Related papers: Fractional calculus via variable-transform-based s…
The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the…
In a recent paper we have suggested that the finite temperature density matrix can be computed efficiently by a combination of polynomial expansion and iterative inversion techniques. We present here significant improvements over this…
Unions of graph multiplier operators are an important class of linear operators for processing signals defined on graphs. We present a novel method to efficiently distribute the application of these operators. The proposed method features…
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
In this paper we are interested in the approximation of fractional powers of self-adjoint positive operators. Starting from the integral representation of the operators, we apply the trapezoidal rule combined with a single-exponential and a…
We introduce a numerical method for the approximation of functions which are analytic on compact intervals, except at the endpoints. This method is based on variable transforms using particular parametrized exponential and…
The Chapman-Kolmogorov equation with fractional integrals is derived. An integral of fractional order is considered as an approximation of the integral on fractal. Fractional integrals can be used to describe the fractal media. Using…
We review the theory of optimal polynomial and rational Chebyshev approximations, and Zolotarev's formula for the sign function over the range (\epsilon \leq |z| \leq1). We explain how rational approximations can be applied to large sparse…
We propose a spectral method by using the Jacobi functions for computing eigenvalue gaps and their distribution statistics of the fractional Schr\"{o}dinger operator (FSO). In the problem, in order to get reliable gaps distribution…
We develop a computationally efficient and robust algorithm for generating pseudo-random samples from a broad class of smooth probability distributions in one and two dimensions. The algorithm is based on inverse transform sampling with a…
It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved, due to the…
Quantum generative modeling is emerging as a powerful tool for advancing data analysis in high-energy physics, where complex multivariate distributions are common. However, efficiently learning and sampling these distributions remains…
We consider a fast approximation method for a solution of a certain stochastic non-local pseudodifferential equation. This equation defines a Mat\'ern class random field. The approximation method is based on the spectral compactness of the…
In this paper, we revisit the diffusive representations of fractional integrals established in \cite{diethelm2023diffusive} to explore novel variants of such representations which provide highly efficient numerical algorithms for the…
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate…
We present a numerical spectral method to solve systems of differential equations on an infinite interval $y\in (-\infty, \infty)$ in presence of linear differential operators of the form $Q(y) \left(\partial/\partial_y\right)^b$ (where…
We study some convergence issues for a recent approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. The approach is based on the minimization on an integral functional which arises from…
Equations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses…
In this paper we propose two proximal gradient algorithms for fractional programming problems in real Hilbert spaces, where the numerator is a proper, convex and lower semicontinuous function and the denominator is a smooth function, either…