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It has been recently demonstrated that self-defocusing (SDF) media with the cubic nonlinearity, whose local coefficient grows from the center to periphery fast enough, support stable bright solitons, without the use of any linear potential.…

A third-order weighted essentially non-oscillatory compact least-squares scheme is developed for the finite volume method on structured curvilinear non-uniform grids. The proposed scheme features compact least-squares reconstruction with…

We propose a high order finite difference linear scheme combined with a high order bound preserving maximum-principle-preserving (MPP) flux limiter to solve the incompressible flow system. For such problem with highly oscillatory structure…

We introduce a new scheme adaption strategy for one- and two-dimensional hyperbolic systems of conservation laws. The proposed approach builds upon the adaptive framework introduced in [S. Chu, A. Kurganov, and I. Menshov, Appl. Numer.…

This work is devoted to the numerical simulation of nonlinear Schr\"odinger and Klein-Gordon equations. We present a general strategy to construct numerical schemes which are uniformly accurate with respect to the oscillation frequency.…

For time-dependent PDEs, the numerical schemes can be rendered bound-preserving without losing conservation and accuracy, by a post processing procedure of solving a constrained minimization in each time step. Such a constrained…

In this thesis, advanced design technique in sliding mode control (SMC) is presented with focus on PID (Proportional-Integral-Derivative) type Sliding surfaces based Sliding mode control with improved power rate exponential reaching law for…

We propose an improved version of the PAMPA algorithm where the solution is sought as globally continuous. The scheme is locally conservative, and there is no mass matrix to invert. This method had been developed in a series of papers, see…

Stability and boundedness analysis for vector nonlinear systems with variable delays and coefficients remains challenging due to the conservatism of existing methods. Moreover, estimates of the transient behavior of solution norms remain…

We present a new class of component-wise numerical schemes that are in the family of relaxation formulations, originally introduced by [S. Jin and Z. P. Xin, Comm. Pure Appl. Math., 48(1995), pp. 235-277]. The relaxation framework enables…

In this study, we propose a class of total variation diminishing (TVD) schemes for solving pseudo-monotone variational inequality arises in elasto-hydrodynamic lubrication point contact problem. A limiter based stable hybrid line splittings…

Nonconvex-nonconcave minimax optimization has gained widespread interest over the last decade. However, most existing works focus on variants of gradient descent-ascent (GDA) algorithms, which are only applicable to smooth nonconvex-concave…

We prove the stability with respect to the flux of solutions to initial-boundary value problems for scalar non-autonomous conservation laws in one space dimension. Key estimates are obtained through a careful construction of the solutions.

This paper presents a novel visual-LiDAR odometry and mapping method with low-drift characteristics. The proposed method is based on two popular approaches, ORB-SLAM and A-LOAM, with monocular scale correction and visual-bootstrapped LiDAR…

Studies of topological bands and their associated low-dimensional boundary modes have largely focused on linear systems. This work reports robust dynamical features of three-dimensional (3D) nonlinear systems in connection with intriguing…

Existing methods rarely capture the temporal evolution of solution norms in vector nonlinear DDEs with variable delays and coefficients, often leading to overly conservative boundedness and stability criteria. We develop a framework that…

Trained neural networks (NN) have attractive features for closing governing equations. There are many methods that are showing promise, but all can fail in cases when small errors consequentially violate physical reality, such as a solution…

We consider density estimation under measurement error with the Smoothness-Penalized Deconvolution (SPeD) estimator. The estimator has a tuning parameter regulating the smoothness of the estimate, and proper choice of this parameter is…

We introduce a model of media with the cubic attractive nonlinearity concentrated along a single or double stripe in the two-dimensional (2D) plane. The model can be realized in terms of nonlinear optics (in the spatial and temporal domains…

We consider hyperbolic systems of conservation laws in one spatial dimension. For any limit of front tracking solutions $v$, and for a general weak solution $u\in L^\infty$ with no BV assumption, we prove the following H\"older-type…