Non-linear least squares | EPFL Graph Search

Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are many similarities to linear least squares, but also some significant differences. In economic theory, the non-linear least squares method is applied in (i) the probit regression, (ii) threshold regression, (iii) smooth regression, (iv) logistic link regression, (v) Box–Cox transformed regressors (m(x,\theta_i) = \theta_1 + \theta_2 x^{(\theta_{3})}). Theory Consider a set of m data points, (x_1, y_1), (x_2, y_2), \dots, (x_m, y_m), and a curve (model function) \hat{y} = f(x, \boldsymbol \beta), that in addition to the variable x also depends on n

Nondyadic and nonlinear multiresolution image approximations

Maria Arrate Munoz Barrutia

This thesis focuses on the development of novel multiresolution image approximations. Specifically, we present two kinds of generalization of multiresolution techniques: image reduction for arbitrary scales, and nonlinear approximations using other metrics than the standard Euclidean one. Traditional multiresolution decompositions are restricted to dyadic scales. As first contribution of this thesis, we develop a method that goes beyond this restriction and that is well suited to arbitrary scale-change computations. The key component is a new and numerically exact algorithm for computing inner products between a continuously defined signal and B-splines of any order and of arbitrary sizes. The technique can also be applied for non-uniform to uniform grid conversion, which is another approximation problem where our method excels. Main applications are resampling and signal reconstruction. Although simple to implement, least-squares approximations lead to artifacts that could be reduced if nonlinear methods would be used instead. The second contribution of the thesis is the development of nonlinear spline pyramids that are optimal for lp-norms. First, we introduce a Banach-space formulation of the problem and show that the solution is well defined. Second, we compute the lp-approximation thanks to an iterative optimization algorithm based on digital filtering. We conclude that l1-approximations reduce the artifacts that are inherent to least-squares methods; in particular, edge blurring and ringing. In addition, we observe that the error of l1-approximations is sparser. Finally, we derive an exact formula for the asymptotic Lp-error; this result justifies using the least-squares approximation as initial solution for the iterative optimization algorithm when the degree of the spline is even; otherwise, one has to include an appropriate correction term. The theoretical background of the thesis includes the modelisation of images in a continuous/discrete formalism and takes advantage of the approximation theory of linear shift-invariant operators. We have chosen B-splines as basis functions because of their nice properties. We also propose a new graphical formalism that links B-splines, finite differences, differential operators, and arbitrary scale changes.

EPFL2002

SEKV-E: Parameter Extractor of Simplified EKV I-V Model for Low-Power Analog Circuits

Christian Enz, Hung-Chi Han

This paper presents the open-source Python-based parameter extractor (SEKV-E) for the simplified EKV (sEKV) model, which enables the modern low-power circuit designs with the inversion coefficient design methodology. The tool extracts the essential sEKV parameters automatically from the given I-V curves using the direct extraction and the multi-stage optimization process. It also handles the overfitting issue because of non-linear least squares. Moreover, this work demonstrates the SEKV-E as a universal tool by widely applying it to different silicon technologies, temperatures, dimensions, and back-gate voltages.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2022

A Least-Squares Method for the Solution of the Non-smooth Prescribed Jacobian Equation

Alexandre Caboussat, Dimitrios Gourzoulidis

We consider a least-squares/relaxation finite element method for the numerical solution of the prescribed Jacobian equation. We look for its solution via a least-squares approach. We introduce a relaxation algorithm that decouples this least-squares problem into a sequence of local nonlinear problems and variational linear problems. We develop dedicated solvers for the algebraic problems based on Newton's method and we solve the differential problems using mixed low-order finite elements. Various numerical experiments demonstrate the accuracy, efficiency and the robustness of the proposed method, compared for instance to augmented Lagrangian approaches.

SPRINGER/PLENUM PUBLISHERS2022

Nondyadic and nonlinear multiresolution image approximations

Maria Arrate Munoz Barrutia

EPFL2002