arXiv subjects

Roman Chertovskih

Publications and source records attributed to Roman Chertovskih.

32 records · Page 2Linked to original sources

arXiv2022

Reconstruction of photospheric velocity fields from highly corrupted data

The analysis of the photospheric velocity field is essential for understanding plasma turbulence in the solar surface, which may be responsible for driving processes such as magnetic reconnection, flares, wave propagation, particle acceleration, and coronal heating. Currently, the only available methods to estimate velocities at the solar photosphere transverse to an observer's line of sight infer flows from differences in image structure in successive observations. Due to data noise, algorithms such as local correlation tracking (LCT) may lead to a vector field with wide gaps where no velocity vectors are provided. In this letter, a novel method for image inpainting of highly corrupted data is proposed and applied to the restoration of horizontal velocity fields in the solar photosphere. The restored velocity field preserves all the vector field components present in the original field. The method shows robustness when applied to both simulated and observational data.

astro-ph.SR↗

arXiv2021

Feedback maximum principle for ensemble control of local continuity equations. An application to supervised machine learning

We consider an optimal control problem for a system of local continuity equations on a space of probability measures. Such systems can be viewed as macroscopic models of ensembles of non-interacting particles or homotypic individuals, representing several different ``populations''. For the stated problem, we propose a necessary optimality condition, which involves feedback controls inherent to the extremal structure, designed via the standard Pontryagin's Maximum Principle conditions. This optimality condition admits a realization as an iterative algorithm for optimal control. As a motivating case, we discuss an application of the derived optimality condition and the consequent numeric method to a problem of supervised machine learning via dynamic systems.

math.OC↗

arXiv2020

Chaotic transients and hysteresis in an $α^{2}$ dynamo model

The presence of chaotic transients in a nonlinear dynamo is investigated through numerical simulations of the 3D magnetohydrodynamic equations. By using the kinetic helicity of the flow as a control parameter, a hysteretic blowout bifurcation is conjectured to be responsible for the transition to dynamo, leading to a sudden increase in the magnetic energy of the attractor. This high-energy hydromagnetic attractor is suddenly destroyed in a boundary crisis when the helicity is decreased. Both the blowout bifurcation and the boundary crisis generate long chaotic transients that are due, respectively, to a chaotic saddle and a relative chaotic attractor.

physics.plasm-ph↗

arXiv2019

Regular path-constrained time-optimal control problems in three-dimensional flow fields

This article concerns a class of time-optimal state constrained control problems with dynamics defined by an ordinary differential equation involving a three-dimensional steady flow vector field. The problem is solved via an indirect method based on the maximum principle in Gamkrelidze's form. The proposed computational method essentially uses a certain regularity condition imposed on the data of the problem. The property of regularity guarantees the continuity of the measure multiplier associated with the state constraint, and ensures the appropriate behavior of the corresponding numerical procedure which, in general, consists in computing the entire field of extremals for the problem in question. Several examples of vector fields are considered to illustrate the computational approach.

math.OC↗

arXiv2019

Time-Optimal Control Problem With State Constraints In A Time-Periodic Flow Field

This article contributes to a framework for a computational indirect method based on the Pontryagin maximum principle to efficiently solve a class of state constrained time-optimal control problems in the presence of a time-dependent flow field. Path-planning in a flow with tidal variations is an emblematic real-life task that serves as an example falling in the class of problems considered here. Under rather natural assumptions on the flow field, the problem is regular and the measure Lagrange multiplier, associated with the state constraint, is continuous. These properties (regularity and continuity) play a critical role in computing the field of extremals by solving the two-point boundary value problem given by the maximum principle. Moreover, some sufficient conditions preventing the emergence of singular control processes are given. A couple of examples of time-periodic fluid flows are studied and the corresponding optimal solutions are found.

math.OC↗

arXiv2019

Computation of kinematic and magnetic $α$-effect and eddy diffusivity tensors by Padé approximation

We present examples of Padé approximation of the $α$-effect and eddy viscosity/diffusivity tensors in various flows. Expressions for the tensors derived in the framework of the standard multiscale formalism are employed. Algebraically the simplest case is that of a two-dimensional parity-invariant six-fold rotation-symmetric flow, where eddy viscosity is negative, indicating intervals of large-scale instability of the flow. Turning to the kinematic dynamo problem for three-dimensional flows of an incompressible fluid, we explore application of Padé approximants for computation of tensors of magnetic $α$-effect and, for parity-invariant flows, of magnetic eddy diffusivity. We construct Padé approximants of the tensors expanded in power series in the inverse molecular diffusivity $1/η$ around $1/η=0$. This yields the values of the dominant growth rate due to the action of the $α$-effect or eddy diffusivity to satisfactory accuracy for $η$, several dozen times smaller than the threshold, above which the power series is convergent. For one sample flow, we observe eddy diffusivity tending to negative infinity when $η$ tends from above to the point of the onset of small-scale dynamo action in a symmetry-invariant subspace where a neutral small-scale magnetic mode resides. However, 49 first coefficients in the power series in $1/η$ prove insufficient for Padé approximants to reproduce this behaviour. We do computations in Fortran in the standard `double' (real*8) and extended `quadruple' (real*16) precision, as well as perform symbolic calculations in Mathematica.

physics.flu-dyn↗

arXiv2019

Negative magnetic eddy diffusivity due to oscillogenic $α$-effect

We study large-scale kinematic dynamo action of steady mirror-antisymmetric flows of incompressible fluid, that involve small spatial scales only, by asymptotic methods of the multiscale stability theory. It turns out that, due to the magnetic $α$-effect in such flows, the large-scale mean field experiences harmonic oscillations in time on the scale O($\varepsilon t$) without growth or decay. Here $\varepsilon$ is the spatial scale ratio and $t$ is the fast time of the order of the flow turnover time. The interaction of the accompanying fluctuating magnetic field with the flow gives rise to an anisotropic magnetic eddy diffusivity, whose dependence on the direction of the large-scale wave vector generically exhibits a singular behaviour, and thus to negative eddy diffusivity for whichever molecular magnetic diffusivity. Consequently, such flows always act as kinematic dynamos on the time scale O($\varepsilon^2t$); for the directions at which eddy diffusivity is infinite, the large-scale mean-field growth rate is finite on the scale O($\varepsilon^{3/2}t$). We investigate numerically this dynamo mechanism for two sample flows.

physics.flu-dyn↗

arXiv2019

Pointwise vanishing velocity helicity of a flow does not preclude magnetic field generation

Pointwise zero velocity helicity density is shown not to prevent steady flows from acting as kinematic dynamos. We present numerical evidences that such flows can generate both small-scale magnetic fields as well as, by the magnetic $α$-effect or negative eddy diffusivity mechanisms, large-scale ones. The flows are constructed as curls of analytically defined space-periodic steady solenoidal flows, whose vorticity helicity (i.e., kinetic helicity) density is everywhere zero.

physics.flu-dyn↗

arXiv2018

An indirect numerical method for a time-optimal state-constrained control problem in a steady two-dimensional fluid flow

This article concerns the problem of computing solutions to state-constrained optimal control problems whose trajectory is affected by a flow field. This general mathematical framework is particularly pertinent to the requirements underlying the control of Autonomous Underwater Vehicles in realistic scenarii. The key contribution consists in devising a computational indirect method which becomes effective in the numerical computation of extremals to optimal control problems with state constraints by using the maximum principle in Gamkrelidze's form in which the measure Lagrange multiplier is ensured to be continuous. The specific problem of time-optimal control of an Autonomous Underwater Vehicle in a bounded space set, subject to the effect of a flow field and with bounded actuation, is used to illustrate the proposed approach. The corresponding numerical results are presented and discussed.

math.OC↗

arXiv2018

Highly corrupted image inpainting through hypoelliptic diffusion

We present a new image inpainting algorithm, the Averaging and Hypoelliptic Evolution (AHE) algorithm, inspired by the one presented in [SIAM J. Imaging Sci., vol. 7, no. 2, pp. 669--695, 2014] and based upon a semi-discrete variation of the Citti-Petitot-Sarti model of the primary visual cortex V1. The AHE algorithm is based on a suitable combination of sub-Riemannian hypoelliptic diffusion and ad-hoc local averaging techniques. In particular, we focus on reconstructing highly corrupted images (i.e. where more than the 80% of the image is missing), for which we obtain reconstructions comparable with the state-of-the-art.

cs.CV↗

arXiv2018

Cortical-inspired image reconstruction via sub-Riemannian geometry and hypoelliptic diffusion

In this paper we review several algorithms for image inpainting based on the hypoelliptic diffusion naturally associated with a mathematical model of the primary visual cortex. In particular, we present one algorithm that does not exploit the information of where the image is corrupted, and others that do it. While the first algorithm is able to reconstruct only images that our visual system is still capable of recognize, we show that those of the second type completely transcend such limitation providing reconstructions at the state-of-the-art in image inpainting. This can be interpreted as a validation of the fact that our visual cortex actually encodes the first type of algorithm.

cs.CV↗

arXiv2017

Backward stochastic differential equation approach to modeling of gene expression

In this article, we introduce a novel backward method to model stochastic gene expression and protein level dynamics. The protein amount is regarded as a diffusion process and is described by a backward stochastic differential equation (BSDE). Unlike many other SDE techniques proposed in the literature, the BSDE method is backward in time; that is, instead of initial conditions it requires the specification of endpoint ("final") conditions, in addition to the model parametrization. To validate our approach we employ Gillespie's stochastic simulation algorithm (SSA) to generate (forward) benchmark data, according to predefined gene network models. Numerical simulations show that the BSDE method is able to correctly infer the protein level distributions that preceded a known final condition, obtained originally from the forward SSA. This makes the BSDE method a powerful systems biology tool for time reversed simulations, allowing, for example, the assessment of the biological conditions (e.g. protein concentrations) that preceded an experimentally measured event of interest (e.g. mitosis, apoptosis, etc.).

physics.bio-ph↗

arXiv2015

Large-scale weakly nonlinear perturbations of convective magnetic dynamos in a rotating layer

We present a new mechanism for generation of large-scale magnetic field by thermal convection which does not involve the alpha-effect. We consider weakly nonlinear perturbations of space-periodic steady convective magnetic dynamos in a rotating layer that were identified in our previous work. The perturbations have a spatial scale in the horizontal direction that is much larger than the period of the perturbed convective magnetohydrodynamic state. Following the formalism of the multiscale stability theory, we have derived the system of amplitude equations governing the evolution of the leading terms in the expansion of the perturbations in power series in the scale ratio. This asymptotic analysis is more involved than in the cases considered earlier, because the kernel of the operator of linearisation has zero-mean neutral modes whose origin lies in the spatial invariance of the perturbed regime, the operator reduced on the generalised kernel has two Jordan normal form blocks of size two, and simplifying symmetries of the perturbed state are now missing. Numerical results for the amplitude equations show that a large-scale perturbation, periodic in slow horizontal variable, either converges to a short-scale neutral stability mode with amplitudes tending to constant values, or it blows up at a finite slow time.

physics.flu-dyn↗

arXiv2014

Hypoelliptic diffusion and human vision: a semi-discrete new twist

This paper presents a semi-discrete alternative to the theory of neurogeometry of vision, due to Citti, Petitot and Sarti. We propose a new ingredient, namely working on the group of translations and discrete rotations $SE(2,N)$. The theoretical side of our study relates the stochastic nature of the problem with the Moore group structure of $SE(2,N)$. Harmonic analysis over this group leads to very simple finite dimensional reductions. We then apply these ideas to the inpainting problem which is reduced to the integration of a completely parallelizable finite set of Mathieu-type diffusions (indexed by the dual of $SE(2,N)$ in place of the points of the Fourier plane, which is a drastic reduction). The integration of the the Mathieu equations can be performed by standard numerical methods for elliptic diffusions and leads to a very simple and efficient class of inpainting algorithms. We illustrate the performances of the method on a series of deeply corrupted images.

math.AP↗