arXiv subjects

Minh-Binh Tran

Publications and source records attributed to Minh-Binh Tran.

At least 55 records · Page 3Linked to original sources

arXiv2020

Identification of linear dynamical systems and machine learning

The topic of identification of dynamic systems, has been at the core of modern control , following the fundamental works of Kalman. Realization Theory has been one of the major outcomes in this domain, with the possibility of identifying a dynamic system from an input-output relationship. The recent development of machine learning concepts has rejuvanated interest for identification. In this paper, we review briefly the results of realization theory, and develop some methods inspired by Machine Learning concepts.

math.OC↗

arXiv2020

A reaction network approach to the theory of acoustic wave turbulence

We propose a new approach to study the long time dynamics of the wave kinetic equation in the statistical description of acoustic turbulence. The approach is based on rewriting the discrete version of the wave kinetic equation in the form of a chemical reaction network, then employing techniques used to study the Global Attractor Conjecture to investigate the long time dynamics of the newly obtained chemical system. We show that the solution of the chemical system converges to an equilibrium exponentially in time. In addition, a resonance broadening modification of the acoustic wave kinetic equation is also studied with the same technique. For the near-resonance equation, if the resonance broadening frequency is larger than a threshold, the solution of the system goes to infinity as time evolves.

math-ph↗

arXiv2020

Boltzmann-type collision operators for Bogoliubov excitations of Bose-Einstein condensates: A unified framework

Starting from the Bogoliubov diagonalization for the Hamiltonian of a weakly interacting Bose gas under the presence of a Bose-Enstein Condensate, we derive the kinetic equation for the Bogoliubov excitations. Without dropping any of the commutators, we find three collisional processes. One of them describes the $1\leftrightarrow2$ interactions between the condensate and the excited atoms. The other two describe the $2\leftrightarrow2$ and $1\leftrightarrow3$ interactions between the excited atoms themselves.

cond-mat.quant-gas↗

arXiv2019

On shock waves from the inhomogeneous Boltzmann equation

We revisit the problem on the inner structure of shock waves in simple gases modelized by the Boltzmann kinetic equation. In \cite{pomeau1987shock}, a self-similarity approach was proposed for infinite total cross section resulting from a power law interaction, but this self-similar form does not have finite energy. Motivated by the work of Pomeau, Bobylev and Cercignani started the rigorous study of the solutions of the spatial homogeneous Boltzmann equation, focusing on those which do not have finite energy \cite{bobylev2002self,bobylev2003eternal}. In the present work, we provide a correction to the self-similar form, so that the solutions are more physically sound in the sense that the energy is no longer infinite and that the perturbation brought by the shock does not grow at large distances of it on the cold side in the soft potential case.

math-ph↗

arXiv2019

On the energy cascade of 3-wave kinetic equations: Beyond Kolmogorov-Zakharov solutions

In weak turbulence theory, the Kolmogorov-Zakharov spectra is a class of time-independent solutions to the kinetic wave equations. In this paper, we construct a new class of time-dependent isotropic solutions to the decaying turbulence problems (whose solutions are energy conserved), with general initial conditions. These solutions exhibit the interesting property that the energy is cascaded from small wavenumbers to large wavenumbers. We can prove that starting with a regular initial condition whose energy at the infinity wave number $|p|=\infty$ is $0$, as time evolves, the energy is gradually accumulated at $\{|p|=\infty\}$. Finally, all the energy of the system is concentrated at $\{|p|=\infty\}$ and the energy function becomes a Dirac function at infinity $Eδ_{\{|p|=\infty\}}$, where $E$ is the total energy. The existence of this class of solutions is, in some sense, {the first complete rigorous} mathematical proof based on the kinetic description for the energy cascade phenomenon for waves with quadratic nonlinearities. We only represent in this paper the analysis of the statistical description of acoustic waves (and equivalently capillary waves). However, our analysis works for other cases as well.

math-ph↗

arXiv2019

On the wave turbulence theory for stratified flows in the ocean

After the pioneering work of Garrett and Munk, the statistics of oceanic internal gravity waves has become a central subject of research in oceanography. The time evolution of the spectral energy of internal waves in the ocean can be described by a near-resonance wave turbulence equation, of quantum Boltzmann type. In this work, we provide the first rigorous mathematical study for the equation by showing the global existence and uniqueness of strong solutions.

math.AP↗

arXiv2019

On the Wave Turbulence Theory for the Nonlinear Schrödinger Equation with Random Potentials

We derive a new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form with the 4-wave turbulence kinetic equation in the wave turbulence theory. Moreover, we construct a class of self-similar solutions for the porous medium equation. These solutions spread infinitely as time goes to infinity and this fact answers the 'weak turbulence' question for the nonlinear Schrödinger equation with random potentials positively. We also derive Ohm's law for the porous medium equation.

math-ph↗

arXiv2018

Quantum hydrodynamic approximations to the finite temperature trapped Bose gases

For the quantum kinetic system modelling the Bose-Einstein Condensate that accounts for interactions between condensate and excited atoms, we use the Chapman-Enskog expansion to derive its hydrodynamic approximations, include both Euler and Navier-Stokes approximations. The hydrodynamic approximations describe not only the macroscopic behavior of the BEC but also its coupling with the non-condensates, which agrees with Landau's two fluid theory.

cond-mat.quant-gas↗

arXiv2018

Uniform in time lower bound for solutions to a quantum Boltzmann equation of bosons

We consider the quantum Boltzmann equation, which describes the growth of the condensate, or in other words, models the interaction between excited atoms and a condensate. In this work, the full form of Bogoliubov dispersion law is considered, which leads to a detailed study of surface integrals inside the collision operator on energy manifolds. We prove that positive radial solutions of the quantum Boltzmann equation are bounded from below by a Gaussian, uniformly in time.

math.AP↗

arXiv2018

The Cauchy problem and BEC stability for the quantum Boltzmann-Condensation system for bosons at very low temperature

We study a quantum Boltzmann-Condensation system that describes the evolution of the interaction between a well formed Bose-Einstein condensate and the quasi-particles cloud. The kinetic model is valid for a dilute regime at which the temperature of the gas is very low compared to the Bose-Einstein condensation critical temperature. In particular, our system couples the density of the condensate from a Gross-Pitaevskii type equation to the kinetic equation through the dispersion relation in the kinetic model and the corresponding transition probability rate from pre to post collision momentum states. We rigorously show the following three properties (1) the well-posedness of the Cauchy problem for the system in the case of a radially symmetric initial configuration, (2) find qualitative properties of the solution such as instantaneous creation of exponential tails and, (3) prove the uniform condensate stability related to the initial mass ratio between condensed particles and quasi-particles. The stability result from (3) leads to global in time existence of the initial value problem for the quantum Boltzmann-Condensation system.

math.AP↗

arXiv2018

On coupling kinetic and Schrodinger equations

We consider in this paper a system coupling a linear quantum Boltzmann equation and a defocusing cubic nonlinear Schrodinger equation. The Schrodinger equation reflects the dynamics of the wave function of the Bose-Einstein Condensate and the kinetic part of the system describes the evolution of the density function of the thermal cloud. An existence and uniqueness result for the system is supplied. We also prove the convergence to equilibrium of the density function of the thermal cloud and a scattering theory for the wave function of the condensate.

math-ph↗

arXiv2018

Optimal local well-posedness theory for the kinetic wave equation

We prove local existence and uniqueness results for the (space-homogeneous) 4-wave kinetic equation in wave turbulence theory. We consider collision operators defined by radial, but general dispersion relations satisfying suitable bounds, and we prove two local well-posedness theorems in nearly critical weighted spaces.

math.AP↗

arXiv2017

On the dynamics of finite temperature trapped Bose gases

The system that describes the dynamics of a Bose-Einstein Condensate (BEC) and the thermal cloud at finite temperature consists of a nonlinear Schrodinger (NLS) and a quantum Boltzmann (QB) equations. In such a system of trapped Bose gases at finite temperature, the QB equation corresponds to the evolution of the density distribution function of the thermal cloud and the NLS is the equation of the condensate. The quantum Boltzmann collision operator in this temperature regime is the sum of two operators $C_{12}$ and $C_{22}$, which describe collisions of the condensate and the non-condenstate atoms and collisions between non-condensate atoms. Above the BEC critical temperature, the system is reduced to an equation containing only $C_{22}$, which possesses a blow-up positive radial solution with respect to the $L^\infty$ norm (cf. \cite{EscobedoVelazquez:2015:FTB}). On the other hand, at the very low temperature regime, the system becomes an equation of $C_{12}$, with a different (much higher order) transition probability, which has a unique global positive radial solution with weighted $L^1$ norm (cf. \cite{AlonsoGambaBinh}). In the current temperature regime, we first decouple the QB and NLS equations, then show a global existence and uniqueness result for positive radial solutions to the spatially homogeneous kinetic system. Different from the case considered in \cite{EscobedoVelazquez:2015:FTB}, due to the presence of the BEC, the collision integrals are associated to sophisticated energy manifolds rather than spheres, since the particle energy is approximated by the Bogoliubov dispersion law. Moreover, the mass of the full system is not conserved while it is conserved for the case considered in \cite{EscobedoVelazquez:2015:FTB}. A new theory is then supplied.

math-ph↗

arXiv2017

On the kinetic equation in Zakharov's wave turbulence theory for capillary waves

The wave turbulence equation is an effective kinetic equation that describes the dynamics of wave spectrum in weakly nonlinear and dispersive media. Such a kinetic model has been derived by physicists in the sixties, though the well-posedness theory remains open, due to the complexity of resonant interaction kernels. In this paper, we provide a global unique radial strong solution, the first such a result, to the wave turbulence equation for capillary waves.

math.AP↗

arXiv2017

An Improved Iterative HDG Approach for Partial Differential Equations

We propose and analyze an iterative high-order hybridized discontinuous Galerkin (iHDG) discretization for linear partial differential equations. We improve our previous work (SIAM J. Sci. Comput. Vol. 39, No. 5, pp. S782--S808) in several directions: 1) the improved iHDG approach converges in a finite number of iterations for the scalar transport equation; 2) it is unconditionally convergent for both the linearized shallow water system and the convection-diffusion equation; 3) it has improved stability and convergence rates; 4) we uncover a relationship between the number of iterations and time stepsize, solution order, meshsize and the equation parameters. This allows us to choose the time stepsize such that the number of iterations is approximately independent of the solution order and the meshsize; and 5) we provide both strong and weak scalings of the improved iHDG approach up to $16,384$ cores. A connection between iHDG and time integration methods such as parareal and implicit/explicit methods are discussed. Extensive numerical results are presented to verify the theoretical findings.

math.NA↗

arXiv2017

A kinetic model for very low temperature dilute Bose gases

We review recent work on a kinetic model for very low temperature dilute Bose gases. The brief derivation, expressions for hydrodynamics modes, and the comparison with a experiment on a BEC of $^{87}{\rm Rb}$ atoms are presented.

cond-mat.quant-gas↗

arXiv2017

iHDG: An Iterative HDG Framework for Partial Differential Equations

We present a scalable iterative solver for high-order hybridized discontinuous Galerkin (HDG) discretizations of linear partial differential equations. It is an interplay between domain decomposition methods and HDG discretizations, and hence inheriting advances from both sides. In particular, the method can be viewed as a Gauss-Seidel approach that requires only independent element-by-element and face-by-face local solves in each iteration. As such, it is well-suited for current and future computing systems with massive concurrencies. Unlike conventional Gauss-Seidel schemes which are purely algebraic, the convergence of iHDG, thanks to the built-in HDG numerical flux, does not depend on the ordering of unknowns. We rigorously show the convergence of the proposed method for the transport equation, the linearized shallow water equation and the convection-diffusion equation. For the transport equation, the method is convergent regardless of mesh size $h$ and solution order $p$, and furthermore the convergence rate is independent of the solution order. For the linearized shallow water and the convection-diffusion equations we show that the convergence is conditional on both $h$ and $p$. Extensive steady and time-dependent numerical results for the 2D and 3D transport equations, the linearized shallow water equation, and the convection-diffusion equation are presented to verify the theoretical findings.

math.NA↗

arXiv2016

eHDG:An Exponentially Convergent Iterative Solver for HDG Discretizations of Hyperbolic Partial Differential Equations

We present a scalable and efficient iterative solver for high-order hybridized discontinuous Galerkin (HDG) discretizations of hyperbolic partial differential equations. It is an interplay between domain decomposition methods and HDG discretizations. In particular, the method is a fixed-point approach that requires only independent element-by-element local solves in each iteration. As such, it is well-suited for current and future computing systems with massive concurrencies. We rigorously show that the proposed method is exponentially convergent in the number of iterations for transport and linearized shallow water equations. Furthermore, the convergence is independent of the solution order. Various 2D and 3D numerical results for steady and time-dependent problems are presented to verify our theoretical findings.

math.NA↗