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We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of…
In this paper, we propose two discontinuous dynamical systems in continuous time with guaranteed prescribed finite-time local convergence to strict local minima of a given cost function. Our approach consists of exploiting a Lyapunov-based…
The multimesh finite element method enables the solution of partial differential equations on a computational mesh composed by multiple arbitrarily overlapping meshes. The discretization is based on a continuous--discontinuous function…
The purpose of this paper is to propose and analyze a multi-step iterative algorithm to solve a convex optimization problem and a fixed point problem posed on a Hadamard space. The convergence properties of the proposed algorithm are…
Recently there have been exciting developments in Monte Carlo methods, with the development of new MCMC and sequential Monte Carlo (SMC) algorithms which are based on continuous-time, rather than discrete-time, Markov processes. This has…
Peridynamics is a nonlocal generalization of continuum mechanics theory which adresses discontinuous problems without using partial derivatives and replacing its by an integral operator. As a consequence, it finds applications in the…
In recent years, various subspace algorithms have been developed to handle large-scale optimization problems. Although existing subspace Newton methods require fewer iterations to converge in practice, the matrix operations and full…
Adaptivity and local mesh refinement are crucial for the efficient numerical simulation of wave phenomena in complex geometry. Local mesh refinement, however, can impose a tiny time-step across the entire computational domain when using…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
We revisit adaptive time stepping, one of the classical topics of numerical analysis and computational engineering. While widely used in application and subject of many theoretical works, a complete understanding is still missing. Apart…
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…
Bayesian optimization methods allocate limited sampling budgets to maximize expensive-to-evaluate functions. One-step-lookahead policies are often used, but computing optimal multi-step-lookahead policies remains a challenge. We consider a…
Multidimensional minimax single facility location problems with Chebyshev distance are examined within the framework of idempotent algebra. A new algebraic solution based on an extremal property of the eigenvalues of irreducible matrices is…
We present a practical algorithm to approximate the exponential of skew-Hermitian matrices up to round-off error based on an efficient computation of Chebyshev polynomials of matrices and the corresponding error analysis. It is based on…
Discontinuous Galerkin methods are developed for solving the Vlasov-Maxwell system, methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy. Such properties in…
Multi-adaptive Galerkin methods are extensions of the standard continuous and discontinuous Galerkin methods for the numerical solution of initial value problems for ordinary or partial differential equations. In particular, the…
In a recent paper Lima, Panario and Wang have provided a new method to multiply polynomials in Chebyshev basis which aims at reducing the total number of multiplication when polynomials have small degree. Their idea is to use Karatsuba's…
In this paper, we present the numerical solution of two-phase flow problems of engineering significance with a space-time finite element method that allows for local temporal refinement. Our basis is the method presented in [3], which…
Convergence failure and slow convergence rates are among the biggest challenges with solving the system of non-linear equations numerically. Although mitigated, such issues still linger when using strictly small time steps and…
An efficient algorithm for time propagation of the time-dependent Kohn-Sham equations is presented. The algorithm is based on dividing the Hamiltonian into small time steps and assuming that it is constant over these steps. This allows for…