@article {25229,
title = {Variational data assimilation for transient blood flow simulations},
journal = { International Journal for Numerical Methods in Biomedical Engineering},
volume = {35},
year = {2018},
month = {10/2018},
pages = {e3152},
publisher = {John Wiley \& Sons},
abstract = {Several cardiovascular diseases are caused from localised abnormal blood flow\ such as in the case of stenosis or aneurysms. Prevailing theories propose that the development is caused by abnormal wall-shear stress in focused\ areas. \ Computational fluid mechanics have arisen as a promising tool for\ a more precise and quantitative analysis, in particular because the\ anatomy is often readily available even by standard imaging techniques such as magnetic resolution and computed tomography angiography. \ However,\ computational fluid mechanics rely on accurate initial and boundary conditions which\ is difficult to obtain. \ In this paper we address the problem of recovering high resolution information from noisy, low-resolution\ measurements of blood flow using variational data assimilation based on a transient Navier-Stokes model.\ Numerical experiments are performed in both 2D and 3D and with pulsatile flow relevant for physiological flow in cerebral aneurysms.\ The results demonstrate that, with suitable regularisation, the model accurately reconstructs flow, even in the presence of significant noise.\ },
keywords = {adjoint equations, blood flow, Finite element method, Navier-Stokes, optimal control, variational data assimilation},
author = {Simon W {Funke} and Nordaas, Magne and Evju, {\O}yvind and Martin Sandve {Aln{\ae}s} and Mardal, Kent-Andre}
}
@phdthesis {25522,
title = {Operator preconditioning for PDE-constrained optimisation and multiscale problems},
volume = {PhD},
year = {2017},
school = {University of Oslo},
address = {University of Oslo},
abstract = {Many physical processes are described with partial differential equations (PDE), with important examples such as thermal conduction, Maxwell laws for electrodynamics and Navier-Stokes equations for viscous fluids.In many applications, we do not want to find just any solution for differential equation, but an optimal solution in a specific sense: We have one or more parameters in the PDE model that we can tune, and we want to find the parameter values that are most beneficial. A classic example is to determine the shape of an airfoil, so that the airflow around the wing results in the best possible flight characteristics.A related problem is the reconstruction of physical processes based on incomplete measurements. In such a problem, we seek the solution to the differential equation that best fits the measurement data.Both examples above can be formulated as optimization problems with PDE constraints. In almost all applications, we can not calculate the exact solution for a PDE or a PDE-constrained optimization problem. Instead, we must compute a numerical approximate solution. For PDE-constrained optimization, this usually involves solving large sparse saddle point systems.In his thesis work Magne has investigated methods to solve such problems in an efficient way. A key technique is operator preconditioning. This involves identifying a mathematical structure in the problem that can be exploited to derive effective solution algorithms. The thesis consists of three papers. The first two paper concerns parameter-robust preconditioning techniques for saddle point systems. In the third paper, a PDE-constrained optimization method is applied to solve a data assimilation problem for cerebral blood flow.},
author = {Nordaas, Magne}
}
@article {24799,
title = {Preconditioners for saddle point systems with trace constraints coupling 2D and 1D domains},
journal = {SIAM Journal of Scientific Computing},
volume = {38},
year = {2016},
publisher = {SIAM},
author = {Kuchta, Miroslav and Nordaas, Magne and Joris C. G. {Verschaeve} and Mortensen, Mikael and Mardal, Kent-Andre}
}
@article {24805,
title = {Robust preconditioners for PDE-constrained optimization with limited observations},
journal = {BIT Numerical Mathematics},
volume = {57},
number = {2},
year = {2016},
pages = {405-431},
publisher = {Springer},
author = {Mardal, Kent-Andre and Nielsen, Bj{\o}rn Fredrik and Nordaas, Magne}
}
@misc {24815,
title = {Variational data assimilation for blood flow simulations},
howpublished = { MS at SIAM UQ, Zurich},
year = {2016},
author = {Mardal, Kent-Andre and Simon W {Funke} and Nordaas, Magne and Evju, {\O}yvind and Martin Sandve {Aln{\ae}s}}
}
@misc {24019,
title = {Robust preconditioners for PDE-constrained optmization with limited data},
howpublished = {2015 SIAM CSE, Salt Lake City},
year = {2015},
type = {Minisymposium },
author = {Nordaas, Magne and Mardal, Kent-Andre and Nielsen, Bj{\o}rn Fredrik}
}
@misc {Simula.simula.2950,
title = {PDE-Constrained Optimisation in Hilbert Spaces},
howpublished = {FEniCS{\textquoteright}14 workshop, Paris},
year = {2014},
month = {June},
abstract = {The solution of optimisation problems constrained by partial differential equations becomes increasingly more feasible due to the increase of computational resources and the development of solution algorithms. The computational cost and the large number of degrees of freedoms in PDE applications raise a particular challenge for partical algorithms. A key property for an efficient optimisation algorithm is that the required number of iterations is independent on the local mesh refinement and its element size. However, a straight-forward application on standard optimisation algorithms on the reduced optimisation problem results in mesh-size dependent iteration numbers and a poor performance on non-homogoenous meshes. In this talk we present on how good convergence is obtained by rethinking these optimisation algorithms to respect the underlying inner products and induced norms of the involved function spaces. Based on this idea we develop a new optimisation framework specifically designed for PDE-constrained optimisation which honors the inner product of the underlying Hilbert spaces. We show that this strategy is equivalent to a variance of preconditioned conjugate gradient methods or for the BFGS a user defined initial estimate of the Hessian matrix. Parallel , currently supports unconstrained optimisation algorithms and integrates seemlesly with FEniCS and dolfin-adjoint. Numerical results demonstrate the effectiveness of the framework.},
keywords = {Conference},
author = {Simon W {Funke} and Nordaas, Magne}
}
@misc {Simula.simula.3092,
title = {Robust Preconditioners for PDE-Constrained Optimization With Limited Observations},
howpublished = {European Multigrid Conference 2014},
year = {2014},
month = {September},
abstract = {Regularization-robust preconditioners for PDE-constrained optimization problems have been successfully developed. These methods, however, typically assume that observation data is available throughout the entire domain of the state equation. For many inverse problems, this is an unrealistic assumption. We propose and analyze preconditioners for PDE-constrained optimization problems with limited observation data, e.g. when observations are only available at the boundary of the computational domain. Our methods are robust with respect to both the regularization parameter and the mesh size. That is, the number of required MINRES iterations is bounded uniformly, regardless of the size of the two parameters. The theoretical findings are illuminated by several numerical results.},
keywords = {Conference},
author = {Nordaas, Magne}
}