Related papers: Results and questions on a nonlinear approximation…
A nonlinear partial differential equation is a nonlinear relationship between an unknown function and how it changes due to two or more input variables. A numerical method reduces such an equation to arithmetic for quick visualization, but…
A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure…
A variational approach is used to develop a robust numerical procedure for solving the nonlinear Poisson-Boltzmann equation. Following Maggs et al., we construct an appropriate constrained free energy functional, such that its…
The Poisson-Boltzmann equation is widely used to model molecular electrostatics; however, it is usually solved in linearised form because the sinh nonlinearity is challenging, limiting its applicability in highly charged systems such as…
The paper proposes and justifies a new algorithm of the proximal Newton type to solve a broad class of nonsmooth composite convex optimization problems without strong convexity assumptions. Based on advanced notions and techniques of…
The classical Ka\v{c}anov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation,…
In this paper we study about the existence of solutions of certain kind of non-linear differential and differential-difference equations. We give partial answer to a problem which was asked by chen et al. in [13].
As an alternative to PINNs, a Deep Ritz framework is proposed to solve fully nonlinear PDEs. A least-squares algorithm is advocated to decouple the nonlinearities from the variational features of several fully nonlinear PDEs. A splitting…
We introduce an $r-$adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we…
In this paper we propose a new method to stabilise non-symmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilised finite element method. Both stabilisation of the element…
To extract the approximate solutions in the case of nonlinear fractional order differential equations with the homogeneous and nonhomogeneous boundary conditions, the weighted residual method is embedded here. We exploit three methods such…
We introduce a direct numerical treatment of nonlinear higher-index differential-algebraic equations by means of overdetermined polynomial least-squares collocation. The procedure is not much more computationally expensive than standard…
In this paper we present applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. According to variational approach in the general case we have the solution as a…
This paper proposes a nonmonotone proximal quasi-Newton algorithm for unconstrained convex multiobjective composite optimization problems. To design the search direction, we minimize the max-scalarization of the variations of the Hessian…
The paper develops new methods of non-parametric estimation a compound Poisson distribution. Such a problem arise, in particular, in the inference of a Levy process recorded at equidistant time intervals. Our key estimator is based on…
Nonlinear least-squares problems are a special class of unconstrained optimization problems in which their gradient and Hessian have special structures. In this paper, we exploit these structures and proposed a matrix-free algorithm with a…
In this paper we extend the hierarchical model reduction framework based on reduced basis techniques for the application to nonlinear partial differential equations. The major new ingredient to accomplish this goal is the introduction of…
Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental…
We study approximation of non-autonomous linear differential equations with variable delay over infinite intervals. We use piecewise constant argument to obtain a corresponding discrete difference equation. The study of numerical…
We present a new adaptive collocation scheme for solving partial differential equations based on Local Coupled Multiquadrics (LCMQs) within a covers-and-nodes framework. The method, referred to as the Adaptive Ch Method, automatically…