Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 2 December 2020Accepted: 02 August 2021Published online: 18 October 2021Keywordsmultiscale, diffeomorphism, multigrid, image registration, algorithmAMS Subject Headings65M12, 68U10, 94A08, 65K10Publication DataISSN (print): 1540-3459ISSN (online): 1540-3467Publisher: Society for Industrial and Applied MathematicsCODEN: mmsubt

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 28 May 2020Accepted: 26 August 2021Published online: 23 November 2021KeywordsCauchy--Born rule, finite temperature, stability of crystalline solids, atomistic model, point defectAMS Subject Headings74B20, 74E15, 74G65, 74N99Publication DataISSN (print): 1540-3459ISSN (online): 1540-3467Publisher: Society for Industrial and Applied MathematicsCODEN: mmsubt

We present a novel weak formulation and discretization for discovering governing equations from noisy measurement data. This method of learning differential equations from data fits into a new class of algorithms that replace pointwise derivative approximations with linear transformations and variance reduction techniques. Compared to the standard SINDy algorithm presented in [S. L. Brunton, J. L. Proctor, and J. N. Kutz, Proc. Natl. Acad. Sci. USA, 113 (2016), pp. 3932-3937], our so-called weak SINDy (WSINDy) algorithm allows for reliable model identification from data with large noise (often with ratios greater than 0.1) and reduces the error in the recovered coefficients to enable accurate prediction. Moreover, the coefficient error scales linearly with the noise level, leading to high-accuracy recovery in the low-noise regime. Altogether, WSINDy combines the simplicity and efficiency of the SINDy algorithm with the natural noise reduction of integration, as demonstrated in [H. Schaeffer and S. G. McCalla, Phys. Rev. E, 96 (2017), 023302], to arrive at a robust and accurate method of sparse recovery.

On the Conservation Properties in Multiple Scale Coupling and Simulation for Darcy Flow with Hyperbolic-Transport in Complex Flows

The mechanism of gas transport across cell membranes remains a topic of considerable interest, particularly regarding the extent to which lipids vs. specific membrane proteins provide conduction pathways. Studies of transmembrane (CO2) transport often rely on data collected under controlled conditions, using pH-sensitive microelectrodes at the extracellular surface to record changes due to extracellular CO2 diffusion and reactions. Although recent detailed computational models can predict a qualitatively correct behavior, a mismatch between the dynamical ranges of the predicted and observed pH curves raises the question whether the discrepancy may be due to a bias introduced by the pH electrode itself. More specifically, it is reasonable to ask whether bringing the electrode tip near or in contact with the membrane creates a local microenvironment between the electrode tip and the membrane, so that the measured data refer to the microenvironment rather than to the free surface. Here, we introduce a detailed computational model, designed to address this question. We find that, as long as a zone of free diffusion exists between the tip and the membrane, the microenvironment behaves effectively as the free membrane. However, according to our model, when the tip contacts the membrane, partial quenching of extracellular diffusion by the electrode rim leads to a significant increase in the pH dynamics under the electrode, matching values measured in physiological experiments. The computational schemes for the model predictions are based on semi-discretization by a finite-element method, and an implicit-explicit time integration scheme to capture the different time scales of the system.

We present a novel algorithm based on the ensemble Kalman filter to solve\ninverse problems involving multiscale elliptic partial differential equations.\nOur method is based on numerical homogenization and finite element\ndiscretization and allows to recover a highly oscillatory tensor from\nmeasurements of the multiscale solution in a computationally inexpensive\nmanner. The properties of the approximate solution are analysed with respect to\nthe multiscale and discretization parameters, and a convergence result is shown\nto hold. A reinterpretation of the solution from a Bayesian perspective is\nprovided, and convergence of the approximate conditional posterior distribution\nis proved with respect to the Wasserstein distance. A numerical experiment\nvalidates our methodology, with a particular emphasis on modelling error and\ncomputational cost.\n

In this paper, we propose and analyze a gradient flow based model for electronic structure calculations. First, based on an extended gradient flow proposed in this paper, we propose a Kohn--Sham gradient flow based model. We prove that our gradient flow based model is orthogonality preserving, the extended gradient has an exponential decay over time $t$, and the equilibrium point is a local minimizer of the Kohn--Sham energy functional. Then we propose a midpoint scheme to carry out the temporal discretization, which is proven to be orthogonality preserving, too. Based on the midpoint scheme, we design a practical orthogonality preserving iteration scheme which can deal with the propagation of the gradient flow based model and prove that the scheme produces approximations that converge to a local minimizer with some convergence rate under some reasonable assumptions. Finally, we report a number of numerical experiments that validate our theoretical results.

In this paper, we study the approximation and existence of vacuum states in the multiscale gas flows governed by the Cauchy problem of compressible Euler equations containing a small parameter $\eta$ in the initial density. The system of compressible Euler equations is reduced to a hyperbolic resonant system at the vacuum so that the weak solution of the Riemann problem is not suitable as the building block of the Glimm (or Godunov) scheme to establish the existence of vacuum states. We construct a new type of approximate solutions, the weak solutions of the regularized Riemann problem for the leading-order system derived from asymptotic expansions around vacuum states. Such an approximate solution obtained by solving the pressureless Euler equations with generalized Riemann data consists of constant states separated by a composite hyperbolic wave. We show the stability of the regularized Riemann solution, together with the numerical simulations, under the small perturbations of initial data. Adopting the approximate solution as the building block of the generalized Glimm scheme, we prove the existence of the vacuum states by showing the stability and consistency of the scheme as $\eta \rightarrow 0$. The numerical simulation indicates that for any small fixed $t>0$, the approximate solutions converge to the exact solutions of the Cauchy problem in $L^1$ as $\eta \rightarrow 0$. The results of this paper can be applied to some hyperbolic resonant systems of balance laws.

The study of active matter consisting of many self-propelled (active) swimmers in an imposed flow is important for many applications. Self-propelled swimmers may represent both living and artificia...

This work is devoted to the numerical simulation of the Bhatnagar--Gross--Krook (BGK) equation for two species in the fluid limit using a particle method. Thus, we are interested in a gas mixture consisting of two species without chemical reactions assuming that the number of particles of each species remains constant. We consider the kinetic two species model proposed by Klingenberg, Pirner, and Puppo in [Kinetic Rel. Models, 10 (2017), pp. 445--465], which separates the intra- and interspecies collisions. We want to study numerically the influence of the two relaxation terms, one corresponding to intraspecies and the other to interspecies collisions. For this, we use the method of micro-macro decomposition. First, we derive an equivalent model based on the micro-macro decomposition (see Bennoune, Lemou, and Mieussens [J. Comput. Phys., 227 (2008), pp. 3781--3803] and Crestetto, Crouseilles, and Lemou [Kinetic Rel. Models, 5 (2012), pp. 787--816]). The kinetic micro part is solved by a particle method, whereas the fluid macro part is discretized by a standard finite volume scheme. The main advantages of this approach are that (i) the noise inherent to the particle method is reduced compared to a standard (without micro-macro decomposition) particle method, and (ii) the computational cost of the method is reduced in the fluid limit since a small number of particles is then sufficient.