic for generic twist angles because of the incommensurability of the layers. The Bistritzer–MacDonald PDE model, which is periodic with respect to the bilayer’s moiré pattern, has recently been shown to rigorously describe these dynamics in a parameter regime. In this work, we first prove that the dynamics of the tight-binding model of incommensurate twisted bilayer graphene can be approximated by computations on finite domains. The main ingredient of this proof is a speed of propagation estimate proved using Combes–Thomas estimates. We then provide extensive numerical computations, which clarify the range of validity of the Bistritzer–MacDonald model.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"21 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925365","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 988-1010, June 2024. Abstract. Spatial memory is key in animal movement modeling, but it has been challenging to explicitly model learning to describe memory acquisition. In this paper, we study novel cognitive consumer-resource models with different consumer learning mechanisms and investigate their dynamics. These models consist of two PDEs in composition with one ODE such that the spectrum of the corresponding linearized operator at a constant steady state is unclear. We describe the spectra of the linearized operators and analyze the eigenvalue problems to determine the stability of the constant steady states. We then perform bifurcation analysis by taking the perceptual diffusion rate as the bifurcation parameter. It is found that steady-state and Hopf bifurcations can both occur in these systems, and the bifurcation points are given so that the stability region can be determined. Moreover, rich spatial and spatiotemporal patterns can be generated in such systems via different types of bifurcation. Our effort establishes a new approach to tackling a hybrid model of PDE-ODE composition and provides a deeper understanding of cognitive movement-driven consumer-resource dynamics.
{"title":"Local Perception and Learning Mechanisms in Resource-Consumer Dynamics","authors":"Qingyan Shi, Yongli Song, Hao Wang","doi":"10.1137/23m1598593","DOIUrl":"https://doi.org/10.1137/23m1598593","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 988-1010, June 2024. <br/> Abstract. Spatial memory is key in animal movement modeling, but it has been challenging to explicitly model learning to describe memory acquisition. In this paper, we study novel cognitive consumer-resource models with different consumer learning mechanisms and investigate their dynamics. These models consist of two PDEs in composition with one ODE such that the spectrum of the corresponding linearized operator at a constant steady state is unclear. We describe the spectra of the linearized operators and analyze the eigenvalue problems to determine the stability of the constant steady states. We then perform bifurcation analysis by taking the perceptual diffusion rate as the bifurcation parameter. It is found that steady-state and Hopf bifurcations can both occur in these systems, and the bifurcation points are given so that the stability region can be determined. Moreover, rich spatial and spatiotemporal patterns can be generated in such systems via different types of bifurcation. Our effort establishes a new approach to tackling a hybrid model of PDE-ODE composition and provides a deeper understanding of cognitive movement-driven consumer-resource dynamics.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"21 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925136","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 940-960, June 2024. Abstract. We propose and analyze a new methodology based on linear-quadratic regulation (LQR) for stabilizing falling liquid films via blowing and suction at the base. LQR methods enable rapidly responding feedback control by precomputing a gain matrix, but they are only suitable for systems of linear ordinary differential equations (ODEs). By contrast, the Navier–Stokes equations that describe the dynamics of a thin liquid film flowing down an inclined plane are too complex to stabilize with standard control-theoretical techniques. To bridge this gap, we use reduced-order models—the Benney equation and a weighted-residual integral boundary layer model—obtained via asymptotic analysis to derive a multilevel control framework. This framework consists of an LQR feedback control designed for a linearized and discretized system of ODEs approximating the reduced-order system, which is then applied to the full Navier–Stokes system. The control scheme is tested via direct numerical simulation (DNS) and compared to analytical predictions of linear stability thresholds and minimum required actuator numbers. Comparing the strategy between the two reduced-order models, we show that in both cases we can successfully stabilize towards a uniform flat film across their respective ranges of valid parameters, with the more accurate weighted-residual model outperforming the Benney-derived controls. The weighted-residual controls are also found to work successfully far beyond their anticipated range of applicability. The proposed methodology increases the feasibility of transferring robust control techniques towards real-world systems and is also generalizable to other forms of actuation.
{"title":"Linear Quadratic Regulation Control for Falling Liquid Films","authors":"Oscar A. Holroyd, Radu Cimpeanu, Susana N. Gomes","doi":"10.1137/23m1548475","DOIUrl":"https://doi.org/10.1137/23m1548475","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 940-960, June 2024. <br/>Abstract. We propose and analyze a new methodology based on linear-quadratic regulation (LQR) for stabilizing falling liquid films via blowing and suction at the base. LQR methods enable rapidly responding feedback control by precomputing a gain matrix, but they are only suitable for systems of linear ordinary differential equations (ODEs). By contrast, the Navier–Stokes equations that describe the dynamics of a thin liquid film flowing down an inclined plane are too complex to stabilize with standard control-theoretical techniques. To bridge this gap, we use reduced-order models—the Benney equation and a weighted-residual integral boundary layer model—obtained via asymptotic analysis to derive a multilevel control framework. This framework consists of an LQR feedback control designed for a linearized and discretized system of ODEs approximating the reduced-order system, which is then applied to the full Navier–Stokes system. The control scheme is tested via direct numerical simulation (DNS) and compared to analytical predictions of linear stability thresholds and minimum required actuator numbers. Comparing the strategy between the two reduced-order models, we show that in both cases we can successfully stabilize towards a uniform flat film across their respective ranges of valid parameters, with the more accurate weighted-residual model outperforming the Benney-derived controls. The weighted-residual controls are also found to work successfully far beyond their anticipated range of applicability. The proposed methodology increases the feasibility of transferring robust control techniques towards real-world systems and is also generalizable to other forms of actuation.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"43 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942552","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 961-987, June 2024. Abstract. The quantitative modeling and design of modern short-pulse fiber lasers cannot be performed with averaged models because of large variations in the pulse parameters within each round trip. Instead, lumped models obtained by concatenating models for the various components of the laser are required. Since the optical pulses in lumped models are periodic, their linear stability is investigated using the monodromy operator, which is the linearization of the roundtrip operator about the pulse. A gradient-based optimization method is developed to discover periodic pulses. The computation of the gradient of the objective function involves numerical computation of the action of both the roundtrip operator and the adjoint of the monodromy operator. A novel Fourier split-step method is introduced to compute solutions of the linearization of the nonlinear, nonlocal, stiff equation that models optical propagation in the fiber amplifier. This method is derived by linearizing the two solution operators in a split-step method for the nonlinear equation. The spectrum of the monodromy operator consists of the essential spectrum, for which there is an analytical formula, and the eigenvalues. There is a multiplicity two eigenvalue at [math], which is due to phase and translation invariance. The remaining eigenvalues are determined from a matrix discretization of the monodromy operator. Simulation results verify the accuracy of the numerical methods; show examples of periodically stationary pulses, their spectra, and their eigenfunctions; and discuss their stability.
{"title":"Floquet Stability of Periodically Stationary Pulses in a Short-Pulse Fiber Laser","authors":"Vrushaly Shinglot, John Zweck","doi":"10.1137/23m1598106","DOIUrl":"https://doi.org/10.1137/23m1598106","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 961-987, June 2024. <br/> Abstract. The quantitative modeling and design of modern short-pulse fiber lasers cannot be performed with averaged models because of large variations in the pulse parameters within each round trip. Instead, lumped models obtained by concatenating models for the various components of the laser are required. Since the optical pulses in lumped models are periodic, their linear stability is investigated using the monodromy operator, which is the linearization of the roundtrip operator about the pulse. A gradient-based optimization method is developed to discover periodic pulses. The computation of the gradient of the objective function involves numerical computation of the action of both the roundtrip operator and the adjoint of the monodromy operator. A novel Fourier split-step method is introduced to compute solutions of the linearization of the nonlinear, nonlocal, stiff equation that models optical propagation in the fiber amplifier. This method is derived by linearizing the two solution operators in a split-step method for the nonlinear equation. The spectrum of the monodromy operator consists of the essential spectrum, for which there is an analytical formula, and the eigenvalues. There is a multiplicity two eigenvalue at [math], which is due to phase and translation invariance. The remaining eigenvalues are determined from a matrix discretization of the monodromy operator. Simulation results verify the accuracy of the numerical methods; show examples of periodically stationary pulses, their spectra, and their eigenfunctions; and discuss their stability.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"64 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940043","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 915-939, June 2024. Abstract. Foraging is crucial for animals to survive. Many species forage in groups, as individuals communicate to share information about the location of available resources. For example, eusocial foragers, such as honey bees and many ants, recruit members from their central hive or nest to a known foraging site. However, the optimal level of communication and recruitment depends on the overall group size, the distribution of available resources, and the extent of interference between multiple individuals attempting to forage from a site. In this paper, we develop a discrete-time Markov chain model of eusocial foragers, who communicate information with a certain probability. We compare the stochastic model and its corresponding infinite population limit. We find that foraging efficiency tapers off when recruitment probability is too high, a phenomenon that does not occur in the infinite population model, even though it occurs for any finite population size. The marginal inefficiency at high recruitment probability increases as the population increases, similar to a boundary layer. In particular, we prove there is a significant gap between the foraging efficiency of finite and infinite population models in the extreme case of complete communication. We also analyze this phenomenon by approximating the stationary distribution of foragers over sites in terms of mean escape times from multiple quasi-steady states. We conclude that, for any finite group of foragers, an individual who has found a resource should only sometimes recruit others to the same resource. We discuss the relationship between our analysis and multiagent multiarm bandit problems.
{"title":"Finite Population Size Effects on Optimal Communication for Social Foragers","authors":"Hyunjoong Kim, Yoichiro Mori, Joshua B. Plotkin","doi":"10.1137/23m1590007","DOIUrl":"https://doi.org/10.1137/23m1590007","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 915-939, June 2024. <br/> Abstract. Foraging is crucial for animals to survive. Many species forage in groups, as individuals communicate to share information about the location of available resources. For example, eusocial foragers, such as honey bees and many ants, recruit members from their central hive or nest to a known foraging site. However, the optimal level of communication and recruitment depends on the overall group size, the distribution of available resources, and the extent of interference between multiple individuals attempting to forage from a site. In this paper, we develop a discrete-time Markov chain model of eusocial foragers, who communicate information with a certain probability. We compare the stochastic model and its corresponding infinite population limit. We find that foraging efficiency tapers off when recruitment probability is too high, a phenomenon that does not occur in the infinite population model, even though it occurs for any finite population size. The marginal inefficiency at high recruitment probability increases as the population increases, similar to a boundary layer. In particular, we prove there is a significant gap between the foraging efficiency of finite and infinite population models in the extreme case of complete communication. We also analyze this phenomenon by approximating the stationary distribution of foragers over sites in terms of mean escape times from multiple quasi-steady states. We conclude that, for any finite group of foragers, an individual who has found a resource should only sometimes recruit others to the same resource. We discuss the relationship between our analysis and multiagent multiarm bandit problems.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"65 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940084","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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