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Iteratively Adaptive Identification of Parameters Solely Using Short-Term Data of Transient Dynamics
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| Pengarang | : | Huan-Fei Ma, Wei Lin |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 1) |
| Halaman | : | 393-411 |
| Abstrak | : | Identifying parameters in nonlinear dynamical systems, based solely on observational time series data, is of paramount importance to modeling, understanding, and predicting complex systems. While the adaptive synchronization technique has been successfully used for various identification tasks in the existing literature, its rigorousness hinges crucially on the prerequisite of the linear independence (LI) behaviors of the data converging to the limit set or attractor. This prerequisite is not always satisfied in real applications, since only the short-term data of transient dynamics produced by the systems under consideration is available. Here, to conquer this challenge, we introduce an iteratively adaptive framework designed to operate efficiently on short-term data of transient dynamics. We conduct theoretical analysis to demonstrate its rigorousness, and we offer a few representative examples from toy models to a real-world dataset to illustrate its practical usefulness and efficacy. In particular, an introduction of deliberately designed control or random perturbations renders the LI behaviors more attainable solely on transient dynamics. Consequently, successful identification of a substantial number of parameters in complex systems using short-term data becomes achievable, which significantly broadens the potential application scenarios of our work. |
Dynamic Behaviors of a Periodic System with Threshold Policy-Guided Periodic and Intermittent Therapy of Tumor
| Pengarang | : | Biao Tang, Yanni Xiao, Jianhong Wu |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 1) |
| Halaman | : | 366-392 |
| Abstrak | : | Recent clinical studies provide strong evidence that adaptive tumor therapy is a promising threshold policy-guided periodic and intermittent treatment for prolonging treatment efficacy particularly for prostate cancer, despite the observed tumor size fluctuations in the patients. Understanding plausible patterns of tumor size fluctuations under different treatment scenarios is important to evaluate the therapy outcomes. Here we use state-of-the-art modeling and analytic frameworks of periodic switching systems and state-dependent switching systems to develop a dynamic model that mimics the adaptive therapy, and we describe dynamical behaviors of the model system with a particular focus on the patterns of periodic fluctuation. Under the threshold policy, patients may not be given therapies during the a priori treatment period. We show that this threshold-guided treatment will give rise to a new type of periodic solutions with complex structures, characterized as \((m,k)T\)-periodic solutions. We examine, both theoretically and numerically, the existence, as well as the local and global stability of such periodic solutions, including the boundary periodic solutions with one vanished compartment and positive \((m, k)T\)-periodic solutions. It is hoped this study also shows the great potential in developing a class of models to describe threshold-triggered periodic and intermittent control in many fields, which in turn may generate many new dynamic behaviors of nonsmooth dynamic systems. |
Synchronized Optimal Transport for Joint Modeling of Dynamics Across Multiple Spaces
| Pengarang | : | Zixuan Cang, Yanxiang Zhao |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 1) |
| Halaman | : | 341-365 |
| Abstrak | : | Optimal transport has been an essential tool for reconstructing dynamics from complex data. With the increasingly available multifaceted data, a system can often be characterized across multiple spaces. Therefore, it is crucial to maintain coherence in the dynamics across these diverse spaces. To address this challenge, we introduce synchronized optimal transport (SyncOT), a novel approach to jointly model dynamics that represent the same system through multiple spaces. Given the correspondence between the spaces, SyncOT minimizes the aggregated cost of the dynamics induced across all considered spaces. The problem is discretized into a finite-dimensional convex problem using a staggered grid. Primal-dual algorithm-based approaches are then developed to solve the discretized problem. Various numerical experiments demonstrate the capabilities and properties of SyncOT and validate the effectiveness of the proposed algorithms. |
Radiative Transport in a Periodic Structure with Band Crossings
| Pengarang | : | Kunlun Qi, Li Wang, Alexander B. Watson |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 1) |
| Halaman | : | 314-340 |
| Abstrak | : | We use the Wigner transformation and asymptotic analysis to systematically derive the semiclassical model for the Schrödinger equation in arbitrary spatial dimensions, with any periodic structure. Our particular emphasis lies in addressing the diabatic effect, i.e., the impact of Bloch band crossings. We consider both deterministic and random scenarios. In the former case, we derive a coupled Liouville system, revealing lower-order interactions among different Bloch bands. In the latter case, a coupled system of radiative transport equations emerges, with the scattering cross section induced by the random inhomogeneities. As a specific application, we deduce the effective dynamics of a wave packet in graphene with randomness. |
Asymptotic Relaxation of Moment Equations for a Multi-species, Homogeneous BGK Model
| Pengarang | : | Evan Habbershaw, Ryan S. Glasby, Jeffrey R. Haack, Cory D. Hauck, Steven M. Wise |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 1) |
| Halaman | : | 294-313 |
| Abstrak | : | Multi-species BGK models describe the dynamics of rarefied gases with constituent particles of different elements or compounds with potentially nontrivial velocity distributions. In this paper, moment equations for the bulk velocities, energies, and temperatures of a spatially homogeneous multi-species BGK model are examined. A key challenge in analyzing these equations is the fact that the collision frequencies are allowed to depend on the species temperatures, which allows for more realistic simulations of dilute gas flow. Therefore, a positive lower bound is established for the species temperatures. With this lower bound, a global existence and uniqueness of solutions to the coupled velocity-energy ODE system is established. The lower bound also enables a proof of exponential decay to a unique steady-state solution. Numerical results are presented to demonstrate how the bulk velocities and temperatures relax for large times. |
An Inverse Boundary Value Problem for the Inhomogeneous Porous Medium Equation
| Pengarang | : | Catalin I. Carstea, Tuhin Ghosh, Gen Nakamura |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 1) |
| Halaman | : | 278-293 |
| Abstrak | : | In this paper we establish uniqueness in the inverse boundary value problem for the porous medium equation ????−∇⋅(?∇??)=0, which is a degenerate parabolic type quasilinear PDE. We assume that ?>1, which is sometimes referred to as the slow diffusion case. Under these assumptions we show that the corresponding Dirichlet-to-Neumann map determines the two coefficients ? and ?. Our approach relies on using a Laplace transform to turn the original equation into a coupled family of nonlinear elliptic equations, indexed by the frequency parameter (1/? in our definition) of the transform. A careful analysis of the asymptotic expansion in powers of ?, as ?→∞, of the solutions to the transformed equation, with special boundary data, allows us to obtain sufficient information to deduce the uniqueness result. |
Solving Stationary Nonlinear Fokker–Planck Equations via Sampling
| Pengarang | : | Lei Li, Yijia Tang, Jingtong Zhang |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 1) |
| Halaman | : | 249-277 |
| Abstrak | : | Solving the stationary nonlinear Fokker–Planck equations is important in applications and examples include the Poisson–Boltzmann equation and the two-layer neural networks. Making use of the connection between the interacting particle systems and the nonlinear Fokker–Planck equations, we propose solving the stationary solution by sampling from the ?-body Gibbs distribution. This avoids simulation of the ?-body system for a long time and the requirement of uniform propagation of chaos from direct simulation of the particle systems. While the sampling strategy could be used for any given temperature, we establish the convergence of the Gibbs measure to the stationary solution when the interaction kernel is bounded (not necessarily continuous) and the temperature is high enough. Numerical experiments are performed for the Poisson–Boltzmann equation and the two-layer neural networks to validate the method and the theory. |
Geometry Based Approximation of Waves in Complex Domains
| Pengarang | : | Davide Pradovera, Monica Nonino, Ilaria Perugia |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 1) |
| Halaman | : | 224-248 |
| Abstrak | : | We consider wave propagation problems over two-dimensional domains with piecewise-linear boundaries, possibly including scatterers. We assume that the wave speed is constant and that the initial conditions and forcing terms are radially symmetric and compactly supported. We propose an approximation of the propagating wave as the sum of some special space-time functions. Each term in this sum identifies a particular field component, modeling the result of a single reflection or diffraction effect. We describe an algorithm for identifying such components automatically based on the domain geometry. To showcase our proposed method, we present several numerical examples, such as waves scattering off wedges and waves propagating through a room in the presence of obstacles. Software implementing our numerical algorithm is made available as open-source code. |
Linearized Calderón Problem: Reconstruction of Unbounded Perturbations in Three Dimensions
| Pengarang | : | Henrik Garde, Markus Hirvensalo |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 1) |
| Halaman | : | 210-223 |
| Abstrak | : | Recently, an algorithm was given in Garde and Hyvönen [SIAM J. Math. Anal., 56 (2024), pp. 3588–3604] for exact direct reconstruction of any ?2 perturbation from linearized data in the two-dimensional linearized Calderón problem. It was a simple forward substitution method based on a two-dimensional Zernike basis. We now consider the three-dimensional linearized Calderón problem in a ball and use a three-dimensional Zernike basis to obtain a method for exact direct reconstruction of any ?3 perturbation from linearized data. The method is likewise a forward substitution, hence making it very efficient to numerically implement. Moreover, the three-dimensional method only makes use of a relatively small subset of boundary measurements for exact reconstruction compared to a full ?2 basis of current densities. |
Far-from-Equilibrium Traveling Pulses in Sloped Semiarid Environments Driven by Autotoxicity Effects
| Pengarang | : | Gabriele Grifo Annalisa Iuorio, Frits Veerman |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 1) |
| Halaman | : | 188-209 |
| Abstrak | : | In this work, we study the influence of autotoxicity on vegetation patterns, by studying the existence and properties of traveling vegetation pulses. To that end, we consider an extension of the one-dimensional Klausmeier model that accounts for the toxicity compounds. Numerical simulations are first conducted to capture the qualitative behaviors of the pulse–type solutions and, then, geometric singular perturbation theory is used to prove the existence of such traveling pulses by constructing the corresponding homoclinic orbits in the associated four-dimensional system. A scaling analysis on the investigated model is performed to identify the asymptotic scaling regime in which traveling pulses can be constructed. Interestingly, due to the autotoxicity extension, the use of geometric blowup techniques can be avoided, in contrast with previous work [P. Carter and A. Doelman, SIAM J. Appl. Math., 78 (2023), pp. 3213–3237]. Some biological observations are extracted from the analytical results and the role of autotoxicity in traveling patterns is emphasized. Finally, the analytically constructed solutions are compared with the numerical ones, leading to a good agreement that confirms the validity of the conducted analysis. Numerical investigations are also carried out in order to gain additional information on vegetation dynamics. |