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Stability of stationary inverse transport equation in diffusion scaling

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arXiv 2017年

作者： Chen, Ke Li, Qin Wang, Li Mathematics Department University of Wisconsin-Madison 480 Lincoln Dr. MadisonWI53705 United States Department of Mathematics Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo 244 Mathematics Building BuffaloNY14206 United States

We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales characterized by the magnitude of a dimensionless parameter-the Knudsen number (Kn). In the diffusive scaling (Kn 1), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well-posedness of inverse transport equation, which raises the question of how uniqueness being lost as Kn → 0. We tackle this problem by examining the stability of inverse problem with varying Kn. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of Knp (p = 1 or 2), and as a result lead to ill-posedness in the zero limit of Kn. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings. Copyright © 2017, The Authors. All rights reserved.

关键词： Parameter estimation

A NEW NUMERICAL APPROACH TO INVERSE TRANSPORT EQUATION WITH ERROR ANALYSIS

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arXiv 2017年

作者： Li, Qin Shu, Ruiwen Wang, Li Mathematics Department University of Wisconsin-Madison 480 Lincoln Dr. MadisonWI53705 United States Department of Mathematics Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo 244 Mathematics Building BuffaloNY14206 United States

The inverse radiative transfer problem finds broad applications in medical imaging, atmospheric science, astronomy, and many other areas. This problem intends to recover the optical properties, denoted as absorption and scattering coefficient of the media, through the source-measurement pairs. A typical computational approach is to form the inverse problem as a PDE-constraint optimization, with the minimizer being the to-be-recovered coefficients. The method is tested to be efficient in practice, but lacks analytical justification: there is no guarantee of the existence or uniqueness of the minimizer, and the error is hard to quantify. In this paper, we provide a different algorithm by levering the ideas from singular decomposition analysis. Our approach is to decompose the measurements into three components, two out of which encode the information of the two coefficients respectively. We then split the optimization problem into two subproblems and use those two components to recover the absorption and scattering coefficients separately. In this regard, we prove the well-posedness of the new optimization, and the error could be quantified with better precision. In the end, we incorporate the diffusive scaling and show that the error is harder to control in the diffusive limit. Copyright © 2017, The Authors. All rights reserved.

关键词： Recovery

Randomizing hypergraphs preserving degree correlation and local clustering

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arXiv 2021年

作者： Nakajima, Kazuki Shudo, Kazuyuki Masuda, Naoki Department of Mathematical and Computing Science Tokyo Institute of Technology Japan Department of Mathematics State University of New York Buffalo United States Computational and Data-Enabled Science and Engineering Program State University of New York Buffalo United States Faculty of Science and Engineering Waseda University Japan

Many complex systems involve direct interactions among more than two entities and can be represented by hypergraphs, in which hyperedges encode higher-order interactions among an arbitrary number of nodes. To analyze structures and dynamics of given hypergraphs, a solid practice is to compare them with those for randomized hypergraphs that preserve some specific properties of the original hypergraphs. In the present study, we propose a family of such reference models for hypergraphs, called the hyper dk-series, by extending the so-called dk-series for dyadic networks to the case of hypergraphs. The hyper dk-series preserves up to the individual node’s degree, node’s degree correlation, node’s redundancy coefficient, and/or the hyperedge’s size depending on the parameter values. Furthermore, we numerically find that higher-order hyper dk-series more accurately preserves the shortest path length and degree distribution of the one-mode projection of the original hypergraph, which the method does not intend to preserve. We also apply the hyper dk-series to numerical simulations of epidemic spreading and evolutionary game dynamics on empirical social hypergraphs. We find that the hyperedge’s size affects these dynamics more than any of the node’s properties and that the node’s degree correlation and redundancy in the empirical hypergraphs promote cooperation. Copyright © 2021, The Authors. All rights reserved.

关键词： Redundancy

Early Warnings for Multistage Transitions in Dynamics on Networks

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arXiv 2022年

作者： MacLaren, Neil G. Kundu, Prosenjit Masuda, Naoki Department of Mathematics State University of New York BuffaloNY14260-2900 United States Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo BuffaloNY14260-5030 United States

Successfully anticipating sudden major changes in complex systems is a practical concern. Such complex systems often form a heterogeneous network, which may show multistage transitions in which some nodes experience a regime shift earlier than others as an environment gradually changes. Here we investigate early warning signals for networked systems undergoing a multistage transition. We found that knowledge of both the ongoing multistage transition and network structure enables us to calculate effective early warning signals for multistage transitions. Furthermore, we found that small subsets of nodes could anticipate transitions as well as or even better than using all the nodes. Even if we fix the network and dynamical system, no single best subset of nodes provides good early warning signals, and a good choice of sentinel nodes depends on the tipping direction and the current stage of the dynamics within a multistage transition, which we systematically characterize. © 2022, CC BY.

关键词： Dynamics

Opinion dynamics on tie-decay networks

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Physical Review Research 2021年第2期3卷 023249-023249页

作者： Kashin Sugishita Mason A. Porter Mariano Beguerisse-Díaz Naoki Masuda Department of Mathematics State University of New York at Buffalo Buffalo New York 14260-2900 USA Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo Buffalo New York 14260-5030 USA and Faculty of Science and Engineering Waseda University 169-8555 Tokyo Japan

In social networks, interaction patterns typically change over time. We study opinion dynamics on tie-decay networks in which tie strength increases instantaneously when there is an interaction and decays exponentially between interactions. Specifically, we formulate continuous-time Laplacian dynamics and a discrete-time DeGroot model of opinion dynamics on these tie-decay networks, and we carry out numerical computations for the continuous-time Laplacian dynamics. We examine the speed of convergence by studying the spectral gaps of combinatorial Laplacian matrices of tie-decay networks. First, we compare the spectral gaps of the Laplacian matrices of tie-decay networks that we construct from empirical data with the spectral gaps for corresponding randomized and aggregate networks. We find that the spectral gaps for the empirical networks tend to be smaller than those for the randomized and aggregate networks. Second, we study the spectral gap as a function of the tie-decay rate and time. Intuitively, we expect small tie-decay rates to lead to fast convergence because the influence of each interaction between two nodes lasts longer for smaller decay rates. Moreover, as time progresses and more interactions occur, we expect eventual convergence. However, we demonstrate that the spectral gap need not decrease monotonically with respect to the decay rate or increase monotonically with respect to time. Our results highlight the importance of the interplay between the times that edges strengthen and decay in temporal networks.

关键词： Network structure Social networks Consensus & non-consensus models

Dimension reduction of dynamical systems on networks with leading and non-leading eigenvectors of adjacency matrices

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arXiv 2022年

作者： Masuda, Naoki Kundu, Prosenjit Department of Mathematics State University of New York BuffaloNY14260-2900 United States Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo BuffaloNY14260-5030 United States

Dimension reduction techniques for dynamical systems on networks are considered to promote our understanding of the original high-dimensional dynamics. One strategy of dimension reduction is to derive a low-dimensional dynamical system whose behavior approximates the observables of the original dynamical system that are weighted linear summations of the state variables at the different nodes. Recently proposed methods use the leading eigenvector of the adjacency matrix of the network as the mixture weights to obtain such observables. In the present study, we explore performances of this type of one-dimensional reductions of dynamical systems on networks when we use non-leading eigenvectors of the adjacency matrix as the mixture weights. Our theory predicts that non-leading eigenvectors can be more efficient than the leading eigenvector and enables us to select the eigenvector minimizing the error. We numerically verify that the optimal non-leading eigenvector outperforms the leading eigenvector for some dynamical systems and networks. We also argue that, despite our theory, it is practically better to use the leading eigenvector as the mixture weights to avoid misplacing the bifurcation point too distantly and to be resistant against dynamical noise. © 2022, CC BY.

关键词： Eigenvalues and eigenfunctions

Metapopulation models imply non-Poissonian statistics of interevent times

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Physical Review Research 2022年第1期4卷 013050-013050页

作者： Elohim Fonseca dos Reis Naoki Masuda Department of Mathematics State University of New York at Buffalo Buffalo 14260-2900 New York USA Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo Buffalo 14260-5030 New York USA Faculty of Science and Engineering Waseda University Tokyo 169-8555 Japan

Interevent times in temporal contact data from humans and animals typically obey heavy-tailed distributions, which impacts contagion and other dynamical processes on networks. We theoretically show that distributions of interevent times heavier-tailed than exponential distributions are a consequence of the most basic metapopulation model used in epidemiology and ecology, in which individuals move from one patch to another according to the simple random walk. Our results hold true irrespective of the network structure and also for more realistic mobility rules such as high-order random walks and the recurrent mobility patterns used for modeling human dynamics.

关键词： Continuous-time random walk Population dynamics Networks & random structures Social networks Diffusion & random walks Markovian processes Non-Markovian processes

Fixation probability in evolutionary dynamics on switching temporal networks

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arXiv 2023年

作者： Bhaumik, Jnanajyoti Masuda, Naoki Department of Mathematics State University of New York BuffaloNY14260-2900 United States Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo BuffaloNY14260-5030 United States

Population structure has been known to substantially affect evolutionary dynamics. Networks that promote the spreading of fitter mutants are called amplifiers of selection, and those that suppress the spreading of fitter mutants are called suppressors of selection. Research in the past two decades has found various families of amplifiers while suppressors still remain somewhat elusive. It has also been discovered that most networks are amplifiers of selection under the birth-death updating combined with uniform initialization, which is a standard condition assumed widely in the literature. In the present study, we extend the birth-death processes to temporal (i.e., time-varying) networks. For the sake of tractability, we restrict ourselves to switching temporal networks, in which the network structure deterministically alternates between two static networks at constant time intervals or stochastically in a Markovian manner. We show that, in a majority of cases, switching networks are less amplifying than both of the two static networks constituting the switching networks. Furthermore, most small switching networks, i.e., networks on six nodes or less, are suppressors, which contrasts to the case of static networks. © 2023, CC BY.

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Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics

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arXiv 2018年

作者： Liu, Jian-Guo Tang, Min Wang, Li Zhou, Zhennan Department of Mathematics Department of Physics Duke University School of Mathematics and Institute of Natural Sciences MOE-LSC Shanghai JiaoTong University Department of Mathematics Computational and Data-Enabled Science and Engineering Program State University of New York Buffalo United States Beijing International Center for Mathematical Research Peking University

In this paper, we study the tumor growth equation along with various models for the nutrient component, including the in vitro model and the in vivo model. At the cell density level, the spatial availability of the tumor density n is governed by the Darcy law via the pressure p\(n\) = nγ. For finite γ, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As γ → ∞, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical *** Codes 35K55, 35B25, 76D27, 92C50 Copyright © 2018, The Authors. All rights reserved.

关键词： Tumors