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The solution of a (stochastic) differential equation can be locally approximated by a (stochastic) expansion. If the vector field of the differential equation is a polynomial, the corresponding expansion is a linear combination of iterated…
The application of error-free transformation (EFT) is recently being developed to solve ill-conditioned problems. It can reduce the number of arithmetic operations required, compared with multiple precision arithmetic, and also be applied…
Recent advances in the field of machine learning open a new era in high performance computing. Applications of machine learning algorithms for the development of accurate and cost-efficient surrogates of complex problems have already…
This contribution is dedicated to the exploration of exponential operator splitting methods for the time integration of evolution equations. It entails the review of previous achievements as well as the depiction of novel results. The…
We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our…
SMT solvers use sophisticated techniques for polynomial (linear or non-linear) integer arithmetic. In contrast, non-polynomial integer arithmetic has mostly been neglected so far. However, in the context of program verification, polynomials…
Most current prevalent iterative methods can be classified into the so-called extended Krylov subspace methods, a class of iterative methods which do not fall into this category are also proposed in this paper. Comparing with traditional…
We introduce PoCET: a free and open-scource Polynomial Chaos Expansion Toolbox for Matlab, featuring the automatic generation of polynomial chaos expansion (PCE) for linear and nonlinear dynamic systems with time-invariant stochastic…
The anisotropic diffusion equation is imperative in understanding cosmic ray diffusion across the Galaxy, the heliosphere, and its interplay with the ambient magnetic field. This diffusion term contributes to the highly stiff nature of the…
In this paper, for solving a class of linear parabolic equations in rectangular domains, we have proposed an efficient Parareal exponential integrator finite element method. The proposed method first uses the finite element approximation…
Simulations of the dynamics generated by partial differential equations (PDEs) provide approximate, numerical solutions to initial value problems. Such simulations are ubiquitous in scientific computing, but the correctness of the results…
We introduce DDE-Solver, a Maple package designed for solving Discrete Differential Equations (DDEs). These equations are functional equations relating algebraically a formal power series F(t, u) with polynomial coefficients in a…
Computational implementations for solving systems of linear equations often rely on a one-size-fits-all approach based on LU decomposition of dense matrices stored in column-major format. Such solvers are typically implemented with the aid…
Forward time step integrators are splitting algorithms with only positive splitting coefficients. When used in solving physical evolution equations, these positive coefficients correspond to positive time steps. Forward algorithms are…
With the stagnation of processor core performance, further reductions in the time-to-solution for geophysical fluid problems are becoming increasingly difficult with standard time integrators. Parallel-in-time exposes and exploits…
In the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system depending on a small parameter, we propose a time-stepping method which is numerically uniformly accurate when the parameter goes to zero. Based on an…
We introduce a new class of arbitrary-order exponential time differencing methods based on spectral deferred correction (ETDSDC) and describe a simple procedure for initializing the requisite matrix functions. We compare the stability and…
The usual explicit finite-difference method of solving partial differential equations is limited in stability because it approximates the exact amplification factor by power-series. By adapting the same exponential-splitting method of…
In this work, we introduce DIRECTGO, a new MATLAB toolbox for derivative-free global optimization. DIRECTGO collects various deterministic derivative-free DIRECT-type algorithms for box-constrained, generally-constrained, and problems with…
IFISS is an established MATLAB finite element software package for studying strategies for solving partial differential equations (PDEs). IFISS3D is a new add-on toolbox that extends IFISS capabilities for elliptic PDEs from two to three…