Orthogonal group | EPFL Graph Search

In mathematics, the orthogonal group in dimension n, denoted \operatorname{O}(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n\times n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted \operatorname{SO}(n). It consists of all orthogonal matrices of determinant 1. This grou

Numerical Methods for First and Second Order Fully Nonlinear Partial Differential Equations

Dimitrios Gourzoulidis

This thesis focuses on the numerical analysis of partial differential equations (PDEs) with an emphasis on first and second-order fully nonlinear PDEs. The main goal is the design of numerical methods to solve a variety of equations such as orthogonal maps, the prescribed Jacobian equation and inequality, the elliptic and parabolic Monge-Ampère equations. For orthogonal map we develop an \emph{operator-splitting/finite element} approach for the numerical solution of the Dirichlet problem. This approach is built on the variational principle, the introduction of an associated flow problem, and a time-stepping splitting algorithm. Moreover, we propose an extension of this method with an \emph{anisotropic mesh adaptation algorithm}. This extension allows us to track singularities of the solution's gradient more accurately. Various numerical experiments demonstrate the accuracy and the robustness of the proposed method for both constant and adaptive mesh. For the prescribed Jacobian equation and the three-dimensional Monge-Ampère equation, we consider a \emph{least-squares/relaxation finite element method} for the numerical solution of the Dirichlet problems. We then introduce a relaxation algorithm that splits the least-square problem, which stems from a reformulation of the original equations, into local nonlinear and variational problems. We develop dedicated solvers for the algebraic problems based on Newton method and we solve the differential problems using mixed low-order finite element method. Overall the least squares approach exhibits appropriate convergence orders in

L^2(\Omega)

and

H^1(\Omega)

error norms for various numerical tests. We also design a \emph{second-order time integration method} for the approximation of a parabolic two-dimensional Monge-Ampère equation. The space discretization of this method is based on low-order finite elements, and the time discretization is achieved by the implicit Crank-Nicolson type scheme. We verify the efficiency of the proposed method on time-dependent and stationary problems. The results of numerical experiments show that the method achieves nearly optimal orders for the

L^2(\Omega)

and

H^1(\Omega)

error norms when smooth solutions are approximated. Finally, we present an adaptive mesh refinement algorithm for the elliptic Monge-Ampere equation based on the residual error estimate. The robustness of the proposed algorithm is verified using various test cases and two different solvers which are inspired by the two previous proposed numerical methods.

EPFL2021

Numerical Methods for First and Second Order Fully Nonlinear Partial Differential Equations

Dimitrios Gourzoulidis

L^2(\Omega)

and

H^1(\Omega)

L^2(\Omega)

and

H^1(\Omega)

EPFL2021