5 research outputs found
Solving Poisson's Equation on the Microsoft HoloLens
We present a mixed reality application (HoloFEM) for the Microsoft HoloLens.
The application lets a user define and solve a physical problem governed by
Poisson's equation with the surrounding real world geometry as input data.
Holograms are used to visualise both the problem and the solution. The finite
element method is used to solve Poisson's equation. Solving and visualising
partial differential equations in mixed reality could have potential usage in
areas such as building planning and safety engineering.Comment: 2 pages, 9 figure
Robust preconditioners for PDE-constrained optimization with limited observations
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.
In this paper we propose and analyze preconditioners for PDE-constrained
optimization problems with limited observation data, e.g. observations are only
available at the boundary of the solution domain. Our methods are robust with
respect to both the regularization parameter and the mesh size. That is, the
condition number of the preconditioned optimality system is uniformly bounded,
independently of the size of these two parameters. We first consider a
prototypical elliptic control problem and thereafter more general
PDE-constrained optimization problems. Our theoretical findings are illuminated
by several numerical results
Variational data assimilation for transient blood flow simulations - Cerebral aneurysms as an illustrative example
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 resonance and computed tomography angiography. However, computational fluid mechanics rely on accurate initial and boundary conditions, which are difficult to obtain. In this paper, we address the problem of recovering high‐resolution information from noisy and low‐resolution physical measurements of blood flow (for example, from phase‐contrast magnetic resonance imaging [PC‐MRI]) using variational data assimilation based on a transient Navier‐Stokes model. Numerical experiments are performed in both 3D (2D space and time) and 4D (3D space and time) 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