In this study, we consider ill-posed time-domain inverse problems for dynamical systems with various boundary conditions and unknown controllers. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables. Radial Basis Functions (RBFs) are assumed as trial functions for th...

The linear systems arising from MFS calculations share certain numerical effects with other systems involving radial basis functions. These effects concern approximation error and stability, which are closely related, and they can already be studied for simple interpolation problems without PDEs. In MFS calculations, they crucially depend on the position and density of the source points and the...

This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The method consists of a Galerkin approximation in the space variables and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussia...

We propose a new iterative algorithm for the retrieval of the microphysical properties of stratospheric and tropospheric aerosols from multiwavelength lidar data. We consider the basic equation as an ill-posed problem and solve the system derived from spline collocation via a Padé iteration. The algorithm takes special care of the fact that the reconstruction of the distribution via spline coll...

In this study, we present numerical methods, based on the optimal quadratic spline collocation (OQSC) methods, for solving the shallow water equations (SWEs) in spherical coordinates. A quadratic spline collocation method approximates the solution of a differential problem by a quadratic spline. In the standard formulation, the quadratic spline is computed by making the residual of the differen...

The flux reconstruction (FR) approach unifies various high-order schemes, including collocation based nodal discontinuous Galerkin (DG) methods, and all spectral difference methods (at least for a linear flux function), within a single framework. Recently a new range of linearly stable FR schemes have been identified, henceforth referred to as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes. In...

We propose a novel direct transcription and solution method for solving nonlinear, continuous-time dynamic optimization problems. Instead of forcing the constraints to be satisfied only at selected number points as in collocation, new approach alternates between minimizing constraining squared norm constraint residuals integrated along whole trajectories. As result, can obtain solutions higher ...

Superconvergence phenomenon of the Legendre spectral collocation method and the p-version finite element method is discussed under the one dimensional setting. For a class of functions that satisfy a regularity condition (M): ‖u‖L∞ ≤ cMk on a bounded domain, it is demonstrated, both theoretically and numerically, that the optimal convergent rate is supergeometric. Furthermore, at proper Gaussia...

For example, investigating the Hamiltonian constraint equation for a black hole with angular momentum we find that, where finite difference solutions require a resolution of 1024×384 (radial×angular) grid points to find a solution of fractional error 10−5 in the ADM mass, a PSC solution achieves the same accuracy with only 12× 4 collocation points. Furthermore, the fractional error is reduced t...