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While nowadays most gradient-based optimization methods focus on exploring the high-dimensional geometric features, the random error accumulated in a stochastic version of any algorithm implementation has not been stressed yet. In this…

The aim of the paper is to demonstrate the use of the Galerkin method for some kind of Volterra equations, determininistic and stochastic as well. The paper consists of two parts: the theoretical and numerical one. In the first part we…

We consider the problem of minimizing the average of a large number of smooth but possibly non-convex functions. In the context of most machine learning applications, each loss function is non-negative and thus can be expressed as the…

We present complexity and numerical results for a new asynchronous parallel algorithmic method for the minimization of the sum of a smooth nonconvex function and a convex nonsmooth regularizer, subject to both convex and nonconvex…

This paper describes an algorithm for selecting parameter values (e.g. temperature values) at which to measure equilibrium properties with Parallel Tempering Monte Carlo simulation. Simple approaches to choosing parameter values can lead to…

We consider model selection and estimation for partial spline models and propose a new regularization method in the context of smoothing splines. The regularization method has a simple yet elegant form, consisting of roughness penalty on…

We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution…

This paper presents several new algorithms for the regularized reconstruction of a surface from its measured gradient field. By taking a matrix-algebraic approach, we establish general framework for the regularized reconstruction problem…

This paper concerns the asymptotic behaviour of solutions of a linear convolution Volterra summation equation with an unbounded forcing term. In particular, we suppose the kernel is summable and ascribe growth bounds to the exogenous…

Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to enforce uniqueness and select a meaningful solution is to introduce a regularizer. While for most…

We study stochastic Volterra equations in Hilbert spaces driven by cylindrical Gaussian noise. We derive a mild formulation for the stochastic Volterra equation, prove the equivalence of mild and strong solutions, the existence and…

In this paper we consider ill-posed inverse problems, both linear and nonlinear, by a heavy ball method in which a strongly convex regularization function is incorporated to detect the feature of the sought solution. We develop ideas on how…

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for…

In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates…

This paper deals with speeding up the convergence of a class of two-step iterative methods for solving linear systems of equations. To implement the acceleration technique, the residual norm associated with computed approximations for each…

Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…

We consider optimization algorithms that successively minimize simple Taylor-like models of the objective function. Methods of Gauss-Newton type for minimizing the composition of a convex function and a smooth map are common examples. Our…

A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive…

In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a…

Large-time behaviour of solutions to stochastic evolution equations driven by two-sided regular Volterra processes is studied. The solution is understood in the mild sense and takes values in a separable Hilbert space. Sufficient conditions…