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The Monte Carlo method is a thriving and mathematically beautiful numerical technique used extensively, nowadays, to deal with many demanding problems in diverse fields. Here, we present an iterative Monte Carlo algorithm to work out very…
In this paper, we study the numerical algorithm for a nonlinear poroelasticity model with nonlinear stress-strain relations. By using variable substitution, the original problem can be reformulated to a new coupled fluid-fluid system, that…
In this paper, we focus on the control of the mean field equilibrium of non linear networks of the Langevin type in the limit of small noise. Using iterative linear approximations, we derive a formula that prescribes a control strategy in…
We analyze a canonical extension of the Stackelberg duopoly to a sequential framework, where each firm strategically anticipates the reactions of all subsequent players. In a triopoly (three-firm) settings, we obtain existence and…
We consider inverse problems for non-linear hyperbolic and elliptic equations and give an introduction to the method based on the multiple linearization, or on the construction of artificial sources, to solve these problems. The method is…
We consider the problem of finding sparse solutions to a system of underdetermined nonlinear system of equations. The methods are based on a Gauss-Newton approach with line search where the search direction is found by solving a linearized…
A new pattern search method for bound constrained optimization is introduced. The proposed algorithm employs the coordinate directions, in a suitable way, with a nonmonotone line search for accepting the new iterate, without using…
In this paper, for solving inconsistent matrix equations we propose a dual-space residual-based randomized extended Kaczmarz method and its version with Nesterov momentum. Without the full column rank assumptions on coefficient matrices, we…
This paper sets up a methodology for approximately solving optimal investment problems using duality methods combined with Monte Carlo simulations. In particular, we show how to tackle high dimensional problems in incomplete markets, where…
In this paper we propose a new finite element method for solving elliptic optimal control problems with pointwise state constraints, including the distributed controls and the Dirichlet or Neumann boundary controls. The main idea is to use…
This paper introduces the Non-linear Partition of Unity Method, a novel technique integrating Radial Basis Function interpolation and Weighted Essentially Non-Oscillatory algorithms. It addresses challenges in high-accuracy approximations,…
We propose a systemic method of applying the auxiliary systems of original equations to find the high order nonlocal symmetries of nonlinear evolution equation. In order to validate the effectiveness of the method, some examples are…
We propose a new method for Monte Carlo solution of non-linear integral equations by combining the Newton-Kantorovich method for solving non-linear equations with the Markov Chain Monte Carlo (MCMC) method for solving linear equations. The…
We propose an implicit iterative algorithm for an exact penalty method arising from inequality constrained optimization problems. A rapidly convergent fixed point method is developed for a regularized penalty functional. The applicability…
The method of this paper is my original creation. A new method for solving linear differential equations is proposed in this paper. The important conclusion of this paper is that arbitrary order linear ordinary differential equations with…
We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic.…
This paper proposes a Newton-type method to solve numerically the eigenproblem of several diagonalizable matrices, which pairwise commute. A classical result states that these matrices are simultaneously diagonalizable. From a suitable…
The problem of decoupling a nonsquare state space system by state feedback with singular input transformation is considered. The problem is solved by conducting a finite search for decouplable square systems, appropriately derived from the…
We propose a pseudo-market solution to resource allocation problems subject to constraints. Our treatment of constraints is general: including bihierarchical constraints due to considerations of diversity in school choice, or scheduling in…
Nonlinearity continuation method, applied to boundary value problems for steady-state Richards equation, gradually approaches the solution through a series of intermediate problems. Originally, the Newton method with simple line search…