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We consider the problem of finding sparse solutions to a system of underdetermined nonlinear system of equations. The methods are based on a Gauss-Newton approach with line search where the search direction is found by solving a linearized…
In this paper, an idea to solve nonlinear equations is presented. During the solution of any problem with Newton's Method, it might happen that some of the unknowns satisfy the convergence criteria where the others fail. The convergence…
When considering an unconstrained minimization problem, a standard approach is to solve the optimality system with a Newton method possibly preconditioned by, e.g., nonlinear elimination. In this contribution, we argue that nonlinear…
Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often…
An extension of the Gauss-Newton algorithm is proposed to find local minimizers of penalized nonlinear least squares problems, under generalized Lipschitz assumptions. Convergence results of local type are obtained, as well as an estimate…
In this paper we consider the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed inverse problems. Under merely Lipschitz condition, we prove that this method together with an a posteriori stopping rule defines an…
Optimization of expensive computer models with the help of Gaussian process emulators in now commonplace. However, when several (competing) objectives are considered, choosing an appropriate sampling strategy remains an open question. We…
Sequential algorithms are popular for experimental design, enabling emulation, optimisation and inference to be efficiently performed. For most of these applications bespoke software has been developed, but the approach is general and many…
The Gaussian process is an indispensable tool for spatial data analysts. The onset of the "big data" era, however, has lead to the traditional Gaussian process being computationally infeasible for modern spatial data. As such, various…
The goal of this note is to point out an erroneous formula for the generalised Hessian of the least squares associated with a system of linear inequalities, that was given in the paper "A finite Newton method for classification" by O.L.…
The generalized Gauss-Newton (GGN) approximation is often used to make practical Bayesian deep learning approaches scalable by replacing a second order derivative with a product of first order derivatives. In this paper we argue that the…
In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise,…
A classical problem in general relativity is the Cauchy problem for the linearised Einstein equation (the initial value problem for gravitational waves) on a globally hyperbolic vacuum spacetime. A well-known result is that it is uniquely…
The textbook Newton's iteration is practically inapplicable on solutions of nonlinear systems with singular Jacobians. By a simple modification, a novel extension of Newton's iteration regains its local quadratic convergence toward…
Gaussian processes retain the linear model either as a special case, or in the limit. We show how this relationship can be exploited when the data are at least partially linear. However from the perspective of the Bayesian posterior, the…
Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and…
These notes are an exposition of Galois Theory from the original Lagrangian and Galoisian point of view. A particular effort was made here to better understand the connection between Lagrange's purely combinatorial approach and Galois…
We propose a simple method that combines neural networks and Gaussian processes. The proposed method can estimate the uncertainty of outputs and flexibly adjust target functions where training data exist, which are advantages of Gaussian…
Semismooth* Newton methods have been proposed in recent years targeting multi-valued inclusion problems and have been successfully implemented to deal with several concrete generalized equations. In this paper, we show that two typical…
We introduce a generalized bootstrap technique for estimators obtained by solving estimating equations. Some special cases of this generalized bootstrap are the classical bootstrap of Efron, the delete-d jackknife and variations of the…