19,407 research outputs found
The Inverse Spectral Transform for the Dunajski hierarchy and some of its reductions, I: Cauchy problem and longtime behavior of solutions
In this paper we apply the formal Inverse Spectral Transform for integrable
dispersionless PDEs arising from the commutation condition of pairs of
one-parameter families of vector fields, recently developed by S. V. Manakov
and one of the authors, to one distinguished class of equations, the so-called
Dunajski hierarchy. We concentrate, for concreteness, i) on the system of PDEs
characterizing a general anti-self-dual conformal structure in neutral
signature, ii) on its first commuting flow, and iii) on some of their basic and
novel reductions. We formally solve their Cauchy problem and we use it to
construct the longtime behavior of solutions, showing, in particular, that
unlike the case of soliton PDEs, different dispersionless PDEs belonging to the
same hierarchy of commuting flows evolve in time in very different ways,
exhibiting either a smooth dynamics or a gradient catastrophe at finite time
Numerical Bifurcation Analysis of PDEs From Lattice Boltzmann Model Simulations: a Parsimonious Machine Learning Approach
We address a three-tier data-driven approach for the numerical solution of the inverse problem in Partial Differential Equations (PDEs) and for their numerical bifurcation analysis from spatio-temporal data produced by Lattice Boltzmann model simulations using machine learning. In the first step, we exploit manifold learning and in particular parsimonious Diffusion Maps using leave-one-out cross-validation (LOOCV) to both identify the intrinsic dimension of the manifold where the emergent dynamics evolve and for feature selection over the parameter space. In the second step, based on the selected features, we learn the right-hand-side of the effective PDEs using two machine learning schemes, namely shallow Feedforward Neural Networks (FNNs) with two hidden layers and single-layer Random Projection Networks (RPNNs), which basis functions are constructed using an appropriate random sampling approach. Finally, based on the learned black-box PDE model, we construct the corresponding bifurcation diagram, thus exploiting the numerical bifurcation analysis toolkit. For our illustrations, we implemented the proposed method to perform numerical bifurcation analysis of the 1D FitzHugh-Nagumo PDEs from data generated by D1Q3 Lattice Boltzmann simulations. The proposed method was quite effective in terms of numerical accuracy regarding the construction of the coarse-scale bifurcation diagram. Furthermore, the proposed RPNN scheme was ∼ 20 to 30 times less costly regarding the training phase than the traditional shallow FNNs, thus arising as a promising alternative to deep learning for the data-driven numerical solution of the inverse problem for high-dimensional PDEs
In-domain control of a heat equation: an approach combining zero-dynamics inverse and differential flatness
This paper addresses the set-point control problem of a heat equation with
in-domain actuation. The proposed scheme is based on the framework of zero
dynamics inverse combined with flat system control. Moreover, the set-point
control is cast into a motion planing problem of a multiple-input, multiple-out
system, which is solved by a Green's function-based reference trajectory
decomposition. The validity of the proposed method is assessed through
convergence and solvability analysis of the control algorithm. The performance
of the developed control scheme and the viability of the proposed approach are
confirmed by numerical simulation of a representative system.Comment: Preprint of an original research pape
Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design
This paper develops a control and estimation design for the one-phase Stefan
problem. The Stefan problem represents a liquid-solid phase transition as time
evolution of a temperature profile in a liquid-solid material and its moving
interface. This physical process is mathematically formulated as a diffusion
partial differential equation (PDE) evolving on a time-varying spatial domain
described by an ordinary differential equation (ODE). The state-dependency of
the moving interface makes the coupled PDE-ODE system a nonlinear and
challenging problem. We propose a full-state feedback control law, an observer
design, and the associated output-feedback control law via the backstepping
method. The designed observer allows estimation of the temperature profile
based on the available measurement of solid phase length. The associated
output-feedback controller ensures the global exponential stability of the
estimation errors, the H1- norm of the distributed temperature, and the moving
interface to the desired setpoint under some explicitly given restrictions on
the setpoint and observer gain. The exponential stability results are
established considering Neumann and Dirichlet boundary actuations.Comment: 16 pages, 11 figures, submitted to IEEE Transactions on Automatic
Contro
A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science
In this paper, we study damped second-order dynamics, which are quasilinear
hyperbolic partial differential equations (PDEs). This is inspired by the
recent development of second-order damping systems for accelerating energy
decay of gradient flows. We concentrate on two equations: one is a damped
second-order total variation flow, which is primarily motivated by the
application of image denoising; the other is a damped second-order mean
curvature flow for level sets of scalar functions, which is related to a
non-convex variational model capable of correcting displacement errors in image
data (e.g. dejittering). For the former equation, we prove the existence and
uniqueness of the solution. For the latter, we draw a connection between the
equation and some second-order geometric PDEs evolving the hypersurfaces which
are described by level sets of scalar functions, and show the existence and
uniqueness of the solution for a regularized version of the equation. The
latter is used in our algorithmic development. A general algorithm for
numerical discretization of the two nonlinear PDEs is proposed and analyzed.
Its efficiency is demonstrated by various numerical examples, where simulations
on the behavior of solutions of the new equations and comparisons with
first-order flows are also documented
Nonlocality and the inverse scattering transform for the Pavlov equation
As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of
integrable dispersionless multidimensional PDEs is non-local, and the proper
choice of integration constants should be the one dictated by the associated
Inverse Scattering Transform (IST). Using the recently made rigorous IST for
vector fields associated with the so-called Pavlov equation
, in this paper we establish the
following. 1. The non-local term arising from its
evolutionary form corresponds to the
asymmetric integral . 2. Smooth and well-localized initial
data evolve in time developing, for , the constraint
, where . 3. Since no smooth and well-localized initial
data can satisfy such constraint at , the initial () dynamics of the
Pavlov equation can not be smooth, although, as it was already established,
small norm solutions remain regular for all positive times. We expect that the
techniques developed in this paper to prove the above results, should be
successfully used in the study of the non-locality of other basic examples of
integrable dispersionless PDEs in multidimensions.Comment: 19 page
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