Related papers: The ultraspherical rectangular collocation method …
This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an…
We consider the numerical solution of partial differential equations with coefficients that are strongly heterogeneous in space. We provide an overview of higher-order localized orthogonal decomposition (LOD) methods for the elliptic…
This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion…
We consider radial complex scaling/perfectly matched layer methods for scalar resonance problems in homogeneous exterior domains. We introduce a new abstract framework to analyze the convergence of domain truncations and discretizations.…
We present a sharp collocated projection method for solving the immiscible, two-phase Navier-Stokes equations in two- and three-dimensions. Our method is built using non-graded adaptive quadtree and octree grids, where all of the fluid…
The polynomial spline collocation method is proposed for solution of Volterra integral equations of the first kind with special piecewise continuous kernels. The Gauss-type quadrature formula is used to approximate integrals during the…
Recently, collocation based radial basis function (RBF) partition of unity methods (PUM) for solving partial differential equations have been formulated and investigated numerically and theoretically. When combined with stable evaluation…
Trust-region (TR) and adaptive regularization using cubics (ARC) have proven to have some very appealing theoretical properties for non-convex optimization by concurrently computing function value, gradient, and Hessian matrix to obtain the…
We extend the piecewise orthogonal collocation method to computing periodic solutions of coupled renewal and delay differential equations. Through a rigorous error analysis, we prove convergence of the relevant finite-element method and…
We examine a variational multiscale method in which the unresolved fine-scales are approximated element-wise using a discontinuous Galerkin method. We establish stability and convergence results for the methodology as applied to the scalar…
A numerical method based on the hybridizable discontinuous Galerkin method in space and backward Euler in time is formulated and analyzed for solving the miscible displacement problem. Under low regularity assumptions, convergence is…
In this paper, we study the stochastic collocation (SC) methods for uncertainty quantification (UQ) in hyperbolic systems of nonlinear partial differential equations (PDEs). In these methods, the underlying PDEs are numerically solved at a…
This paper focuses on scalability and robustness of spectral clustering for extremely large-scale datasets with limited resources. Two novel algorithms are proposed, namely, ultra-scalable spectral clustering (U-SPEC) and ultra-scalable…
In this work, we fully explore three refined convergence structures of the lowest-order rectangular Raviart-Thomas element in solving the Laplace eigenvalue problem. Firstly, the scheme possesses a property of supercloseness between the…
Orthogonal collocation methods are direct approaches for solving optimal control problems (OCP). A high solution accuracy is achieved with few optimization variables, making it more favorable for embedded and real-time NMPC applications.…
The unit-modulus least squares (UMLS) problem has a wide spectrum of applications in signal processing, e.g., phase-only beamforming, phase retrieval, radar code design, and sensor network localization. Scalable first-order methods such as…
Many iterative methods for solving optimization or feasibility problems have been invented, and often convergence of the iterates to some solution is proven. Under favourable conditions, one might have additional bounds on the distance of…
This work proposes a scheme for significantly reducing the computational complexity of discretized problems involving the non-smooth forward propagation of uncertainty by combining the adaptive hierarchical sparse grid stochastic…
The Douglas-Rachford splitting algorithm is a classical optimization method that has found many applications. When specialized to two normal cone operators, it yields an algorithm for finding a point in the intersection of two convex sets.…
Approximate solutions of linear and nonlinear integral equations using methods related to an interpolatory projection involve many integrals which need to be evaluated using a numerical quadrature formula. In this paper, we consider…