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We develop a sparse spectral method for a class of fractional differential equations, posed on $\mathbb{R}$, in one dimension. These equations can include sqrt-Laplacian, Hilbert, derivative and identity terms. The numerical method utilizes…
We study the accuracy with which cosmological parameters can be determined from real space power spectrum of matter density contrast at weakly nonlinear scales using analytical approaches. From power spectra measured in $N$-body simulations…
We present two new quantum algorithms for reaction-diffusion equations that employ the truncated Chebyshev polynomial approximation. This method is employed to numerically solve the ordinary differential equation emerging from the…
Rapid progress in cosmological Large Scale Structure (LSS) surveys motivates precise theoretical predictions. The Effective Field Theory of Large-Scale Structure (EFTofLSS) is routinely applied to data, and requires fast computation of its…
Solving the stationary nonlinear Fokker-Planck equations is important in applications and examples include the Poisson-Boltzmann equation and the two layer neural networks. Making use of the connection between the interacting particle…
Nonlinear Schwarz methods are a type of nonlinear domain decomposition method used as an alternative to Newton's method for solving discretized nonlinear partial differential equations. In this article, the first parallel implementation of…
A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise…
A general result is presented on the lack of second order corrections and on the form of the leading order Floquet quasi-energies in the high-frequency approximation for a two-level driven system. Then, by a perturbative approach in the…
The Source Galerkin method finds approximate solutions to the functional differential equations of field theories in the presence of external sources. While developing this process, it was recognized that approximations of the spectral…
We present a new numerical scheme to treat the non-linear evolution of cosmological power spectra. Governing equations for matter power spectra have been previously derived by a non-perturbative technique with closure approximation.…
We present an algorithm for determining the minimal order differential equations associated to a given Feynman integral in dimensional or analytic regularisation. The algorithm is an extension of the Griffiths-Dwork pole reduction adapted…
Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and…
In theoretical physics, we sometimes have two perturbative expansions of physical quantity around different two points in parameter space. In terms of the two perturbative expansions, we introduce a new type of smooth interpolating function…
We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the…
Cosmological perturbation theory is a powerful tool to predict the statistics of large-scale structure in the weakly non-linear regime, but even at 1-loop order it results in computationally expensive mode-coupling integrals. Here we…
Modified gravity models with scale-dependent linear growth typically exhibit an enhancement in the power spectrum beyond a certain scale. The conventional methods for extracting cosmological information usually involve inferring modified…
We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form \varepsilon zf^{\prime} = F(\varepsilon,z,f) with F…
Implementation of many statistical methods for large, multivariate data sets requires one to solve a linear system that, depending on the method, is of the dimension of the number of observations or each individual data vector. This is…
We investigate a data-driven approach for tuning the filtered spherical harmonics method (\fpn) when solving the radiation transport equation (RTE). The \fpn method extends the classical spherical harmonics approach (\pn) by introducing…
In recent years two Krylov subspace methods have been proposed for solving skew symmetric linear systems, one based on the minimum residual condition, the other on the Galerkin condition. We give new, algorithm-independent proofs that in…