1,137 research outputs found
Parameter identification in a semilinear hyperbolic system
We consider the identification of a nonlinear friction law in a
one-dimensional damped wave equation from additional boundary measurements.
Well-posedness of the governing semilinear hyperbolic system is established via
semigroup theory and contraction arguments. We then investigte the inverse
problem of recovering the unknown nonlinear damping law from additional
boundary measurements of the pressure drop along the pipe. This coefficient
inverse problem is shown to be ill-posed and a variational regularization
method is considered for its stable solution. We prove existence of minimizers
for the Tikhonov functional and discuss the convergence of the regularized
solutions under an approximate source condition. The meaning of this condition
and some arguments for its validity are discussed in detail and numerical
results are presented for illustration of the theoretical findings
Preconditioners for state constrained optimal control problems\ud with Moreau-Yosida penalty function tube
Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the state poses a significant challenge for optimization methods. Our focus here is on the incorporation of the constraints via the Moreau-Yosida regularization technique. This method has been studied recently and has proven to be advantageous compared to other approaches. In this paper we develop preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau-Yosida regularized problem. Numerical results illustrate the competitiveness of this approach. \ud
\ud
Copyright c 2000 John Wiley & Sons, Ltd
Adaptive local minimax Galerkin methods for variational problems
In many applications of practical interest, solutions of partial differential
equation models arise as critical points of an underlying (energy) functional.
If such solutions are saddle points, rather than being maxima or minima, then
the theoretical framework is non-standard, and the development of suitable
numerical approximation procedures turns out to be highly challenging. In this
paper, our aim is to present an iterative discretization methodology for the
numerical solution of nonlinear variational problems with multiple (saddle
point) solutions. In contrast to traditional numerical approximation schemes,
which typically fail in such situations, the key idea of the current work is to
employ a simultaneous interplay of a previously developed local minimax
approach and adaptive Galerkin discretizations. We thereby derive an adaptive
local minimax Galerkin (LMMG) method, which combines the search for saddle
point solutions and their approximation in finite-dimensional spaces in a
highly effective way. Under certain assumptions, we will prove that the
generated sequence of approximate solutions converges to the solution set of
the variational problem. This general framework will be applied to the specific
context of finite element discretizations of (singularly perturbed) semilinear
elliptic boundary value problems, and a series of numerical experiments will be
presented
Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework
We analyze temporal approximation schemes based on overlapping domain
decompositions. As such schemes enable computations on parallel and distributed
hardware, they are commonly used when integrating large-scale parabolic
systems. Our analysis is conducted by first casting the domain decomposition
procedure into a variational framework based on weighted Sobolev spaces. The
time integration of a parabolic system can then be interpreted as an operator
splitting scheme applied to an abstract evolution equation governed by a
maximal dissipative vector field. By utilizing this abstract setting, we derive
an optimal temporal error analysis for the two most common choices of domain
decomposition based integrators. Namely, alternating direction implicit schemes
and additive splitting schemes of first and second order. For the standard
first-order additive splitting scheme we also extend the error analysis to
semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which
also contains numerical experiments. Version 3 and 4: Only comments added.
Version 2, page 2: Clarified statement on stability issues for ADI schemes
with more than two operator
- …