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In this paper, we consider Burgers' equation with uncertain boundary and initial conditions. The polynomial chaos (PC) approach yields a hyperbolic system of deterministic equations, which can be solved by several numerical methods. Here,…
In this paper the approximate solution of the heat exchange problem by A.N. Tikhonov regularization method is presented. The errror estimation of this approximate is obtained.
In this paper, we address the problem of approximating solutions of ill-posed problems using mollification. We quickly review existing mollification regularization methods and provide two new approximate solutions to a general ill-posed…
The well-known analytical solution of Burgers' equation is extended to curvilinear coordinate systems in three-dimensions by a method which is much simpler and more suitable to practical applications than that previously used. The results…
We propose a general algorithm for approximating nonstandard Bayesian posterior distributions. The algorithm minimizes the Kullback-Leibler divergence of an approximating distribution to the intractable posterior distribution. Our method…
In this article we establish strong convergence rates on the whole probability space for explicit full-discrete approximations of stochastic Burgers equations with multiplicative trace-class noise. The key step in our proof is to establish…
We introduce an algorithm to solve linear inverse problems regularized with the total (gradient) variation in a gridless manner. Contrary to most existing methods, that produce an approximate solution which is piecewise constant on a fixed…
This paper focuses on the regularization of backward time-fractional diffusion problem on unbounded domain. This problem is well-known to be ill-posed, whence the need of a regularization method in order to recover stable approximate…
In this paper, we explore a nonlocal inviscid Burgers equation. Fixing a parameter $h$, we prove existence and uniqueness of the local solution of the equation $\InviscidBurgersNonlocal{u}$ with periodic initial condition. We also explore…
We propose an analysis for the stabilized finite element methods proposed in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput., 35(6) 2013,…
In this article a modified Levenberg-Marquardt method coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations is investigated. We show that the proposed method is a convergent…
In this project we investigate the stochastic Burgers' equation with multiplicative space-time white noise on an unbounded spatial domain. We give a random field solution to this equation by defining a process via a kind of Feynman-Kac…
Analytic solutions for Burgers equations with source terms, possibly stiff, represent an important element to assess numerical schemes. Here we present a procedure, based on the characteristic technique to obtain analytic solutions for…
We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently in [Hairer, Weber, Probab. Theory…
In this short note, we formulate the convergence rates of the well known Tikhonov regularization scheme for solving the nonlinear ill-posed problems in Banach spaces. For deriving the convergence rates, we employ the novel smoothness…
In this paper, we investigate an inverse Cauchy problem for a stochastic hyperbolic equation. A Lipschitz type observability estimate is established using a pointwise Carleman identity. By minimizing the constructed Tikhonov-type…
A novel theoretical framework, the inverse problem approach, is proposed to calculate non-perturbative quantities in quantum chromodynamics (QCD). Based on the dispersion relation of quantum field theory, this approach determines unknown…
This paper deals with Lavrentiev regularization for solving linear ill-posed problems, mostly with respect to accretive operators on Hilbert spaces. We present converse and saturation results which are an important part in regularization…
We consider the numerical approximation of the ill-posed data assimilation problem for stationary convection-diffusion equations and extend our previous analysis in [Numer. Math. 144, 451--477, 2020] to the convection-dominated regime.…
In this study collocation method based on the extended B-spline functions for the numerical solutions of the Generalized Burhers Fisher equation is set up. The approximate solution of the equation is constructed with the combination of the…