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Nonlinear Continuous Data Assimilation [article]

Adam Larios, Yuan Pei
2017 arXiv   pre-print
We introduce three new nonlinear continuous data assimilation algorithms. These models are compared with the linear continuous data assimilation algorithm introduced by Azouani, Olson, and Titi (AOT). As a proof-of-concept for these models, we computationally investigate these algorithms in the context of the 1D Kuramoto-Sivashinsky equation. We observe that the nonlinear models experience super-exponential convergence in time, and converge to machine precision significantly faster than the linear AOT algorithm in our tests.
arXiv:1703.03546v1 fatcat:ws7juxf6xvfufhee34z7ou73ii

On the Attractor for the Semi-Dissipative Boussinesq Equations [article]

Animikh Biswas, Ciprian Foias, Adam Larios
2015 arXiv   pre-print
In this article, we study the long time behavior of solutions of a variant of the Boussinesq system in which the equation for the velocity is parabolic while the equation for the temperature is hyperbolic. We prove that the system has a global attractor which retains some of the properties of the global attractors for the 2D and 3D Navier-Stokes equations. Moreover, this attractor contains infinitely many invariant manifolds in which several universal properties of the Batchelor, Kraichnan, Leith theory of turbulence are potentially present.
arXiv:1507.00080v1 fatcat:onpelvtlnramzfonu2dqodveve

Efficient solutions for nonlocal diffusion problems via boundary-adapted spectral methods [article]

Siavash Jafarzadeh, Adam Larios, Florin Bobaru
2019 arXiv   pre-print
We introduce an efficient boundary-adapted spectral method for peridynamic diffusion problems with arbitrary boundary conditions. The spectral approach transforms the convolution integral in the peridynamic formulation into a multiplication in the Fourier space, resulting in computations that scale as O(NlogN). The limitation of regular spectral methods to periodic problems is eliminated using the volume penalization method. We show that arbitrary boundary conditions or volume constraints can
more » ... enforced in this way to achieve high levels of accuracy. To test the performance of our approach we compare the computational results with analytical solutions of the nonlocal problem. The performance is tested with convergence studies in terms of nodal discretization and the size of the penalization parameter in problems with Dirichlet and Neumann boundary conditions.
arXiv:1905.03875v1 fatcat:aqnmqxhaxncqth4rerxbg4pjca

Regularity criteria for the Kuramoto-Sivashinsky equation in dimensions two and three [article]

Adam Larios, Mohammad Mahabubur Rahman, Kazuo Yamazaki
2021 arXiv   pre-print
Indeed, it appears that φ, φ x , φ y , u 1 φ x , u 2 φ y , φ xx , φ xy ≡ φ yx , φ yy , and div(u) = ∆φ are all in ADAM LARIOS*, MOHAMMAD MAHABUBUR RAHMAN, AND KAZUO YAMAZAKI ADAM LARIOS*, MOHAMMAD  ...  MAHABUBUR RAHMAN, AND KAZUO YAMAZAKI ADAM LARIOS*, MOHAMMAD MAHABUBUR RAHMAN, AND KAZUO YAMAZAKI  ... 
arXiv:2112.07634v1 fatcat:qtm7r5lmd5hl7cwvuor5bez6wu

Sensitivity Analysis for the 2D Navier-Stokes Equations with Applications to Continuous Data Assimilation [article]

Adam Larios, Elizabeth Carlson
2020 arXiv   pre-print
We rigorously prove the well-posedness of the formal sensitivity equations with respect to the Reynolds number corresponding to the 2D incompressible Navier-Stokes equations. Moreover, we do so by showing a sequence of difference quotients converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the related data assimilation equations, which utilize the continuous data assimilation algorithm proposed by Azouani, Olson, and Titi. As a result, this
more » ... method of proof provides uniform bounds on difference quotients, demonstrating parameter recovery algorithms that change parameters as the system evolves will not blow-up. We also note that this appears to be the first such rigorous proof of global existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier-Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the Reynolds number.
arXiv:2007.01860v1 fatcat:oke5iufq4rfplixsedcwbh7ib4

A Blow-Up Criterion for the 3D Euler Equations Via the Euler-Voigt Inviscid Regularization [article]

Adam Larios, Edriss S. Titi
2015 arXiv   pre-print
We propose a new blow-up criterion for the 3D Euler equations of incompressible fluid flows, based on the 3D Euler-Voigt inviscid regularization. This criterion is similar in character to a criterion proposed in a previous work by the authors, but it is stronger, and better adapted for computational tests. The 3D Euler-Voigt equations enjoy global well-posedness, and moreover are more tractable to simulate than the 3D Euler equations. A major advantage of these new criteria is that one only
more » ... s to simulate the 3D Euler-Voigt, and not the 3D Euler equations, to test the blow-up criteria, for the 3D Euler equations, computationally.
arXiv:1507.08203v1 fatcat:cb3z4xsg4nfndf4soue4clycvy

Parameter Recovery and Sensitivity Analysis for the 2D Navier-Stokes Equations Via Continuous Data Assimilation [article]

Elizabeth Carlson, Joshua Hudson, Adam Larios
2018 arXiv   pre-print
We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown Reynolds number. We determine the large-time error between the true solution of the 2D Navier-Stokes equations and the assimilated solution due to discrepancy between an approximate Reynolds number and the physical Reynolds number. Additionally, we develop an algorithm that can be run in tandem with the AOT algorithm to recover both the true solution and the Reynolds number
more » ... or equivalently the true viscosity) using only spatially discrete velocity measurements. The algorithm we propose involves changing the viscosity mid-simulation. Therefore, we also examine the sensitivity of the equations with respect to the Reynolds number. We prove that a sequence of difference quotients with respect to the Reynolds number converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the assimilated equations. We also note that this appears to be the first such rigorous proof of existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier-Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the Reynolds number.
arXiv:1812.07646v1 fatcat:3ednd2w43jcyrbelm4xwzql6fy

Global well-posedness of the velocity-vorticity-Voigt model of the 3D Navier-Stokes equations [article]

Adam Larios, Yuan Pei, Leo Rebholz
2018 arXiv   pre-print
The velocity-vorticity formulation of the 3D Navier-Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier-Stokes equations, which we call the 3D velocity-vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity-vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this
more » ... under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier-Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier-Stokes equations based on this inviscid regularization.
arXiv:1802.08766v1 fatcat:hp2l7bzdtnd6dd3shlh5cqlrca

Dynamically learning the parameters of a chaotic system using partial observations [article]

Elizabeth Carlson, Joshua Hudson, Adam Larios, Vincent R. Martinez, Eunice Ng, Jared P. Whitehead
2021 arXiv   pre-print
The LSODA routine switches between nonstiff (Adams type) and stiff (backward differentiation) methods.  ... 
arXiv:2108.08354v1 fatcat:scvovu6cancftk2ahbsugu3hai

Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data [article]

Adam Larios, Yuan Pei
2018 arXiv   pre-print
We propose a data assimilation algorithm for the 2D Navier-Stokes equations, based on the Azouani, Olson, and Titi (AOT) algorithm, but applied to the 2D Navier-Stokes-Voigt equations. Adapting the AOT algorithm to regularized versions of Navier-Stokes has been done before, but the innovation of this work is to drive the assimilation equation with observational data, rather than data from a regularized system. We first prove that this new system is globally well-posed. Moreover, we prove that
more » ... r any admissible initial data, the L^2 and H^1 norms of error are bounded by a constant times a power of the Voigt-regularization parameter α>0, plus a term which decays exponentially fast in time. In particular, the large-time error goes to zero algebraically as α goes to zero. Assuming more smoothness on the initial data and forcing, we also prove similar results for the H^2 norm.
arXiv:1810.10616v1 fatcat:cmsl3bdqljdidp732bnovbsktu

Global Regularity vs. Finite-Time Singularities: Some Paradigms on the Effect of Boundary Conditions and Certain Perturbations [article]

Adam Larios, Edriss S. Titi
2014 arXiv   pre-print
In light of the question of finite-time blow-up vs. global well-posedness of solutions to problems involving nonlinear partial differential equations, we provide several cautionary examples which indicate that modifications to the boundary conditions or to the nonlinearity of the equations can effect whether the equations develop finite-time singularities. In particular, we aim to underscore the idea that in analytical and computational investigations of the blow-up of three-dimensional Euler
more » ... d Navier-Stokes equations, the boundary conditions may need to be taken into greater account. We also examine a perturbation of the nonlinearity by dropping the advection term in the evolution of the derivative of the solutions to the viscous Burgers equation, which leads to the development of singularities not present in the original equation, and indicates that there is a regularizing mechanism in part of the nonlinearity. This simple analytical example corroborates recent computational observations in the singularity formation of fluid equations.
arXiv:1401.1534v1 fatcat:6w76nr6ljnafdnviwq7cvoiwqe

Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study [article]

Adam Larios, Collin Victor
2018 arXiv   pre-print
Data assimilation is a technique for increasing the accuracy of simulations of solutions to partial differential equations by incorporating observable data into the solution as time evolves. Recently, a promising new algorithm for data assimilation based on feedback-control at the PDE level has been proposed in the pioneering work of Azouani, Olson, and Titi (2014). The standard version of this algorithm is based on measurement from data points that are fixed in space. In this work, we consider
more » ... the scenario in which the data collection points move in space over time. We demonstrate computationally that, at least in the setting of the 1D Allen-Cahn reaction diffusion equations, the algorithm converges with significantly fewer measurement points, up to an order or magnitude in some cases. We also provide an application of the algorithm to an inverse problem in the case of a uniform static grid.
arXiv:1812.01686v1 fatcat:ngox2m7crbbkten6mqeijjtdsy

Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations [article]

Adam Larios, Edriss S. Titi
2011 arXiv   pre-print
We prove existence, uniqueness, and higher-order global regularity of strong solutions to a particular Voigt-regularization of the three-dimensional inviscid resistive Magnetohydrodynamic (MHD) equations. Specifically, the coupling of a resistive magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid resistive MHD system. The results hold in both the whole space ^3 and in the context of periodic boundary conditions. Weak solutions for this
more » ... rized model are also considered, and proven to exist globally in time, but the question of uniqueness for weak solutions is still open. Since the main purpose of this line of research is to introduce a reliable and stable inviscid numerical regularization of the underlying model we, in particular, show that the solutions of the Voigt regularized system converge, as the regularization parameter α0, to strong solutions of the original inviscid resistive MHD, on the corresponding time interval of existence of the latter. Moreover, we also establish a new criterion for blow-up of solutions to the original MHD system inspired by this Voigt regularization. This type of regularization, and the corresponding results, are valid for, and can also be applied to, a wide class of hydrodynamic models.
arXiv:1104.0358v1 fatcat:ecohbyjwyje7rh5dqyglaoqmhu

Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations [article]

Matthew Gardner, Adam Larios, Leo G. Rebholz, Duygu Vargun, Camille Zerfas
2020 arXiv   pre-print
We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity, and nudging applied to the velocity only. We prove that under a typical finite element spatial discretization and backward Euler temporal discretization, application of CDA preserves the unconditional long-time stability property of the velocity-vorticity method and provides optimal long-time accuracy. These
more » ... operties hold if nudging is applied only to the velocity, and if nudging is also applied to the vorticity then the optimal long-time accuracy is achieved more rapidly in time. Numerical tests illustrate the theory, and show its effectiveness on an application problem of channel flow past a flat plate.
arXiv:2006.07295v1 fatcat:c54efutzijhptjqmci3uhj25ny

Continuous data assimilation for the magnetohydrodynamic equations in 2D using one component of the velocity and magnetic fields [article]

Animikh Biswas, Joshua Hudson, Adam Larios, Yuan Pei
2017 arXiv   pre-print
We propose several continuous data assimilation (downscaling) algorithms based on feedback control for the 2D magnetohydrodynamic (MHD) equations. We show that for sufficiently large choices of the control parameter and resolution and assuming that the observed data is error-free, the solution of the controlled system converges exponentially (in L^2 and H^1 norms) to the reference solution independently of the initial data chosen for the controlled system. Furthermore, we show that a similar
more » ... ult holds when controls are placed only on the horizontal (or vertical) variables, or on a single Elsässer variable, under more restrictive conditions on the control parameter and resolution. Finally, using the data assimilation system, we show the existence of abridged determining modes, nodes and volume elements.
arXiv:1704.02082v1 fatcat:j6astjafifboxcvwst22ph3xva
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