Related papers: A theoretical basis for the Harmonic Balance Metho…

Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from…

When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the…

We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent…

The harmonic balance method is the most commonly used method for solving periodic solutions of nonlinear dynamic systems, but the high-order approximation of nonlinear terms requires sophisticated formula derivation, which limits its…

This work addresses a central challenge in the numerical analysis of the cutoff spatially homogeneous Boltzmann equation: the development of rigorously justified, accurate numerical schemes. We present (i) a novel Fourier spectral method…

We deduce a procedure to apply balanced truncation to parameter-dependent differential-algebraic systems. For that we solve multiple projected Lyapunov equations for different parameter values to compute the Gramians that are required for…

This paper describes a method of calculating the transforms, currently obtained via Fourier and reverse Fourier transforms. The method allows calculating efficiently the transforms of a signal having an arbitrary dimension of the digital…

We consider two approaches to balanced truncation of stochastic linear systems, which follow from different generalizations of the reachability Gramian of deterministic systems. Both preserve mean-square asymptotic stability, but only the…

This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial conditions. The class of…

We introduce a new method to approximate Euclidean correlation functions by exponential sums. The Truncated Hankel Correlator (THC) method builds a Hankel matrix from the full correlator data available and truncates the eigenspectrum of…

The adoption of cardiovascular simulations for diagnosis and surgical planning on a patient-specific basis requires the development of faster methods than the existing state-of-the-art techniques. To address this need, we leverage the…

We propose an alternative approach that avoids the nonlinear equations for the Fourier coefficients that appear in the method of harmonic balance. We apply it to two simple illustrative examples.

We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an…

This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse…

To facilitate the numerical analysis of particle methods, we derive truncation error estimates for the approximate operators in a generalized particle method. Here, a generalized particle method is defined as a meshfree numerical method…

This note is devoted to inverse spectral problems for canonical Hamiltonian systems on the half-line. An approach to inverse spectral problems based on the use of truncated Toeplitz operators has been especially effective in the case when…

In this paper, we discuss some limitations of the modified equations approach as a tool for stability analysis for a class of explicit linear schemes to scalar partial derivative equations. We show that the infinite series obtained by…

Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the…

We propose a new numerical method to solve the linearized problem of travel time tomography with incomplete data. Our method is based on the technique of the truncation of the Fourier series with respect to a special basis of L2. This way…

This article considers a Cauchy problem of Helmholtz equations whose solution is well known to be exponentially unstable with respect to the inputs. In the framework of variational quasi-reversibility method, a Fourier truncation is applied…