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Authors:Tao Tang, Boyi Wang, Jiang Yang Pages: 773 - 792 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 3, Page 773-792, June 2022. In this paper, we aim to study the motions of interfaces and coarsening rates governed by the time-fractional Cahn--Hilliard equation (TFCHE). It is observed by many numerical experiments that the microstructure evolution described by the TFCHE displays quite different dynamical processes compared with the classical Cahn--Hilliard equation, in particular, regarding motions of interfaces and coarsening rates. By using the method of matched asymptotic expansions, we first derive the sharp interface limit models. Then we can theoretically analyze the motions of interfaces with respect to different timescales. For instance, for the TFCHE with the constant diffusion mobility, the sharp interface limit model is a fractional Stefan problem at the timescale $t=O(1)$. However, on the timescale $t=O(\varepsilon^{-\frac1\alpha})$, the sharp interface limit model is a fractional Mullins--Sekerka model. Similar asymptotic regime results are also obtained for the case with one-sided degenerated mobility. Moreover, the scaling invariant property of the sharp interface models suggests that the TFCHE with constant mobility preserves an $\alpha/3$ coarsening rate, and a crossover of the coarsening rates from $\frac{\alpha}{3}$ to $\frac\alpha4$ is obtained for the case with one-sided degenerated mobility, in good agreement with the numerical experiments. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-05-03T07:00:00Z DOI: 10.1137/21M1427863 Issue No:Vol. 82, No. 3 (2022)

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Authors:Youngmin Park, Prashant Singh, Thomas G. Fai Pages: 793 - 820 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 3, Page 793-820, June 2022. We study the dynamics of membrane vesicle motor transport into dendritic spines, which are bulbous intracellular compartments in neurons that play a key role in transmitting signals between neurons. We consider the stochastic analogue of the vesicle transport model in [Park and Fai, Bull. Math. Biol., 82 (2020), pp. 1--31]. The stochastic version, which may be considered as an agent-based model, relies mostly on the action of individual myosin motors to produce vesicle motion. To aid in our analysis, we coarse-grain this agent-based model using a master equation combined with a partial differential equation describing the probability of local motor positions. We confirm through convergence studies that the coarse-graining captures the essential features of bistability in velocity (observed in experiments) and waiting-time distributions to switch between steady-state velocities. Interestingly, these results allow us to reformulate the translocation problem in terms of the mean first passage time for a run-and-tumble particle moving on a finite domain with absorbing boundaries at the two ends. We conclude by presenting numerical and analytical calculations of vesicle translocation. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-05-12T07:00:00Z DOI: 10.1137/21M1434180 Issue No:Vol. 82, No. 3 (2022)

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Authors:A. Liu, F. M. G. Magpantay Pages: 381 - 407 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 381-407, April 2022. The stability of equilibria and asymptotic behaviors of trajectories are often the primary focuses of mathematical modeling. However, many interesting phenomena that we would like to model, such as the “honeymoon period” of a disease after the onset of mass vaccination programs, are transient dynamics. Honeymoon periods can last for decades and can be important public health considerations. In many fields of science, especially in ecology, there is growing interest in a systematic study of transient dynamics. In this work, we attempt to provide a technical definition of “long transient dynamics” such as the honeymoon period and explain how these behaviors arise in systems of ordinary differential equations. We define a transient center, a point in state space that causes long transient behaviors, and derive some of its properties. In the end, we define reachable transient centers, which are transient centers that can be reached from initializations that do not need to be near the transient center. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-03-09T08:00:00Z DOI: 10.1137/20M1367131 Issue No:Vol. 82, No. 2 (2022)

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Authors:Sebastian Acosta, Gunther Uhlmann, Jian Zhai Pages: 408 - 426 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 408-426, April 2022. We consider the ultrasound imaging problem governed by a nonlinear wave equation of Westervelt type with variable wave speed. We show that the coefficient of nonlinearity can be recovered uniquely from knowledge of the Dirichlet-to-Neumann map. Our proof is based on a second order linearization and the use of Gaussian beam solutions to reduce the problem to the inversion of a weighted geodesic ray transform. We propose an inversion algorithm and report the results of a numerical implementation to solve the nonlinear ultrasound imaging problem in a transmission setting in the frequency domain. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-03-14T07:00:00Z DOI: 10.1137/21M1431813 Issue No:Vol. 82, No. 2 (2022)

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Authors:J. Dardé, N. Hyvönen, T. Kuutela, T. Valkonen Pages: 427 - 449 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 427-449, April 2022. Electrical impedance tomography is an imaging modality for extracting information on the interior structure of a physical body from boundary measurements of current and voltage. This work studies a new robust way of modeling the contact electrodes used for driving current patterns into the examined object and measuring the resulting voltages. The idea is to not define the electrodes as strict geometric objects on the measurement boundary but only to assume approximate knowledge about their whereabouts and let a boundary admittivity function determine the actual locations of the current inputs. Such an approach enables reconstructing the boundary admittivity, i.e., the locations and strengths of the contacts, at the same time and with analogous methods as the interior admittivity. The functionality of the new model is verified by two-dimensional numerical experiments based on water tank data. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-03-15T07:00:00Z DOI: 10.1137/21M1396125 Issue No:Vol. 82, No. 2 (2022)

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Authors:Mats K. Brun, Elyes Ahmed, Jan M. Nordbotten, Nils ChrStenseth Pages: 450 - 475 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 450-475, April 2022. This paper concerns the modeling and numerical simulation of the process of speciation. In particular, given conditions for which one or more speciation events within an ecosystem occur, our aim is to develop the necessary modeling and simulation tools. Care is also taken to establish a solid mathematical foundation on which our modeling framework is built. This is the subject of the first half of the paper. The second half is devoted to developing a multiscale framework for eco-evolutionary modeling, where the relevant scales are that of species and individual/population, respectively. The species level model we employ can be considered as an extension of the classical Lotka--Volterra model, where in addition to the species abundance, the model also governs the evolution of the species mean traits and species trait covariances and in this sense generalizes the purely ecological Lotka--Volterra model to an eco-evolutionary model. Although the model thus allows for evolving species, it does not (by construction) allow for the branching of species, i.e., speciation events. The reason for this is related to that of separate scales; the unit of species is too coarse to capture the fine-scale dynamics of a speciation event. Instead, the branching species should be regarded as a population of individuals moving along a selection of trait axes (i.e., trait-space). For this, we employ a trait-specific population density model governing the dynamics of the population density as a function of evolutionary traits. At this scale there is no a priori definition of species, but both species and speciation may be defined a posteriori as, e.g., local maxima and saddle points of the population density, respectively. Hence, a system of interacting species can be described at the species level, while for branching species a population level description is necessary. Our multiscale framework thus consists of coupling the species and population level models where speciation events are detected in advance and then resolved at the population scale until the branching is complete. Moreover, since the population level model is formulated as a PDE, we first establish the well-posedness in the time-discrete setting and then derive the a posteriori error estimates, which provides a fully computable upper bound on an energy-type error, including also for the case of general smooth distributions (which will be useful for the detection of speciation events). Several numerical tests validate our framework in practice. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-03-22T07:00:00Z DOI: 10.1137/21M1405228 Issue No:Vol. 82, No. 2 (2022)

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Authors:Philippe Marchner, Xavier Antoine, Christophe Geuzaine, Hadrien Bériot Pages: 476 - 501 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 476-501, April 2022. This article is devoted to the derivation and assessment of local absorbing boundary conditions (ABCs) for numerically solving heterogeneous time-harmonic acoustic problems. To this end, we develop a strategy inspired by the work of Engquist and Majda to build local approximations of the Dirichlet-to-Neumann operator for heterogeneous media, which is still an open problem. We focus on three simplified but characteristic examples of increasing complexity to highlight the strengths and weaknesses of the proposed ABCs: the propagation in a duct with a longitudinal variation of the speed of sound, the propagation in a nonuniform mean flow using a convected wave operator, and the propagation in a duct with a transverse variation of the speed of sound and density. For each case, we follow the same systematic approach to construct a family of local ABCs and explain their implementation in a high-order finite element context. Numerical simulations allow us to validate the accuracy of the ABCs and to give recommendations for the tuning of their parameters. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-03-22T07:00:00Z DOI: 10.1137/21M1414929 Issue No:Vol. 82, No. 2 (2022)

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Authors:Alessandro Ciallella, Emilio N. M. Cirillo, Barbara Vantaggi Pages: 502 - 525 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 502-525, April 2022. We show that residence time measures can be used to identify the geometrical and transmission properties of a defect along a path. The model we study is based on a one-dimensional simple random walk. The sites of the lattice are regular, i.e., the jumping probabilities are the same in each site, except for a site, called defect, where the jumping probabilities are different. At each side of the lattice an absorbing site is present. We show that by measuring the fraction of particles crossing the channel and/or the typical time they need to cross it, it is possible to identify the main features of the lattice and of the defect site, namely, the jumping probabilities at the regular and at the defect site and the position of the defect in the lattice. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-03-24T07:00:00Z DOI: 10.1137/20M1380284 Issue No:Vol. 82, No. 2 (2022)

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Authors:Maciej Buze, Thoms E. Woolley, Angela Mihai Pages: 526 - 548 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 526-548, April 2022. We present a stochastic modeling framework for atomistic propagation of a Mode I surface crack, with atoms interacting according to the Lennard--Jones interatomic potential at zero temperature. Specifically, we invoke the Cauchy--Born rule and the maximum entropy principle to infer probability distributions for the parameters of the interatomic potential. We then study how uncertainties in the parameters propagate to the quantities of interest relevant to crack propagation, namely, the critical stress intensity factor and the lattice trapping range. For our numerical investigation, we rely on an automated version of the so-called numerical-continuation enhanced flexible boundary NCFlex algorithm. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-03-29T07:00:00Z DOI: 10.1137/21M1416436 Issue No:Vol. 82, No. 2 (2022)

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Authors:Michal Beneš, Miroslav Kolář, Daniel Ševčovič Pages: 549 - 575 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 549-575, April 2022. In this article we investigate a system of geometric evolution equations describing a curvature driven motion of a family of three-dimensional curves in the normal and binormal directions. Evolving curves may be the subject of mutual interactions having both local or nonlocal character where the entire curve may influence evolution of other curves. Such an evolution and interaction can be found in applications. We explore the direct Lagrangian approach for treating the geometric flow of such interacting curves. Using the abstract theory of nonlinear analytic semiflows, we are able to prove local existence, uniqueness, and continuation of classical Hölder smooth solutions to the governing system of nonlinear parabolic equations. Using the finite volume method, we construct an efficient numerical scheme solving the governing system of nonlinear parabolic equations. Additionally, a nontrivial tangential velocity is considered allowing for redistribution of discretization nodes. We also present several computational studies of the flow combining the normal and binormal velocity and considering nonlocal interactions. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-03-29T07:00:00Z DOI: 10.1137/21M1417181 Issue No:Vol. 82, No. 2 (2022)

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Authors:Shuang Liu, Yuan Lou, Pengfei Song Pages: 576 - 601 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 576-601, April 2022. A new monotonocity of principal eigenvalues in time-periodic patch environments is established. As an application, a patch model for two competing species in spatio-temporally varying environments is investigated. When two species are identical except for their relaxation time, the species with the shorter relaxation time will always drive the other one to extinction. When two species are identical except for their diffusion rates, our results suggest that the faster diffusing species could be favored for some intermediate range of relaxation time, while the slower diffusing species will be favored for both short and long relaxation time. In general, short relaxation time and slow diffusion rate tend to help species gain advantage in competition. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-04-05T07:00:00Z DOI: 10.1137/20M1320973 Issue No:Vol. 82, No. 2 (2022)

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Authors:Ru-Yu Lai, Kui Ren, Ting Zhou Pages: 602 - 624 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 602-624, April 2022. Motivated by applications in imaging nonlinear optical absorption by photoacoustic tomography, we study in this work inverse coefficient problems for a semilinear radiative transport equation and its diffusion approximation with internal data that are functionals of the coefficients and the solutions to the equations. Based on the techniques of first- and second-order linearization, we derive uniqueness and stability results for the inverse problems. For uncertainty quantification purposes, we also establish the stability of the reconstruction of the absorption coefficients with respect to the change in the scattering coefficient. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-04-14T07:00:00Z DOI: 10.1137/21M1436178 Issue No:Vol. 82, No. 2 (2022)

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Authors:Toby L. Kirk, Jack Evans, Colin P. Please, S. Jonathan Chapman Pages: 625 - 653 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 625-653, April 2022. In mathematical models of lithium-ion batteries, the highly heterogeneous porous electrodes are frequently approximated as comprising spherical particles of uniform size, leading to the commonly used single-particle model (SPM) when transport in the electrolyte is assumed to be fast. Here electrode heterogeneity is modeled by extending this to a distribution of particle sizes. Unimodal and bimodal particle-size distributions (PSD) are considered. For a unimodal PSD, the effect of the spread of the distribution on the cell dynamics is investigated, and choice of effective particle radius when approximating by an SPM assessed. Asymptotic techniques are used to derive a correction to the SPM valid for narrow, but realistic, PSDs. In addition, it is shown that the heterogeneous internal states of all particles (relevant when modeling degradation, for example) can be efficiently computed after the fact. For a bimodal PSD, the results are well approximated by a double-particle model (DPM), with one size representing each mode. Results for lithium iron phosphate with a bimodal PSD show that the DPM captures an experimentally observed double plateau in the discharge curve, suggesting it is entirely due to bimodality. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-04-14T07:00:00Z DOI: 10.1137/20M1344305S Issue No:Vol. 82, No. 2 (2022)

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Authors:Shanshan Chen, Junping Shi, Zhisheng Shuai, Yixiang Wu Pages: 654 - 676 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 654-676, April 2022. Threshold values in population dynamics can be formulated as spectral bounds of matrices, determining the dichotomy of population persistence and extinction. For a square matrix $\rho A + Q$, where $A$ is an essentially nonnegative matrix describing population dispersal among patches in a heterogeneous environment and $Q$ is a real diagonal matrix encoding within-patch population dynamics, the monotonicity of its spectral bound with respect to dispersal rate/coupling strength/travel frequency $\rho$ has been established by Karlin and generalized by Altenberg while investigating the reduction principle in evolution biology and evolution dispersal in patchy landscapes. In this paper, we provide two new proofs rooted in our investigation of persistence in spatial population dynamics. The first one is an analytic derivation utilizing a graph-theoretic approach based on Kirchhoff's matrix-tree theorem; the second one employs the Collatz--Wielandt formula from matrix theory and complex analysis arguments. This monotonicity result has numerous applications in persistence and stability analysis of complex biological systems in heterogeneous environments. We illustrate this by applying it to well-known ecological models of single species, predator-prey, and competition. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-04-21T07:00:00Z DOI: 10.1137/20M1345220 Issue No:Vol. 82, No. 2 (2022)

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Authors:Michael Vynnycky, Sean McKee Pages: 677 - 693 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 677-693, April 2022. We revisit the analysis of a transient nonlinear reaction-diffusion equation describing two-particle coalescence or, alternatively, annihilation in an infinite domain. On nondimensionalizing the governing equations, we find that the problem is, in general, free of dimensionless parameters, indicating that the original work, which was purported to be for the slow reaction limit, is instead better interpreted as being for small times. By means of analysis using similarity-like variables, we also find that the delta-peaked initial condition used in the original analysis leads to an unexpected and easily overlooked restriction on the reaction order, which cannot be greater than 3; the same applies for two-term regular perturbation expansions that are valid for short times, and which we also determine. The full equations are then solved numerically for all times and the solutions are found to agree well with the analytical results that are available for short times. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-04-21T07:00:00Z DOI: 10.1137/21M140701X Issue No:Vol. 82, No. 2 (2022)

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Authors:James Dalby, Patrick E. Farrell, Apala Majumdar, Jingmin Xia Pages: 694 - 719 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 694-719, April 2022. We study a model system with nematic and magnetic order, within a channel geometry modeled by an interval, $[-D, D]$. The system is characterized by a tensor-valued nematic order parameter ${{Q}}$ and a vector-valued magnetization ${{M}}$, and the observable states are modeled as stable critical points of an appropriately defined free energy which includes a nemato-magnetic coupling term, characterized by a parameter $c$. We (i) derive $L^\infty$ bounds for ${{Q}}$ and ${{M}}$; (ii) prove a uniqueness result in specified parameter regimes; (iii) analyze order reconstruction solutions, possessing domain walls, and their stabilities as a function of $D$ and $c$ and; (iv) perform numerical studies that elucidate the interplay of $c$ and $D$ for multistability. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-04-27T07:00:00Z DOI: 10.1137/21M1400171 Issue No:Vol. 82, No. 2 (2022)

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Authors:Qingle Meng, Zhengjian Bai, Huaian Diao, Hongyu Liu Pages: 720 - 749 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 720-749, April 2022. We consider the time-harmonic elastic wave scattering from a general (possibly anisotropic) inhomogeneous medium with an embedded impenetrable obstacle. We show that the impenetrable obstacle can be effectively approximated by an isotropic elastic medium with a particular choice of material parameters. We derive sharp estimates to rigorously verify such an effective approximation. Our study is strongly motivated by the related studies of two challenging inverse elastic problems including the inverse boundary problem with partial data and the inverse scattering problem of recovering mediums with buried obstacles. The proposed effective medium theory readily yields some interesting applications of practical significance to these inverse problems. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-04-27T07:00:00Z DOI: 10.1137/21M1431369 Issue No:Vol. 82, No. 2 (2022)

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Authors:Rodney O. Fox, Frédérique Laurent Pages: 750 - 771 Abstract: SIAM Journal on Applied Mathematics, Volume 82, Issue 2, Page 750-771, April 2022. A tractable solution is proposed to a classical problem in kinetic theory, namely, given any set of realizable velocity moments up to order $2n$, a closure for the moment of order $2n+1$ is constructed for which the moment system found from the free-transport term in the one-dimensional (1-D) kinetic equation is globally hyperbolic and in conservative form. In prior work, the hyperbolic quadrature method of moments (HyQMOM) was introduced to close this moment system up to fourth order ($n \le 2$). Here, HyQMOM is reformulated and extended to arbitrary even-order moments. The HyQMOM closure is defined based on the properties of the monic orthogonal polynomials $Q_n$ that are uniquely defined by the velocity moments up to order $2n-1$. Thus, HyQMOM is strictly a moment closure and does not rely on the reconstruction of a velocity distribution function with the same moments. On the boundary of moment space, $n$ double roots of the characteristic polynomial $P_{2n+1}$ of the Jacobian matrix of the system are the roots of $Q_n$, while in the interior, $P_{2n+1}$ and $Q_n$ share $n$ roots. The remaining $n+1$ roots of $P_{2n+1}$ bound and separate the roots of $Q_n$. An efficient algorithm, based on the Chebyshev algorithm, for computing the moment of order $2n+1$ from the moments up to order $2n$ is developed. The analytical solution to a 1-D Riemann problem is used to demonstrate convergence of the HyQMOM closure with increasing $n$. Citation: SIAM Journal on Applied Mathematics PubDate: 2022-04-28T07:00:00Z DOI: 10.1137/21M1406143 Issue No:Vol. 82, No. 2 (2022)