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We present a novel framework for PDE-constrained $r$-adaptivity of high-order meshes. The proposed method formulates mesh movement as an optimization problem, with an objective function defined as a convex combination of a mesh quality…
This paper proposes an explicit computational method for solving a three-dimensional system of nonlinear elastodynamic sine-Gordon equations subject to appropriate initial and boundary conditions. The time derivative is approximated by…
In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the…
This paper develops a novel deep learning approach for solving evolutionary equations, which integrates sequential learning strategies with an enhanced hard constraint strategy featuring trainable parameters, addressing the low…
We propose a novel finite-difference time-domain (FDTD) scheme for the solution of the Maxwell's equations in which linear dispersive effects are present. The method uses high-order accurate approximations in space and time for the…
We consider the adaptive-rank integration of {2D and 3D} time-dependent advection-diffusion partial differential equations (PDEs) with variable coefficients. We employ a standard finite-difference method for spatial discretization coupled…
This paper is concerned with the theory, construction and application of variable-stepsize implicit Peer two-step methods that are super-convergent for variable stepsizes, i.e., preserve their classical order achieved for uniform stepsizes…
First order optimization algorithms play a major role in large scale machine learning. A new class of methods, called adaptive algorithms, were recently introduced to adjust iteratively the learning rate for each coordinate. Despite great…
Exponential time differencing methods is a power tool for high-performance numerical simulation of computationally challenging problems in condensed matter physics, fluid dynamics, chemical and biological physics, where mathematical models…
This paper presents a novel adaptive-sparse polynomial dimensional decomposition (PDD) method for stochastic design optimization of complex systems. The method entails an adaptive-sparse PDD approximation of a high-dimensional stochastic…
This paper presents an efficient high-order sharp-interface method for solving the three-dimensional (3D) Poisson equation with Dirichlet boundary conditions on a nonuniform Cartesian grid with irregular domain boundaries. The new approach…
Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and…
We study an efficient strategy based on finite elements to value spread options on commodities whose underlying assets follow a dynamic described by a certain class of two-dimensional Levy models by solving their associated partial…
We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\geq 1$. Their construction is accomplished by both the…
Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in [7]. We…
In this study, we first present an improved version of the classical sixth-order combined compact difference (CCD6) scheme to enhance the convective stability of advection equations through an increased dispersion accuracy. This improved…
Recently, adaptive control systems with relaxed persistent excitation (PE) conditions have been proposed to guarantee true parameter convergence and improve the transient response. However, in some cases, sufficient control performance and…
In this paper, we will present advanced discretization methods for solving retarded potential integral equations. We employ a $C^{\infty}$-partition of unity method in time and a conventional boundary element method for the spatial…
There is a rising interest in Spatio-temporal systems described by Partial Differential Equations (PDEs) among the control community. Not only are these systems challenging to control, but the sizing and placement of their actuation is an…
In this work, we propose and develop efficient and accurate numerical methods for solving the Kirchhoff-Love plate model in domains with complex geometries. The algorithms proposed here employ curvilinear finite-difference methods for…