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Concept# System of linear equations

Summary

In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin{cases}
3x+2y-z=1\
2x-2y+4z=-2\
-x+\frac{1}{2}y-z=0
\end{cases}
is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple
:(x,y,z)=(1,-2,-2),
since it makes all three equations valid. The word "system" indicates that the equations should be considered collectively, rather than individually.
In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer

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MATH-251(c): Numerical analysis

Le cours présente des méthodes numériques pour la résolution de problèmes mathématiques comme des systèmes d'équations linéaires ou non linéaires, approximation de fonctions, intégration et dérivation, équations différentielles.

ME-523: Nonlinear Control Systems

Les systèmes non linéaires sont analysés en vue d'établir des lois de commande. On présente la stabilité au sens de Lyapunov, ainsi que des méthodes de commande géométrique (linéarisation exacte). Divers exemples illustrent la théorie (exercices papier crayon et simulations à l'ordinateur).

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L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

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We present the first accelerated randomized algorithm for solving linear systems in Euclidean spaces. One essential problem of this type is the matrix inversion problem. In particular, our algorithm can be specialized to invert positive definite matrices in such a way that all iterates (approximate solutions) generated by the algorithm are positive definite matrices themselves. This opens the way for many applications in the field of optimization and machine learning. As an application of our general theory, we develop the first accelerated (deterministic and stochastic) quasi-Newton updates. Our updates lead to provably more aggressive approximations of the inverse Hessian, and lead to speed-ups over classical non-accelerated rules in numerical experiments. Experiments with empirical risk minimization show that our rules can accelerate training of machine learning models.

In this thesis, we study two distinct problems.
The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1-form 'a'. The study of this differential operator is linked to the study of the multiplication by a two form, that is the system of linear equations where we take a k-form and apply the exterior wedge product by 'da', the exterior derivative of 'a'. We establish links between the partial differential equation and the linear system.
The second problem is a generalization of the symmetric gradient and the curl equation. The equation of a symmetric gradient consists of taking a vector field, apply the gradient and then add the transpose of the gradient, whereas in the curl equation we subtract the transpose of the gradient. Both can be seen as an equation of the form A * grad u + (grad u)t * A, where A is a symmetric matrix for the case of the symmetric gradient and skew symmetric for the curl equation. We generalize to the case where A verifies no symmetry assumption and more significantly add a Dirichlet condition on the boundary.

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Inverse reconstruction from images is a central problem in many scientific and engineering disciplines. Recent progress on differentiable rendering has led to methods that can efficiently differentiate the full process of image formation with respect to millions of parameters to solve such problems via gradient-based optimization. At the same time, the availability of cheap derivatives does not necessarily make an inverse problem easy to solve. Mesh-based representations remain a particular source of irritation: an adverse gradient step involving vertex positions could turn parts of the mesh inside-out, introduce numerous local self-intersections, or lead to inadequate usage of the vertex budget due to distortion. These types of issues are often irrecoverable in the sense that subsequent optimization steps will further exacerbate them. In other words, the optimization lacks robustness due to an objective function with substantial non-convexity. Such robustness issues are commonly mitigated by imposing additional regularization, typically in the form of Laplacian energies that quantify and improve the smoothness of the current iterate. However, regularization introduces its own set of problems: solutions must now compromise between solving the problem and being smooth. Furthermore, gradient steps involving a Laplacian energy resemble Jacobi's iterative method for solving linear equations that is known for its exceptionally slow convergence. We propose a simple and practical alternative that casts differentiable rendering into the framework of preconditioned gradient descent. Our preconditioner biases gradient steps towards smooth solutions without requiring the final solution to be smooth. In contrast to Jacobi-style iteration, each gradient step propagates information among all variables, enabling convergence using fewer and larger steps. Our method is not restricted to meshes and can also accelerate the reconstruction of other representations, where smooth solutions are generally expected. We demonstrate its superior performance in the context of geometric optimization and texture reconstruction.