arXiv subjects

Vo Anh Khoa

Publications and source records attributed to Vo Anh Khoa.

32 records · Page 2Linked to original sources

arXiv2018

Application of the cut-off projection to solve a backward heat conduction problem in a two-slab composite system

The main goal of this paper is applying the cut-off projection for solving one-dimensional backward heat conduction problem in a two-slab system with a perfect contact. In a constructive manner, we commence by demonstrating the Fourier-based solution that contains the drastic growth due to the high-frequency nature of the Fourier series. Such instability leads to the need of studying the projection method where the cut-off approach is derived consistently. In the theoretical framework, the first two objectives are to construct the regularized problem and prove its stability for each noise level. Our second interest is estimating the error in $L^2$-norm. Another supplementary objective is computing the eigen-elements. All in all, this paper can be considered as a preliminary attempt to solve the heating/cooling of a two-slab composite system backward in time. Several numerical tests are provided to corroborate the qualitative analysis.

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arXiv2017

Corrector homogenization estimates for a non-stationary Stokes-Nernst-Planck-Poisson system in perforated domains

We consider a non-stationary Stokes-Nernst-Planck-Poisson system posed in perforated domains. Our aim is to justify rigorously the homogenization limit for the upscaled system derived by means of two-scale convergence in \cite{RMK12}. In other words, we wish to obtain the so-called corrector homogenization estimates that specify the error obtained when upscaling the microscopic equations. Essentially, we control in terms of suitable norms differences between the micro- and macro-concentrations and between the corresponding micro- and macro-concentration gradients. The major challenges that we face are the coupled flux structure of the system, the nonlinear drift terms and the presence of the microstructures. Employing various energy-like estimates, we discuss several scalings choices and boundary conditions.

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arXiv2017

The Cauchy problem of coupled elliptic sine-Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing

Developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering. In this paper, we explore the regularization method for the coupled elliptic sine-Gordon equations along with Cauchy data. The system of equations originates from the static case of the coupled hyperbolic sine-Gordon equations modeling the coupled Josephson junctions in superconductivity, and so far it addresses the Josephson $π$-junctions. In general, the Cauchy problem is not well-posed, and herein the Hadamard-instability occurs drastically. Generalizing the kernel-based regularization method, we propose a stable approximate solution. Confirmed by the error estimate, this solution strongly converges to the exact solution in $L^2$-norm. The main concern of this paper is also with the way to compute the regularized solution formed by an alike integral equation. We employ the proposed techniques that successfully approximated the highly oscillatory integral, and apply the Picard-like iteration to organize an efficient and reliable tool of computations. The results are viewed as the improvement as well as the generalization of many previous works. The paper is also accompanied by a numerical example that demonstrates the potential of this idea.

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arXiv2017

Correctors justification for a Smoluchowski--Soret--Dufour model posed in perforated domains

We study a coupled thermo-diffusion system that accounts for the dynamics of hot colloids in periodically heterogeneous media. Our model describes the joint evolution of temperature and colloidal concentrations in a saturated porous structure, where the Smoluchowski interactions are responsible for aggregation and fragmentation processes in the presence of Soret-Dufour type effects. Additionally, we allow for deposition and depletion on internal micro-surfaces. In this work, we derive corrector estimates quantifying the rate of convergence of the periodic homogenization limit process performed in \cite{KAM14} via two-scale convergence arguments. The major technical difficulties in the proof are linked to the estimates between nonlinear processes of aggregation and deposition and to the convergence arguments of the \emph{a priori} information of the oscillating weak solutions and cell functions in high dimensions. Essentially, we circumvent the arisen difficulties by a suitable use of the energy method and of fine integral estimates controlling interactions at the level of micro-surfaces.

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arXiv2016

A generalized filter regularization result for some nonlinear evolution equations in Hilbert spaces

Despite the strong focus of regularization on ill-posed problems, the general construction of such methods has not been fully explored. Moreover, many previous studies cannot be clearly adapted to handle more complex scenarios, albeit the greatly increasing concerns on the improvement of wider classes. In this note, we rigorously study a general theory for filter regularized operators in a Hilbert space for nonlinear evolution equations which have occurred naturally in different areas of science. The starting point lies in problems that are in principle ill-posed with respect to the initial/final data\textendash these basically include the Cauchy problem for nonlinear elliptic equations and the backward-in-time nonlinear parabolic equations. We derive general filters that can be used to stabilize those problems. Essentially, we establish the corresponding well-posed problem whose solution converges to the solution of the ill-posed problem. The approximation can be confirmed by the error estimates in the Hilbert space. This work improves very much many papers in the same line of field.

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arXiv2016

A high-order corrector estimate for a semi-linear elliptic system in perforated domains

We derive in this note a high-order corrector estimate for the homogenization of a microscopic semi-linear elliptic system posed in perforated domains. The major challenges are the presence of nonlinear volume and surface reaction rates. This type of correctors justifies mathematically the convergence rate of formal asymptotic expansions for the two-scale homogenization settings. As main tool, we follow the standard approach by the energy-like method to investigate the error estimate between the micro and macro concentrations and micro and macro concentration gradients. This work aims at generalizing the results reported in [2, 7].

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arXiv2016

Reconstruction of the electric field of the Helmholtz equation in 3D

In this paper, we rigorously investigate the truncation method for the Cauchy problem of Helmholtz equations which is widely used to model propagation phenomena in physical applications. The method is a well-known approach to the regularization of several types of ill-posed problems, including the model postulated by Regi\' nska and Regi\' nski \cite{RR06}. Under certain specific assumptions, we examine the ill-posedness of the non-homogeneous problem by exploring the representation of solutions based on Fourier mode. Then the so-called regularized solution is established with respect to a frequency bounded by an appropriate regularization parameter. Furthermore, we provide a short analysis of the nonlinear forcing term. The main results show the stability as well as the strong convergence confirmed by the error estimates in $L^2$-norm of such regularized solutions. Besides, the regularization parameters are formulated properly. Finally, some illustrative examples are provided to corroborate our qualitative analysis.

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arXiv2015

Asymptotic analysis of a semi-linear elliptic system in perforated domains: well-posedness and correctors for the homogenization limit

In this study, we prove results on the weak solvability and homogenization of a microscopic semi-linear elliptic system posed in perforated media. The model presented here explores the interplay between stationary diffusion and both surface and volume chemical reactions in porous media. Our interest lies in deriving homogenization limits (upscaling) for alike systems and particularly in justifying rigorously the obtained averaged descriptions. Essentially, we prove the well-posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations and then use the structure of the two scale expansions to derive corrector estimates delimitating this way the convergence rate of the asymptotic approximates to the macroscopic limit concentrations. Our techniques include Moser-like iteration techniques, a variational formulation, two-scale asymptotic expansions as well as energy-like estimates.

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arXiv2015

Stability and error estimates of a general modified quasi-boundary value method via a semi-linear backward parabolic equation

Regularization methods have been recently developed to construct stable approximate solutions to classical partial differential equations considered as final value problems. In this paper, we investigate the backward parabolic problem with locally Lipschitz source: $\partial_{t}u+μ\left(t\right)\mathcal{A}u\left(t\right)=f\left(t,u\right)$ where $\mathcal{A}:\mathcal{D}\left(\mathcal{A}\right)\subset\mathcal{H}\to\mathcal{H}$ is a positive, self-adjoint and unbounded linear operator on the Hilbert space $\mathcal{H}$. The problem arises in many applications, but it is in general ill-posed. The ill-posedness is caused by catastrophic growth in the representation of solution, then independence of solution on data makes computational procedures impossible. Therefore, we contribute to this interesting field the study of the stable approximation solution via the modified quasi-boundary value method based on nonlinear spectral theory, which slightly extends the results in many works. Our main focus is to qualitatively prove the stability and convergence rate where semi-group technicalities are highly useful.

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arXiv2015

A modified integral equation method of the nonlinear elliptic equation with globally and locally Lipschitz source

The paper is devoted to investigating a Cauchy problem for nonlinear elliptic PDEs in the abstract Hilbert space. The problem is hardly solved by computation since it is severely ill-posed in the sense of Hadamard. We shall use a modified integral equation method to regularize the nonlinear problem with globally and locally Lipschitz source terms. Convergence estimates are established under priori assumptions on exact solution. A numerical test is provided to illustrate that the proposed method is feasible and effective. These results extend some earlier works on a Cauchy problem for elliptic equations.

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arXiv2015

On an inverse problem in the parabolic equation arising from groundwater pollution problem

In this paper, we consider an inverse problem to determine a source term in a parabolic equation, where the data are obtained at a certain time. In general, this problem is ill-posed, therefore the Tikhonov regularization method is proposed to solve the problem. In the theoretical results, a priori error estimate between the exact solution and its regularized solutions is obtained. We also propose both methods, a priori and a posteriori parameter choice rules. In addition, the proposed methods have been verified by numerical experiments to estimate the errors between the regularized solutions and exact solutions. Eventually, from the numerical results it shows that the a posteriori parameter choice rule method gives a better the convergence speed in comparison with the a priori parameter choice rule method in some specific applications.

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arXiv2014

On the regularization of solution of an inverse ultraparabolic equation associated with perturbed final data

In this paper, we study the inverse problem for a class of abstract ultraparabolic equations which is well-known to be ill-posed. We employ some elementary results of semi-group theory to present the formula of solution, then show the instability cause. Since the solution exhibits unstable dependence on the given data functions, we propose a new regularization method to stabilize the solution. then obtain the error estimate. A numerical example shows that the method is efficient and feasible. This work slightly extends to the earlier results in Zouyed et al. \cite{key-9} (2014).

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arXiv2014

A numerical approach to approximation for an ultraparabolic equation

We study the following ultraparabolic equation \[ \frac{\partial}{\partial t}u\left(t,s\right)+\frac{\partial}{\partial s}u\left(t,s\right)+\mathcal{L}u\left(t,s\right)=f\left(u\left(t,s\right),t,s\right),\quad\left(t,s\right)\in\left(0,T\right)\times\left(0,T\right), \] where $\mathcal{L}$ is a positive-definite, self-adjoint operator with compact inverse and $f$ is a nonlinear function. Mathematically, the bibliography on initial-boundary value problems for ultraparabolic equations is not extensive although the problems have many applications related to option pricing, multi parameter Brownian motion, population dynamics and so forth. In this paper, we present the approximate solution by virtue of finite difference scheme and Fourier series. For the linear case, we give the approximate solution and obtain a stability result. For the nonlinear case, we use an iterative scheme by linear approximation to get the approximate solution and obtain error estimates. Some numerical examples are given to demonstrate the efficiency of the method.

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arXiv2014

Approximation of mild solutions of the linear and nonlinear elliptic equations

In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form \[ \frac{\partial^{2}}{\partial t^{2}}u\left(t\right)=\mathcal{A}u\left(t\right)+f\left(t,u\left(t\right)\right),\quad t\in\left[0,T\right], \] where $\mathcal{A}$ is a positive-definite, self-adjoint operator with compact inverse. As we know, these problems are well-known to be ill-posed. On account of the orthonormal eigenbasis and the corresponding eigenvalues related to the operator, the method of separation of variables is used to show the solution in series representation. Thereby, we propose a modified method and show error estimations in many accepted cases. For illustration, two numerical examples, a modified Helmholtz equation and an elliptic sine-Gordon equation, are constructed to demonstrate the feasibility and efficiency of the proposed method.

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