Related papers: Least squares solvers for ill-posed PDEs that are …
Common computational problems, such as parameter estimation in dynamic models and PDE constrained optimization, require data fitting over a set of auxiliary parameters subject to physical constraints over an underlying state. Naive…
In this paper, a two-step regularization method is used to solve an ill-posed spherical pseudo-differential equation in the presence of noisy data. For the first step of regularization we approximate the data by means of a spherical…
A convincing feature of least-squares finite element methods is the built-in a posteriori error estimator for any conforming discretization. In order to generalize this property to discontinuous finite element ansatz functions, this paper…
After a general introduction about the regularization by noise phenomenon in the degenerate setting, the first part of this PhD thesis focuses at establishing the Schauder estimates, a useful analytical tool to prove also the well-posedness…
This paper studies well-posedness and parameter sensitivity of the Square Root LASSO (SR-LASSO), an optimization model for recovering sparse solutions to linear inverse problems in finite dimension. An advantage of the SR-LASSO (e.g., over…
We investigate errors in tangents and adjoints of implicit functions resulting from errors in the primal solution due to approximations computed by a numerical solver. Adjoints of systems of linear equations turn out to be unconditionally…
This paper addresses problems on the robust structural design of complex networks. More precisely, we address the problem of deploying the minimum number of dedicated sensors, i.e., those measuring a single state variable, that ensure the…
For the problem of high-dimensional sparse linear regression, it is known that an $\ell_0$-based estimator can achieve a $1/n$ "fast" rate on the prediction error without any conditions on the design matrix, whereas in absence of…
This paper concerns cheaply computable formulas and bounds for the condition number of the TLS problem. For a TLS problem with data $A$, $b$, two formulas are derived that are simpler and more compact than the known results in the…
Whenever we use devices to take measurements, calibration is indispensable. While the purpose of calibration is to reduce bias and uncertainty in the measurements, it can be quite difficult, expensive, and sometimes even impossible to…
Algorithmic stability is a central concept in statistics and learning theory that measures how sensitive an algorithm's output is to small changes in the training data. Stability plays a crucial role in understanding generalization,…
We analyze a simple prefiltered variation of the least squares estimator for the problem of estimation with biased, semi-parametric noise, an error model studied more broadly in causal statistics and active learning. We prove an oracle…
Functional partial least squares (FPLS) is commonly used for fitting scalar-on-function regression models. For the sake of accuracy, FPLS demands that each realization of the functional predictor is recorded as densely as possible over the…
In this paper, we address the accuracy of the results for the overdetermined full rank linear least squares problem. We recall theoretical results obtained in Arioli, Baboulin and Gratton, SIMAX 29(2):413--433, 2007, on conditioning of the…
Least absolute deviation regression is applied using a fixed number of points for all values of the index to estimate the index and scale parameter of the stable distribution using regression methods based on the empirical characteristic…
Least squares is by far the simplest and most commonly applied computational method in many fields. In almost all applications, the least squares objective is rarely the true objective. We account for this discrepancy by parametrizing the…
This paper considers fundamental limits for solving sparse inverse problems in the presence of Poisson noise with physical constraints. Such problems arise in a variety of applications, including photon-limited imaging systems based on…
We define and analyse a least-squares finite element method for a first-order reformulation of the obstacle problem. Moreover, we derive variational inequalities that are based on similar but non-symmetric bilinear forms. A priori error…
Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model that must be estimated. However, high…
We show that adaptive least-squares finite element methods driven by the canonical least-squares functional converge under weak conditions on PDE operator, mesh-refinement, and marking strategy. Contrary to prior works, our plain…