350 research outputs found
Inverse problems for linear hyperbolic equations using mixed formulations
We introduce in this document a direct method allowing to solve numerically
inverse type problems for linear hyperbolic equations. We first consider the
reconstruction of the full solution of the wave equation posed in - a bounded subset of - from a partial
distributed observation. We employ a least-squares technique and minimize the
-norm of the distance from the observation to any solution. Taking the
hyperbolic equation as the main constraint of the problem, the optimality
conditions are reduced to a mixed formulation involving both the state to
reconstruct and a Lagrange multiplier. Under usual geometric optic conditions,
we show the well-posedness of this mixed formulation (in particular the inf-sup
condition) and then introduce a numerical approximation based on space-time
finite elements discretization. We prove the strong convergence of the
approximation and then discussed several examples for and . The
problem of the reconstruction of both the state and the source term is also
addressed
Inexact inner-outer Golub-Kahan bidiagonalization method: A relaxation strategy
We study an inexact inner-outer generalized Golub-Kahan algorithm for the
solution of saddle-point problems with a two-times-two block structure. In each
outer iteration, an inner system has to be solved which in theory has to be
done exactly. Whenever the system is getting large, an inner exact solver is,
however, no longer efficient or even feasible and iterative methods must be
used. We focus this article on a numerical study showing the influence of the
accuracy of an inner iterative solution on the accuracy of the solution of the
block system. Emphasis is further given on reducing the computational cost,
which is defined as the total number of inner iterations. We develop relaxation
techniques intended to dynamically change the inner tolerance for each outer
iteration to further minimize the total number of inner iterations. We
illustrate our findings on a Stokes problem and validate them on a mixed
formulation of the Poisson problem.Comment: 25 pages, 9 figure
The INTERNODES method for applications in contact mechanics and dedicated preconditioning techniques
The mortar finite element method is a well-established method for the numerical solution of partial differential equations on domains displaying non-conforming interfaces. The method is known for its application in computational contact mechanics. However, its implementation remains challenging as it relies on geometrical projections and unconventional quadrature rules. The INTERNODES (INTERpolation for NOn-conforming DEcompositionS) method, instead, could overcome the implementation difficulties thanks to flexible interpolation techniques. Moreover, it was shown to be at least as accurate as the mortar method making it a very promising alternative for solving problems in contact mechanics. Unfortunately, in such situations the method requires solving a sequence of ill-conditioned linear systems. In this paper, preconditioning techniques are designed and implemented for the efficient solution of those linear systems
Domain Decomposition for Stochastic Optimal Control
This work proposes a method for solving linear stochastic optimal control
(SOC) problems using sum of squares and semidefinite programming. Previous work
had used polynomial optimization to approximate the value function, requiring a
high polynomial degree to capture local phenomena. To improve the scalability
of the method to problems of interest, a domain decomposition scheme is
presented. By using local approximations, lower degree polynomials become
sufficient, and both local and global properties of the value function are
captured. The domain of the problem is split into a non-overlapping partition,
with added constraints ensuring continuity. The Alternating Direction
Method of Multipliers (ADMM) is used to optimize over each domain in parallel
and ensure convergence on the boundaries of the partitions. This results in
improved conditioning of the problem and allows for much larger and more
complex problems to be addressed with improved performance.Comment: 8 pages. Accepted to CDC 201
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