This paper reviews popular acceleration techniques to converge the non-linear self-consistent field equations appearing in quantum chemistry calculations with localized basis sets. The different methodologies, as well as their advantages and limitations are discussed within the same framework. Several illustrative examples of calculations are presented. This paper attempts to describe recent achievements and remaining challenges in this field.

The numerical minimization of the functional $\mathcal{F}\left(u\right)={\int }_{\mathrm{\Omega }}\varphi \left(x,{\nu }_u\right)\left|Du\right|+{\int }_{\partial \mathrm{\Omega }}\mu ud{\mathcal{H}}^{n-1}-{\int }_{\mathrm{\Omega }}\kappa udx$, $u\in BV\left(\mathrm{\Omega };\left\{-1,1\right\}\right)$ is addressed. The function $\varphi$ is continuous, has linear growth, and is convex and positively homogeneous of degree one in the second variable. We prove that $\mathcal{F}$ can be equivalently minimized on the convex set $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ and then regularized with a sequence ${\left\{{\mathcal{F}}_{ϵ}\left(u\right)\right\}}_{ϵ}$, of stricdy convex functionals defined on $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$. Then both $\mathcal{F}$ and ${\mathcal{F}}_{ϵ}$, can be discretized by continuous linear finite elements. The convexity property of the functionals on $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ is useful in the numerical minimization...

We address the numerical minimization of the functional $\mathcal{F}\left(v\right)={\int }_{\mathrm{\Omega }}\left|Dv\right|+{\int }_{\partial \mathrm{\Omega }}\mu vd{\mathcal{H}}^{n-1}-{\int }_{\mathrm{\Omega }}xvdx$, for $v\in BV\left(\mathrm{\Omega };\left\{-1,1\right\}\right)$. We note that $\mathcal{F}$ can be equivalently minimized on the larger, convex, set $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ and that, on that space, $\mathcal{F}$ may be regularized with a sequence $\mathit{\left\{}\mathcal{F}{}_{ϵ}\right(v)=\int {}_{\mathrm{\Omega }}\sqrt{{ϵ}^2+{\left|Dv\right|}^2}+\int {}_{\partial \mathrm{\Omega }}\mu vd\mathcal{H}{}^{n-1}-\int {}_{\mathrm{\Omega }}xvdx\mathit{\}}{}_{ϵ}$of regular functionals. Then both $\mathcal{F}$ and ${\mathcal{F}}_{ϵ}$ can be discretized by continuous linear finite elements. The convexity of the functionals in $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ is useful for the numerical minimization of $\mathcal{F}$. We prove the $\mathrm{\Gamma }-L^1\left(\mathrm{\Omega }\right)$-convergence of the discrete functionals to $\mathcal{F}$ and present a few numerical examples.

In this paper we consider the data assimilation problem for a timedependent transport problem in a slab when the initial condition is not known. The spaces of traces are introduced, the solvability of the original initial-boundary value transport problem is studied. The properties of the control operator are investigated, the solvability of the data assimilation problem is proved. The class of iterative methods for solving the problem is considered, and the convergence conditions are studied. The...

In this note, we employ nonlinear on-line parameter estimation methods based on adaptive neural network approximators for detecting changes due to actuator faults in a class of second order distributed parameter systems. The motivating example is a cantilevered beam actuated via a pair of piezoceramic patches. We examine changes in the control input term, which provide a simple and practical model of actuator failures. Using Lyapunov redesign methods, a stable learning scheme for fault diagnosis...

The purpose of this article is to show the great interest of the use of propagation (or pruning) techniques, inside classical interval Branch-and-Bound algorithms. Therefore, a propagation technique based on the construction of the calculus tree is entirely explained and some properties are presented without the need of any formalism (excepted interval analysis). This approach is then validated on a real example: the optimal design of an electrical rotating machine.

The purpose of this article is to show the great interest of the use of propagation (or pruning) techniques, inside classical interval Branch-and-Bound algorithms. Therefore, a propagation technique based on the construction of the calculus tree is entirely explained and some properties are presented without the need of any formalism (excepted interval analysis). This approach is then validated on a real example: the optimal design of an electrical rotating machine.

We extend thresholding methods for numerical realization of mean curvature flow on obstacles to the anisotropic setting where interfacial energy depends on the orientation of the interface. This type of schemes treats the interface implicitly, which supports natural implementation of topology changes, such as merging and splitting, and makes the approach attractive for applications in material science. The main tool in the new scheme are convolution kernels developed in previous studies that approximate...