Related papers: Local high-order regularization and applications t…
Interpolation models are critical for a wide range of applications, from numerical optimization to artificial intelligence. The reliability of the provided interpolated value is of utmost importance, and it is crucial to avoid the…
A regularization algorithm allowing random noise in derivatives and inexact function values is proposed for computing approximate local critical points of any order for smooth unconstrained optimization problems. For an objective function…
This paper addresses a multi-scale finite element method for second order linear elliptic equations with arbitrarily rough coefficient. We propose a local oversampling method to construct basis functions that have optimal local…
Integral operators of Abel type of order a > 0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a…
In this paper the hp-version of the boundary element method is applied to the electric field integral equation on a piecewise plane (open or closed) Lipschitz surface. The underlying meshes are supposed to be quasi-uniform. We use…
Discrete regularization methods are often applied for obtaining stable approximate solutions for ill-posed operator equations $Tx=y$, where $T: X\to Y$ is a bounded operator between Hilbert spaces with non-closed range $R(T)$ and $y\in…
Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than the ambient dimension of the object to be estimated. A line of recent work has studied…
Optimization problems with composite functions consist of an objective function which is the sum of a smooth and a (convex) nonsmooth term. This particular structure is exploited by the class of proximal gradient methods and some of their…
We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe…
Some prominent discretisation methods such as finite elements provide a way to approximate a function of $d$ variables from $n$ values it takes on the nodes $x_i$ of the corresponding mesh. The accuracy is $n^{-s_a/d}$ in $L^2$-norm, where…
This paper considers optimization of smooth nonconvex functionals in smooth infinite dimensional spaces. A H\"older gradient descent algorithm is first proposed for finding approximate first-order points of regularized polynomial…
In this work we consider the problem of semi-active damping optimization of mechanical systems with fixed damper positions. Our goal is to compute a damping that is locally optimal with respect to the $\mathcal{H}_\infty$-norm of the…
Regularization plays an important role in solving ill-posed problems by adding extra information about the desired solution, such as sparsity. Many regularization terms usually involve some vector norm, e.g., $L_1$ and $L_2$ norms. In this…
The main purpose of the paper is to study sharp estimates of approximation of periodic functions in the H\"older spaces $H_p^{r,\alpha}$ for all $0<p\le\infty$ and $0<\alpha\le r$. By using modifications of the classical moduli of…
This paper develops a fully discrete soft thresholding polynomial approximation over a general region, named Lasso hyperinterpolation. This approximation is an $\ell_1$-regularized discrete least squares approximation under the same…
Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we…
We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which…
Regularization techniques are widely employed in optimization-based approaches for solving ill-posed inverse problems in data analysis and scientific computing. These methods are based on augmenting the objective with a penalty function,…
We present and analyze an approximation scheme for a class of highly oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as examples. The scheme is based on polynomial interpolation combined with suitable pre- and…
A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that…