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Register a new userApproximation of null controls for semilinear heat equations using a least-squares approach
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Arnaud Munch
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The null distributed controllability of the semilinear heat equation $y_t-Delta y + g(y)=f ,1_{omega}$, assuming that $g$ satisfies the growth condition $g(s)/(vert svert log^{3/2}(1+vert svert))rightarrow 0$ as $vert svert rightarrow infty$ and that $g^primein L^infty_{loc}(mathbb{R})$ has been obtained by Fernandez-Cara and Zuazua in 2000. The proof based on a fixed point argument makes use of precise estimates of the observability constant for a linearized heat equation. It does not provide however an explicit construction of a null control. Assuming that $g^primein W^{s,infty}(mathbb{R})$ for one $sin (0,1]$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach, generalizes Newton type methods and guarantees the convergence whatever be the initial element of the sequence. In particular, after a finite number of iterations, the convergence is super linear with a rate equal to $1+s$. Numerical experiments in the one dimensional setting support our analysis.
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The exact distributed controllability of the semilinear wave equation $y_{tt}-y_{xx} + g(y)=f ,1_{omega}$, assuming that $g$ satisfies the growth condition $vert g(s)vert /(vert svert log^{2}(vert svert))rightarrow 0$ as $vert svert rightarrow infty$ and that $g^primein L^infty_{loc}(mathbb{R})$ has been obtained by Zuazua in the nineties. The proof based on a Leray-Schauder fixed point argument makes use of precise estimates of the observability constant for a linearized wave equation. It does not provide however an explicit construction of a null control. Assuming that $g^primein L^infty_{loc}(mathbb{R})$, that $sup_{a,bin mathbb{R},a eq b} vert g^prime(a)-g^{prime}(b)vert/vert a-bvert^r<infty $ for some $rin (0,1]$ and that $g^prime$ satisfies the growth condition $vert g^prime(s)vert/log^{2}(vert svert)rightarrow 0$ as $vert svert rightarrow infty$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach guarantees the convergence whatever the initial element of the sequence may be. In particular, after a finite number of iterations, the convergence is super linear with rate $1+r$. This general method provides a constructive proof of the exact controllability for the semilinear wave equation.
The epsilon alternating least squares ($epsilon$-ALS) is developed and analyzed for canonical polyadic decomposition (approximation) of a higher-order tensor where one or more of the factor matrices are assumed to be columnwisely orthonormal. It is shown that the algorithm globally converges to a KKT point for all tensors without any assumption. For the original ALS, by further studying the properties of the polar decomposition, we also establish its global convergence under a reality assumption not stronger than those in the literature. These results completely address a question concerning the global convergence raised in [L. Wang, M. T. Chu and B. Yu, emph{SIAM J. Matrix Anal. Appl.}, 36 (2015), pp. 1--19]. In addition, an initialization procedure is proposed, which possesses a provable lower bound when the number of columnwisely orthonormal factors is one. Armed with this initialization procedure, numerical experiments show that the $epsilon$-ALS exhibits a promising performance in terms of efficiency and effectiveness.
We introduce and analyze a space-time least-squares method associated to the unsteady Navier-Stokes system. Weak solution in the two dimensional case and regular solution in the three dimensional case are considered. From any initial guess, we construct a minimizing sequence for the least-squares functional which converges strongly to a solution of the Navier-Stokes system. After a finite number of iterates related to the value of the viscosity constant, the convergence is quadratic. Numerical experiments within the two dimensional case support our analysis. This globally convergent least-squares approach is related to the damped Newton method when used to solve the Navier-Stokes system through a variational formulation.
It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation $partial_{tt}y-partial_{xx}y + g(y)=f 1_{omega}$, with Dirichlet boundary conditions, is exactly controllable in $H^1_0(0,1)cap L^2(0,1)$ with controls $fin L^2((0,1)times(0,T))$, for any $T>0$ and any nonempty open subset $omega$ of $(0,1)$, assuming that $gin mathcal{C}^1(R)$ does not grow faster than $betavert xvert ln^{2}vert xvert$ at infinity for some $beta>0$ small enough. The proof, based on the Leray-Schauder fixed point theorem, is however not constructive. In this article, we design a constructive proof and algorithm for the exact controllability of semilinear 1D wave equations. Assuming that $g^prime$ does not grow faster than $beta ln^{2}vert xvert$ at infinity for some $beta>0$ small enough and that $g^prime$ is uniformly Holder continuous on $R$ with exponent $sin[0,1]$, we design a least-squares algorithm yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+s$ after a finite number of iterations.
We conduct a study and comparison of superiorization and optimization approaches for the reconstruction problem of superiorized/regularized least-squares solutions of underdetermined linear equations with nonnegativity variable bounds. Regarding superiorization, the state of the art is examined for this problem class, and a novel approach is proposed that employs proximal mappings and is structurally similar to the established forward-backward optimization approach. Regarding convex optimization, accelerated forward-backward splitting with inexact proximal maps is worked out and applied to both the natural splitting least-squares term/regularizer and to the reverse splitting regularizer/least-squares term. Our numerical findings suggest that superiorization can approach the solution of the optimization problem and leads to comparable results at significantly lower costs, after appropriate parameter tuning. On the other hand, applying accelerated forward-backward optimization to the reverse splitting slightly outperforms superiorization, which suggests that convex optimization can approach superiorization too, using a suitable problem splitting.
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