Kevin Lin
 Associate Professor, Mathematics
 Associate Professor, Applied Mathematics  GIDP
 Associate Professor, Cognitive Science  GIDP
 Associate Professor, StatisticsGIDP
Contact
 (520) 6266628
 Mathematics, Rm. 606
 Tucson, AZ 85721
 klin@math.arizona.edu
Degrees
 Ph.D. Mathematics
 University of California at Berkeley, Berkeley, California, United States
 Random perturbations of SRB measures and numerical studies of chaotic dynamics
 B.S. Computer Science and Engineering
 Massachusetts Institute of Technology, Cambridge, Massachusetts, United States
 M.S. Electrical Engineering and Computer Science
 Massachusetts Institute of Technology, Cambridge, Massachusetts, United States
 CoordinateIndependent Computations on Differential Equations
 B.S. Mathematics
 Massachusetts Institute of Technology, Cambridge, Massachusetts, United States
Interests
Teaching
Dynamical systems; applied mathematics; computing.
Research
Dynamical systems; computation.
Courses
201718 Courses

Dissertation
MATH 920 (Spring 2018) 
Dynamical Systems+Chaos
MATH 557B (Spring 2018) 
Independent Study
MATH 599 (Spring 2018) 
Journal Club
APPL 595B (Spring 2018) 
Dissertation
MATH 920 (Fall 2017) 
Dynamical Systems+Chaos
MATH 557A (Fall 2017) 
Independent Study
MATH 599 (Fall 2017) 
Intro:Stat+Biostatistics
MATH 263 (Fall 2017)
201617 Courses

Dissertation
MATH 920 (Spring 2017) 
Honors Thesis
MATH 498H (Spring 2017) 
Journal Club
APPL 595B (Spring 2017) 
Calculus II
MATH 129 (Fall 2016) 
Dissertation
MATH 920 (Fall 2016) 
Honors Thesis
MATH 498H (Fall 2016) 
Intro Ord Diff Equations
MATH 254 (Fall 2016) 
Journal Club
APPL 595B (Fall 2016)
Scholarly Contributions
Journals/Publications
 Lajoie, G., Lin, K., Thivierge, J., & SheaBrown, E. (2016). Encoding in balanced networks: revisiting spike patterns and chaos in stimulusdriven systems. PLOS Computational Biology.
 Lu, F., Lin, K., & Chorin, A. (2016). "Comparison of continuous and discretetime databased modeling for hypoelliptic systems. Communications in Applied Mathematics and Computational Science, 11, 187216. doi:10.2140/camcos.2016.11.187
 Lu, F., Lin, K., & Chorin, A. (2017). Databased stochastic model reduction for the KuramotoSivashinsky equation. Physica D.
 Barrat, A., Fernandez, B., Lin, K. K., & Young, L. (2013). Modeling temporal networks using random itineraries. Physical Review Letters, 110(15).More infoAbstract: We propose a procedure to generate dynamical networks with bursty, possibly repetitive and correlated temporal behaviors. Regarding any weighted directed graph as being composed of the accumulation of paths between its nodes, our construction uses random walks of variable length to produce timeextended structures with adjustable features. The procedure is first described in a general framework. It is then illustrated in a case study inspired by a transportation system for which the resulting synthetic network is shown to accurately mimic the empirical phenomenology. © 2013 American Physical Society.
 Goodman, J., Lin, K. K., & Morzfeld, M. (2014). Smallnoise analysis and symmetrization of implicit Monte Carlo samplers. Communications in Pure and Applied Mathematics.More infoThis paper is about a new class of sampling algorithms, called implicit samplers, introduced by Chorin and Tu. It was accepted for publication in February 2015.
 Hooker, G., Lin, K. K., & Rogers, B. (2014). Control theory and experimental design in diffusion processes. SIAM Journal on Uncertainty Quantification.More infoThis paper is about using stochastic optimal control methods to design adaptive experimental protocols so as to improve the information yield of dynamic experiments (such as those done in neuroscience and in microorganism population biology). It was accepted for publication in January 2015.
 Lajoie, G., Lin, K. K., & SheaBrown, E. (2013). Chaos and reliability in balanced spiking networks with temporal drive. Physical Review E  Statistical, Nonlinear, and Soft Matter Physics, 87(5).More infoAbstract: Biological information processing is often carried out by complex networks of interconnected dynamical units. A basic question about such networks is that of reliability: If the same signal is presented many times with the network in different initial states, will the system entrain to the signal in a repeatable way? Reliability is of particular interest in neuroscience, where large, complex networks of excitatory and inhibitory cells are ubiquitous. These networks are known to autonomously produce strongly chaotic dynamics  an obvious threat to reliability. Here, we show that such chaos persists in the presence of weak and strong stimuli, but that even in the presence of chaos, intermittent periods of highly reliable spiking often coexist with unreliable activity. We elucidate the local dynamical mechanisms involved in this intermittent reliability, and investigate the relationship between this phenomenon and certain timedependent attractors arising from the dynamics. A conclusion is that chaotic dynamics do not have to be an obstacle to precise spike responses, a fact with implications for signal coding in large networks. © 2013 American Physical Society.
 Lin, K. K., Wedgwood, K. C., Coombes, S., & Young, L. (2013). Limitations of perturbative techniques in the analysis of rhythms and oscillations. Journal of Mathematical Biology, 66(12), 139161.More infoPMID: 22290314;Abstract: Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are "sufficiently weak", an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shearinduced chaos, i. e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of "sticky" phasespace structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience. © 2012 The Author(s).
 Wedgwood, K. C., Lin, K. K., Thul, R., & Coombes, S. (2013). Phaseamplitude descriptions of neural oscillator models. Journal of Mathematical Neuroscience, 3(1), 122.More infoPMID: 23347723;PMCID: PMC3582465;Abstract: Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to weak perturbations is now particularly well advanced, and has allowed for the development of a theory of weakly connected neural networks. However, the strongattraction assumption may well not be the natural one for many neural oscillator models. For example, the popular conductance based MorrisLecar model is known to respond to periodic pulsatile stimulation in a chaotic fashion that cannot be adequately described with a phase reduction. In this paper, we generalise the phase description that allows one to track the evolution of distance from the cycle as well as phase on cycle. We use a classical technique from the theory of ordinary differential equations that makes use of a moving coordinate system to analyse periodic orbits. The subsequent phaseamplitude description is shown to be very well suited to understanding the response of the oscillator to external stimuli (which are not necessarily weak). We consider a number of examples of neural oscillator models, ranging from planar through to high dimensional models, to illustrate the effectiveness of this approach in providing an improvement over the standard phasereduction technique. As an explicit application of this phaseamplitude framework, we consider in some detail the response of a generic planar model where the strongattraction assumption does not hold, and examine the response of the system to periodic pulsatile forcing. In addition, we explore how the presence of dynamical shear can lead to a chaotic response. © 2013 K.C.A.
 Lin, K. K., & Young, L. (2010). Dynamics of periodically kicked oscillators. Journal of Fixed Point Theory and Applications, 7(2), 291312.More infoAbstract: We review some recent results surrounding a general mechanism for producing chaotic behavior in periodically kicked oscillators. The key geometric ideas are illustrated via a simple linear shear model. © 2010 Springer Basel AG.
 Lin, K. K., & Young, L. (2010). Nonequilibrium steady states for certain Hamiltonian models. Journal of Statistical Physics, 139(4), 630657.More infoAbstract: We report the results of a numerical study of nonequilibrium steady states for a class of Hamiltonian models. In these models of coupled matterenergy transport, particles exchange energy through collisions with pinneddown rotating disks. In Commun. Math. Phys. 262, 237267, 2006, Eckmann and Young studied 1D chains and showed that certain simple formulas give excellent approximations of energy and particle density profiles. Keeping the basic mode of interaction in Commun. Math. Phys. 262, 237267, 2006, we extend their prediction scheme to a number of new settings: 2D systems on different lattices, driven by a variety of boundary (heat bath) conditions including the use of thermostats. Particleconserving models of the same type are shown to behave similarly. The second half of this paper examines memory and finitesize effects, which appear to impact only minimally the profiles of the models tested in Commun. Math. Phys. 262, 237267, 2006. We demonstrate that these effects can be significant or insignificant depending on the local geometry. Dynamical mechanisms are proposed, and in the case of directional bias in particle trajectories due to memory, correction schemes are derived and shown to give accurate predictions. © 2010 Springer Science+Business Media, LLC.
 Balint, P., Lin, K. K., & Young, L. (2009). Ergodicity and energy distributions for some boundary driven integrable Hamiltonian chains. Communications in Mathematical Physics, 294(1), 199228.More infoAbstract: We consider systems of moving particles in 1dimensional space interacting through energy storage sites. The ends of the systems are coupled to heat baths, and resulting steady states are studied. When the two heat baths are equal, an explicit formula for the (unique) equilibrium distribution is given. The bulk of the paper concerns nonequilibrium steady states, i.e., when the chain is coupled to two unequal heat baths. Rigorous results including ergodicity are proved. Numerical studies are carried out for two types of bath distributions. For chains driven by exponential baths, our main finding is that the system does not approach local thermodynamic equilibrium as system size tends to infinity. For bath distributions that are sharply peaked Gaussians, in spite of the nearintegrable dynamics, transport properties are found to be more normal than expected. © 2009 SpringerVerlag.
 Goodman, J. B., & Lin, K. K. (2009). Coupling control variates for Markov chain Monte Carlo. Journal of Computational Physics, 228(19), 71277136.More infoAbstract: We show that Markov couplings can be used to improve the accuracy of Markov chain Monte Carlo calculations in some situations where the steadystate probability distribution is not explicitly known. The technique generalizes the notion of control variates from classical Monte Carlo integration. We illustrate it using two models of nonequilibrium transport. © 2009 Elsevier Inc. All rights reserved.
 Lin, K. K., SheaBrown, E., & Young, L. (2009). Reliability of coupled oscillators. Journal of Nonlinear Science, 19(5), 497545.More infoAbstract: We study the reliability of phase oscillator networks in response to fluctuating inputs. Reliability means that an input elicits essentially identical responses upon repeated presentations, regardless of the network's initial condition. Single oscillators are well known to be reliable. We show in this paper that unreliable behavior can occur in a network as small as a coupled oscillator pair in which the signal is received by the first oscillator and relayed to the second with feedback. A geometric explanation based on shearinduced chaos at the onset of phaselocking is proposed. We treat larger networks as decomposed into modules connected by acyclic graphs, and give a mathematical analysis of the acyclic parts. Moreover, for networks in this class, we show how the source of unreliability can be localized, and address questions concerning downstream propagation of unreliability once it is produced. © 2009 Springer Science+Business Media, LLC.
 Lin, K. K., SheaBrown, E., & Young, L. (2009). Reliability of layered neural oscillator networks. Communications in Mathematical Sciences, 7(1), 239247.More infoAbstract: We study the reliability of large networks of pulsecoupled oscillators in response to fluctuating stimuli. Reliability means that a stimulus elicits essentially identical responses upon repeated presentations. We view the problem on two scales: neuronal reliability, which concerns the repeatability of spike times of individual oscillators embedded within a network, and pooledresponse reliability, which addresses the repeatability of the total output from the network. We find that individual embedded oscillators can be reliable or unreliable depending on network conditions, whereas pooled responses of sufficiently large networks are mostly reliable. © 2009 International Press.
 Lin, K. K., SheaBrown, E., & Young, L. (2009). Spiketime reliability of layered neural oscillator networks. Journal of Computational Neuroscience, 27(1), 135160.More infoPMID: 19156509;Abstract: We study the reliability of layered networks of coupled 'type I' neural oscillators in response to fluctuating input signals. Reliability means that a signal elicits essentially identical responses upon repeated presentations, regardless of the network's initial condition. We study reliability on two distinct scales: neuronal reliability, which concerns the repeatability of spike times of individual neurons embedded within a network, and pooledresponse reliability, which concerns the repeatability of total synaptic outputs from a subpopulation of the neurons in a network. We find that neuronal reliability depends strongly both on the overall architecture of a network, such as whether it is arranged into one or two layers, and on the strengths of the synaptic connections. Specifically, for the type of singleneuron dynamics and coupling considered, singlelayer networks are found to be very reliable, while twolayer networks lose their reliability with the introduction of even a small amount of feedback. As expected, pooled responses for large enough populations become more reliable, even when individual neurons are not. We also study the effects of noise on reliability, and find that noise that affects all neurons similarly has much greater impact on reliability than noise that affects each neuron differently. Qualitative explanations are proposed for the phenomena observed. © Springer Science+Business Media, LLC 2009.
 Lin, K. K., & Young, L. (2008). Shearinduced chaos. Nonlinearity, 21(5), 899922.More infoAbstract: Guided by a geometric understanding developed in earlier works of Wang and Young, we carry out numerical studies of shearinduced chaos in several parallel but different situations. The settings considered include periodic kicking of limit cycles, random kicks at Poisson times and continuoustime driving by white noise. The forcing of a quasiperiodic model describing two coupled oscillators is also investigated. In all cases, positive Lyapunov exponents are found in suitable parameter ranges when the forcing is suitably directed. © 2008 IOP Publishing Ltd and London Mathematical Society.
 Lin, K. K., & Young, L. (2007). Correlations in nonequilibrium steady states of random halves models. Journal of Statistical Physics, 128(3), 607639.More infoAbstract: We present the results of a detailed study of energy correlations at steady state for a 1D model of coupled energy and matter transport. Our aim is to discovervia theoretical arguments, conjectures, and numerical simulationshow spatial covariances scale with system size, their relations to local thermodynamic quantities, and the randomizing effects of heat baths. Among our findings are that shortrange covariances respond quadratically to local temperature gradients, and longrange covariances decay linearly with macroscopic distance. These findings are consistent with exact results for the simple exclusion and KMP models. © 2007 Springer Science+Business Media, LLC.
 Lin, K. K. (2006). Entrainment and chaos in a pulsedriven hodgkinhuxley oscillator. SIAM Journal on Applied Dynamical Systems, 5(2), 179204.More infoAbstract: The HodgkinHuxley model describes action potential generation in certain types of neurons and is a standard model for conductancebased, excitable cells. Following the early work of Winfree and Best, this paper explores the response of a spontaneously spiking HodgkinHuxley neuron model to a periodic pulsatile drive. The response as a function of drive period and amplitude is systematically characterized. A wide range of qualitatively distinct responses are found, including entrainment to the input pulse train and persistent chaos. These observations are consistent with a theory of kicked oscillators developed by Q. Wang and L.S. Young. In addition to general features predicted by WangYoung theory, it is found that most combinations of drive period and amplitude lead to entrainment instead of chaos. This preference for entrainment over chaos is explained by the structure of the HodgkinHuxley phaseresetting curve. © 2006 Society for Industrial and Applied Mathematics.
 Lin, K. K. (2005). Convergence of invariant densities in the smallnoise limit. Nonlinearity, 18(2), 659683.More infoAbstract: Let ρ0 be an invariant probability density of a deterministic dynamical system f and ρε the invariant probability density of a random perturbation of f by additive noise of amplitude ε. Suppose ρ0 is stochastically stable in the sense that ρε → 0 as ε → 0. Through a systematic numerical study of concrete examples, I show that: The rate of convergence of ρε to 7rho;ε0 as ε → 0 is frequently governed by power laws: ∥ρε  ρ0∥1 ∼ ε γ for some γ > 0. When the deterministic system / exhibits exponential decay of correlations, a simple heuristic can correctly predict the exponent γ based on the structure of ρ0. The heuristic fails for systems with some 'intermittency', i.e. systems which do not exhibit exponential decay of correlations. For these examples, the convergence of ρε to ρ0 as ε → 0 continues to be governed by power laws but the heuristic provides only an upper bound on the power law exponent γ. Furthermore, this numerical study requires the computation of ∥ρε  ρ0∥ 1 for 1.52.5 decades of ε and provides an opportunity to discuss and compare standard numerical methods for computing invariant probability densities in some depth. © 2005 IOP Publishing Ltd and London Mathematical Society.
 Guillopé, L., Lin, K. K., & Zworski, M. (2004). The Selberg Zeta Function for Convex CoCompact Schottky Groups. Communications in Mathematical Physics, 245(1), 149176.More infoAbstract: We give a new upper bound on the Selberg zeta function for a convex cocompact Schottky group acting on the hyperbolic space ℍn+1 : in strips parallel to the imaginary axis the zeta function is bounded by exp(Csδ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp(Csn+1), and it gives new bounds on the number of resonances (scattering poles) of Γ\Hn+1. The proof of this result is based on the application of holomorphic L2techniques to the study of the determinants of the Ruelle transfer operators and on the quasiselfsimilarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider γ\Hn+1 as the simplest model of quantum chaotic scattering.
 Lin, K. K. (2002). Numerical study of quantum resonances in chaotic scattering. Journal of Computational Physics, 176(2), 295329.More infoAbstract: This paper presents numerical evidence that in quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like hDKE+1/2 as h → 0. Here, KE denotes the subset of the energy surface {H = E} which stays bounded for all time under the flow generated by the classical Hamiltonian H and D(KE) denotes its fractal dimension. Since the number of bound states in a quantum system with n degrees of freedom scales like hn, this suggests that the quantity (D(KE) + 1)/2 represents the effective number of degrees of freedom in chaotic scattering problems. The calculations were performed using a recursive refinement technique for estimating the dimension of fractal repellors in classical Hamiltonian scattering, in conjunction with tools from modern quantum chemistry and numerical linear algebra. © 2002 Elsevier Science (USA).
 Lin, K. K., & Zworski, M. (2002). Quantum resonances in chaoti scattering. Chemical Physics Letters, 355(12), 201205.More infoAbstract: This Letter summarizes numerical results from [J. Comp. Phys. (to appear)] which show that in quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like h (D(KE)+1)/2. Here, K E denotes the subset of the classical energy surface H = E which stays bounded for all time under the flow of H and D(K E) denotes its fractal dimension. Since the number of bound states in a quantum system with n degrees of freedom scales like h n, this suggests that the quantity (D(K E) + 1)/2 represents the effective number of degrees of freedom in chaotic scattering problems. © 2002 Published by Elsevier Science B.V.
Presentations
 Lin, K. (2017, April). Control Theory and Experimental Design in Diffusion Processes. ASU Mathematical Biology Seminar. Arizona State.
 Lin, K. (2017, April). Implicit sampling in the smallnoise limit. UC Santa Cruz Applied Math Seminar. UC Santa Cruz.
 Lin, K. (2017, August). Discretetime approach to stochastic parameterization of chaotic dynamics. PIMS Workshop on Nonlinear Stochastic Dynamics. University of Alberta.
 Lin, K. (2017, March). Empirical approaches to the MoriZwanzig formalism. SIAM Computational Science and Engineering Conference. Atlanta, GA: SIAM.
 Lin, K. (2017, May). Datadriven modeling and the MoriZwanzig formalism. SIAM Conference on Applications of Dynamical Systems. Snowbird, UT: SIAM.More infoThis talk was part of a minisymposium coorganized by myself and John Harlim (Penn State).
 Lin, K. (2017, May). Discretetime approach to stochastic parameterization and model reduction. University of Chicago Scientific & Statistical Computing Seminar. University of Chicago.
 Lin, K. (2017, November). Implicit sampling in the smallnoise limit. Georgia Tech Applied and Computational Mathematics Seminar. Georgia Tech.
 Lin, K. (2017, November). MoriZwanzig formalism and discretetime stochastic parameterization of chaotic dynamics. West Coast ReducedOrder Modeling Workshop. Lawrence Berkeley National Lab.
 Lin, K. (2016, February). Implicit samplers in the smallnoise limit. Mathematics Colloquium, Tulane University.
 Lin, K. (2016, June). Implicit samplers in the smallnoise limit. Isaac Newton Institute (UK) Seminar. Cambridge University, UK.
 Lin, K. (2016, June). Implicit samplers in the smallnoise limit. Statistical Mechanics Seminar, University of Warwick. University of Warwick (UK).
Poster Presentations
 Lin, K., Fellous, J., & Greene, P. (2017, 11). Spike sorting via source localization. Society for Neuroscience. Washington DC.
 Leach, A., Lin, K., & Morzfeld, M. (2016, December). Symmetrized importance samplers for uncertainty quantification and data assimilation. American Geophysical Union: General Assembly.
 Lin, K. (2016, December). Discretetime datadriven modeling. American Geophysical Union Fall Meeting. San Francisco, CA.