88 research outputs found
On an inverse problem for anisotropic conductivity in the plane
Let be a bounded domain with smooth
boundary and a smooth anisotropic conductivity on .
Starting from the Dirichlet-to-Neumann operator on
, we give an explicit procedure to find a unique domain
, an isotropic conductivity on and the boundary
values of a quasiconformal diffeomorphism which
transforms into .Comment: 9 pages, no figur
New global stability estimates for the Gel'fand-Calderon inverse problem
We prove new global stability estimates for the Gel'fand-Calderon inverse
problem in 3D. For sufficiently regular potentials this result of the present
work is a principal improvement of the result of [G. Alessandrini, Stable
determination of conductivity by boundary measurements, Appl. Anal. 27 (1988),
153-172]
Order reduction approaches for the algebraic Riccati equation and the LQR problem
We explore order reduction techniques for solving the algebraic Riccati
equation (ARE), and investigating the numerical solution of the
linear-quadratic regulator problem (LQR). A classical approach is to build a
surrogate low dimensional model of the dynamical system, for instance by means
of balanced truncation, and then solve the corresponding ARE. Alternatively,
iterative methods can be used to directly solve the ARE and use its approximate
solution to estimate quantities associated with the LQR. We propose a class of
Petrov-Galerkin strategies that simultaneously reduce the dynamical system
while approximately solving the ARE by projection. This methodology
significantly generalizes a recently developed Galerkin method by using a pair
of projection spaces, as it is often done in model order reduction of dynamical
systems. Numerical experiments illustrate the advantages of the new class of
methods over classical approaches when dealing with large matrices
Explicit Stabilised Gradient Descent for Faster Strongly Convex Optimisation
This paper introduces the Runge-Kutta Chebyshev descent method (RKCD) for
strongly convex optimisation problems. This new algorithm is based on explicit
stabilised integrators for stiff differential equations, a powerful class of
numerical schemes that avoid the severe step size restriction faced by standard
explicit integrators. For optimising quadratic and strongly convex functions,
this paper proves that RKCD nearly achieves the optimal convergence rate of the
conjugate gradient algorithm, and the suboptimality of RKCD diminishes as the
condition number of the quadratic function worsens. It is established that this
optimal rate is obtained also for a partitioned variant of RKCD applied to
perturbations of quadratic functions. In addition, numerical experiments on
general strongly convex problems show that RKCD outperforms Nesterov's
accelerated gradient descent
Solution of the time-domain inverse resistivity problem in the model reduction framework Part I. One-dimensional problem with SISO data
none3sixnoneV. Druskin; V. Simoncini; M. ZaslavskyV. Druskin; V. Simoncini; M. Zaslavsk
Adaptive tangential interpolation in rational Krylov subspaces for MIMO model reduction data
Model reduction approaches have been shown to be powerful techniques in the numerical simulation of very large dynamical systems. The presence of multiple inputs and outputs (MIMO systems) makes the reduction process even more challenging. We consider projection-based approaches where the reduction of complexity is achieved by direct projection of the problem onto a rational Krylov subspace of significantly smaller dimension. We present an effective way to treat multiple inputs by dynamically choosing the next direction vectors to expand the space. We apply the new strategy to the approximation of the transfer matrix function and to the solution of the Lyapunov matrix equation. Numerical results confirm that the new approach is competitive with respect to state-of-the-art methods both in terms of CPU time and memory requirements
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