1,062 research outputs found
All-at-Once Solution if Time-Dependent PDE-Constrained Optimisation Problems
Time-dependent partial differential equations (PDEs) play an important role in applied mathematics and many other areas of science. One-shot methods try to compute the solution to these problems in a single iteration that solves for all time-steps at the same time. In this paper, we look at one-shot approaches for the optimal control of time-dependent PDEs and focus on the fast solution of these problems. The use of Krylov subspace solvers together with an efficient preconditioner allows for minimal storage requirements. We solve only approximate time-evolutions for both forward and adjoint problem and compute accurate solutions of a given control problem only at convergence of the overall Krylov subspace iteration. We show that our approach can give competitive results for a variety of problem formulations
All-at-once solution of time-dependent PDE-constrained optimization problems
Time-dependent partial differential equations (PDEs) play an important role in applied mathematics and many other areas of science. One-shot methods try to compute the solution to these problems in a single iteration that solves for all time-steps at the same time. In this paper, we look at one-shot approaches for the optimal control of time-dependent PDEs and focus on the fast solution of these problems. The use of Krylov subspace solvers together with an efficient preconditioner allows for minimal storage requirements. We solve only approximate time-evolutions for both forward and adjoint problem and compute accurate solutions of a given control problem only at convergence of the overall Krylov subspace iteration. We show that our approach can give competitive results for a variety of problem formulations
On the local stability of semidefinite relaxations
We consider a parametric family of quadratically constrained quadratic
programs (QCQP) and their associated semidefinite programming (SDP)
relaxations. Given a nominal value of the parameter at which the SDP relaxation
is exact, we study conditions (and quantitative bounds) under which the
relaxation will continue to be exact as the parameter moves in a neighborhood
around the nominal value. Our framework captures a wide array of statistical
estimation problems including tensor principal component analysis, rotation
synchronization, orthogonal Procrustes, camera triangulation and resectioning,
essential matrix estimation, system identification, and approximate GCD. Our
results can also be used to analyze the stability of SOS relaxations of general
polynomial optimization problems.Comment: 23 pages, 3 figure
PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces
This is the first of two articles in which we give a proof - for a broad
class of four-manifolds - of Witten's conjecture that the Donaldson and
Seiberg-Witten series coincide, at least through terms of degree less than or
equal to c-2, where c is a linear combination of the Euler characteristic and
signature of the four-manifold. This article is a revision of sections 1-3 of
an earlier version of the article dg-ga/9712005, now split into two parts,
while a revision of sections 4-7 of that earlier version appears in a recently
updated dg-ga/9712005. In the present article, we construct virtual normal
bundles for the Seiberg-Witten strata of the moduli space of PU(2) monopoles
and compute their Chern classes.Comment: Journal fur die Reine und Angewandte Mathematik, to appear; 64 page
- …