Related papers: Even Order Explicit Symplectic Geometric Algorithm…
Using gradient descent (GD) with fixed or decaying step-size is a standard practice in unconstrained optimization problems. However, when the loss function is only locally convex, such a step-size schedule artificially slows GD down as it…
This paper proposes some efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear partial differential equations (PDEs). The main idea of these algorithms is to utilize the solutions on the $k$-th…
We introduce a fast direct solver for variable-coefficient elliptic partial differential equations on surfaces based on the hierarchical Poincar\'e-Steklov method. The method takes as input an unstructured, high-order quadrilateral mesh of…
We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the…
In this paper, a piecewise quadratic nonconforming finite element method on rectangular grids for a fourth-order elliptic singular perturbation problem is presented. This proposed method is robustly convergent with respect to the…
We take into account the dynamics of three types of models of rotating galaxies in polar coordinates in a rotating frame. Due to non-axisymmetric potential perturbations, the angular momentum varies with time, and the kinetic energy depends…
Optimization algorithms are pivotal in advancing various scientific and industrial fields but often encounter obstacles such as trapping in local minima, saddle points, and plateaus (flat regions), which makes the convergence to reasonable…
We study symplectic numerical integration of mechanical systems with a Hamiltonian specified in non-canonical coordinates and its application to guiding-center motion of charged plasma particles in magnetic confinement devices. The…
In this paper, we address the challenge of solving large-scale graph-structured nonlinear programs (gsNLPs) in a scalable manner. GsNLPs are problems in which the objective and constraint functions are associated with nodes on a graph and…
This paper studies binary quadratic programs in which the objective is defined by a Euclidean distance matrix, subject to a general polyhedral constraint set. This class of nonconcave maximisation problems includes the capacitated,…
The numerical optimization of continuous functions is a fundamental task in many scientific and engineering domains, ranging from mechanical design to training of artificial intelligence models. Among the most effective and widely used…
This paper presents the Safe Sequential Quadratically Constrained Quadratic Programming (SS-QCQP) algorithm, a first-order method for smooth inequality-constrained nonconvex optimization that guarantees feasibility at every iteration. The…
We present an algorithm for planning trajectories that avoid obstacles and satisfy key-door precedence specifications expressed with a fragment of signal temporal logic. Our method includes a novel exact convex partitioning of the obstacle…
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…
Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of…
Explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable Hamiltonians. For the first time, the article proposes for arbitrary Hamiltonians similar integrators,…
We present a continuous/discontinuous Galerkin method for approximating solutions to a fourth order elliptic PDE on a surface embedded in $\mathbb{R}^3$. A priori error estimates, taking both the approximation of the surface and the…
An algorithm for solving first order ODEs, by systematically determining symmetries of the form [ xi = F(x), eta = P(x) y + Q(x) ], where xi d/dx + eta d/dy is the symmetry generator - is presented. To these {\it linear} symmetries one can…
Quantum error correction (QEC) is an essential concept for any quantum information processing device. Typically, QEC is designed with minimal assumptions about the noise process; this generic assumption exacts a high cost in efficiency and…
New implicit and implicit-explicit time-stepping methods for the wave equation in second-order form are described with application to two and three-dimensional problems discretized on overset grids. The implicit schemes are single step,…