Related papers: Matlab program method of computing Carleman estima…
Carleman linearization is a mathematical technique that transforms nonlinear dynamical systems into infinite-dimensional linear systems, enabling simplified analysis. Initially developed for ordinary differential equations (ODEs) and later…
We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ \partial_t^\alpha u - Lu + \lambda u= f \quad \mathrm{in} \quad (0,T) \times \mathbb{R}^d,$$ where $\partial_t^\alpha u$ is the Caputo…
We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure $\mu(\om)$ of (not necessarily compact) basic semi-algebraic sets$\om\subset\R^n$. We obtain two monotone (non increasing and non…
We develop efficient numerical integration methods for computing an integral whose integrand is a product of a smooth function and the Gaussian function with a small standard deviation. Traditional numerical integration methods applied to…
In the present work, an attempted was made to develop a numerical algorithm by the use of new orthogonal hybrid functions formed from hybrid of piecewise constant orthogonal sample-and-hold functions and piecewise linear orthogonal…
The first objective of the paper is to estimate logarithmic partial derivative for meromorphic functions in several complex variables. Our estimations for logarithmic partial derivatives extend the results of Gundersen \cite{GG2} to the…
A version of the convexification globally convergent numerical method is constructed for a coefficient inverse problem for a wave-like partial differential equation. The presence of the Carleman Weight Function in the corresponding…
In this note we provide an algorithm for computing the fractional integrals of orthogonal polynomials, which is more stable than that using the expression of the polynomials w.r.t. the canonical basis. This algorithm is aimed at solving…
Robust estimation is essential in computer vision, robotics, and navigation, aiming to minimize the impact of outlier measurements for improved accuracy. We present a fast algorithm for Geman-McClure robust estimation, FracGM, leveraging…
We present two approaches for enhancing the accuracy of second order finite difference approximations of two-dimensional semilinear parabolic systems. These are the fourth order compact difference scheme and the fourth order scheme based on…
We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains containing a potential that is critically singular, diverging as an inverse square on all the boundary of the domain. These estimates are…
Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional…
We obtain Calder{\'o}n-Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous coefficients. We then derive the well-posedness of the martingale problem associated to related degenerate operators, and therefore…
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of…
This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method…
This is a survey, which is a continuation of the previous survey of the author about applications of Carleman estimates to Inverse Problems, J. Inverse and Ill-Posed Problems, 21, 477-560, 2013. It is shown here that Tikhonov functionals…
We analyse stability of observability estimates for solutions to wave and Scr\" odinger equations subjected to additive perturbations. The paper generalises the recent averaged observability/control result by allowing for systems consisting…
We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and…
In this paper we established a class of optimal fourth-order methods which is obtained by existing third-order method for solving nonlinear equations for simple roots by using weight functions. Some physical examples are given to illustrate…
The problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well-known to be ill-posed and it is regularized by backward Sobolev-type equations. Error…