Post-Processing Techniques and Wavelet Applications for Hammerstein Integral Equations

This dissertation is focused on the varieties of numerical solutions of nonlinear Hammerstein integral equations. In the first part of this dissertation, several acceleration techniques for post-processed solutions of the Hammerstein equation are discussed. The post-processing techniques are implemented based on interpolation and extrapolation. In this connection, we generalize the results in [29] and [28] to nonlinear integral equations of the Hammerstein type. Post-processed collocation solutions are shown to exhibit better accuracy. Moreover, an extrapolation technique for the Galerkin solution of Hammerstein equation is also obtained. This result appears new even in the setting of the linear Fredholm equation. In the second half of this dissertation, the wavelet-collocation technique of solving nonlinear Hammerstein integral equation is discussed. The main objective is to establish a fast wavelet-collocation method for Hammerstein equation by using a \u27linearization\u27 technique. The sparsity in the Jacobian matrix takes place in the fast wavelet-collocation method for Hammerstein equation with smooth as well as weakly singular kernels. A fast algorithm is based upon the block truncation strategy which was recently proposed in [10]. A multilevel augmentation method for the linearized Hammerstein equation is subsequently proposed which further accelerates the solution process while maintaining the order of convergence. Numerical examples are given throughout this dissertation

Linear prediction of point process times and marks

In this paper, we are interested in linear prediction of a particular kind of stochastic process, namely a marked temporal point process. The observations are event times recorded on the real line, with marks attached to each event. We show that in this case, linear prediction extends straightforwardly from the theory of prediction for stationary stochastic processes. Following classical lines, we derive a Wiener-Hopf-type integral equation to characterise the linear predictor, extending the "model independent origin" of the Hawkes process (Jaisson, 2015) as a corollary. We propose two recursive methods to solve the linear prediction problem and show that these are computationally efficient in known cases. The first solves the Wiener-Hopf equation via a set of differential equations. It is particularly well-adapted to autoregressive processes. In the second method, we develop an innovations algorithm tailored for moving-average processes. A small simulation study on two typical examples shows the application of numerical schemes for estimation of a Hawkes process intensity

A Finite Element Splitting Extrapolation for Second Order Hyperbolic Equations

Splitting extrapolation is an efficient technique for solving large scale scientific and engineering problems in parallel. This article discusses a finite element splitting extrapolation for second order hyperbolic equations with time-dependent coefficients. This method possesses a higher degree of parallelism, less computational complexity, and more flexibility than Richardson extrapolation while achieving the same accuracy. By means of domain decomposition and isoparametric mapping, some grid parameters are chosen according to the problem. The multiparameter asymptotic expansion of the d-quadratic finite element error is also established. The splitting extrapolation formulas are developed from this expansion. An approximation with higher accuracy on a globally fine grid can be computed by solving a set of smaller discrete subproblems on different coarser grids in parallel. Some a posteriori error estimates are also provided. Numerical examples show that this method is efficient for solving discontinuous problems and nonlinear hyperbolic equations

On the Inversion of High Energy Proton

Inversion of the K-fold stochastic autoconvolution integral equation is an elementary nonlinear problem, yet there are no de facto methods to solve it with finite statistics. To fix this problem, we introduce a novel inverse algorithm based on a combination of minimization of relative entropy, the Fast Fourier Transform and a recursive version of Efron's bootstrap. This gives us power to obtain new perspectives on non-perturbative high energy QCD, such as probing the ab initio principles underlying the approximately negative binomial distributions of observed charged particle final state multiplicities, related to multiparton interactions, the fluctuating structure and profile of proton and diffraction. As a proof-of-concept, we apply the algorithm to ALICE proton-proton charged particle multiplicity measurements done at different center-of-mass energies and fiducial pseudorapidity intervals at the LHC, available on HEPData. A strong double peak structure emerges from the inversion, barely visible without it.Comment: 29 pages, 10 figures, v2: extended analysis (re-projection ratios, 2D

Euclidean to Minkowski Bethe–Salpeter amplitude and observables

We propose a method to reconstruct the Bethe-Salpeter amplitude in Minkowski space given the Euclidean Bethe-Salpeter amplitude -- or alternatively the Light-Front wave function -- as input. The method is based on the numerical inversion of the Nakanishi integral representation and computing the corresponding weight function. This inversion procedure is, in general, rather unstable, and we propose several ways to considerably reduce the instabilities. In terms of the Nakanishi weight function, one can easily compute the BS amplitude, the LF wave function and the electromagnetic form factor. The latter ones are very stable in spite of residual instabilities in the weight function. This procedure allows both, to continue the Euclidean BS solution in the Minkowski space and to obtain a BS amplitude from a LF wave function.Comment: 11 pages, 12 figure