Browsing by Author "Huber, Felix"

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Open Access
Efficient algorithms for geodesic shooting in diffeomorphic image registration
(2019) Huber, Felix
Diffeomorphic image registration is a common problem in medical image analysis. Here, one searches for a diffeomorphic deformation that maps one image (the moving or template image) onto another image (the fixed or reference image). We can formulate the search for such a map as a PDE constrained optimization problem. These types of problems are computationally expensive. This gives rise to the need for efficient algorithms. After introducing the PDE constrained optimization problem, we derive the first and second order optimality conditions. We discretize the problem using a pseudo-spectral discretization in space and consider Heun's method and the semi-Lagrangian method for the time integration of the PDEs that appear in the optimality system. To solve this optimization problem, we consider an L-BFGS and an inexact Gauss-Newton-Krylov method. To reduce the cost of solving the linear system that arises in Newton-type methods, we investigate different preconditioners. They exploit the structure of the Hessian, and use algorithms to efficiently compute an approximation to its inverse. Further, we build the preconditioners on a coarse grid to further reduce computational costs. The different methods are evaluated for two-dimensional image data (real and synthetic). We study the spectrum of the different building blocks that appear in the Hessian. It is demonstrated that low rank preconditioners are able to significantly reduce the number of iterations needed to solve the linear system in Newton-type optimizers. We then compare different optimization methods based on their overall performance. This includes the accuracy and time-to-solution. L-BFGS turns out to be the best method, in terms of runtime, if we solve solving for large gradient tolerances. If we are interested in computing accurate solutions with a small gradient norm, an inexact Gauss-Newton-Krylov optimizer with the regularization term as preconditioner performs best.
Open Access
Knowledge-based modeling of simulation behavior for Bayesian optimization
(2024) Huber, Felix; Bürkner, Paul-Christian; Göddeke, Dominik; Schulte, Miriam
Numerical simulations consist of many components that affect the simulation accuracy and the required computational resources. However, finding an optimal combination of components and their parameters under constraints can be a difficult, time-consuming and often manual process. Classical adaptivity does not fully solve the problem, as it comes with significant implementation cost and is difficult to expand to multi-dimensional parameter spaces. Also, many existing data-based optimization approaches treat the optimization problem as a black-box, thus requiring a large amount of data. We present a constrained, model-based Bayesian optimization approach that avoids black-box models by leveraging existing knowledge about the simulation components and properties of the simulation behavior. The main focus of this paper is on the stochastic modeling ansatz for simulation error and run time as optimization objective and constraint, respectively. To account for data covering multiple orders of magnitude, our approach operates on a logarithmic scale. The models use a priori knowledge of the simulation components such as convergence orders and run time estimates. Together with suitable priors for the model parameters, the model is able to make accurate predictions of the simulation behavior. Reliably modeling the simulation behavior yields a fast optimization procedure because it enables the optimizer to quickly indicate promising parameter values. We test our approach experimentally using the multi-scale muscle simulation framework OpenDiHu and show that we successfully optimize the time step widths in a time splitting approach in terms of minimizing the overall error under run time constraints.
Open Access
The sparse grid combination technique for quantities of interest
(2016) Huber, Felix
The curse of dimensionality is a major problem for large scale simulations. One way to tackle this problem is the sparse grid combination technique. While a full grid requires O{h_n^{-d}} grid points the sparse grid combination technique needs significantly less points. In contrast to the traditional combination technique, which combines solution functions themselves, this work puts its focus on the combination technique with quantities of interest and their surpluses. After introducing the concept of surpluses that describe how much the solution changes if the grids are refined, we defined the combination technique as a sum of these surpluses. We show how the concept of surpluses can be utilized to deduce error bounds for the quantity of interest and helps to adapt the combination technique to problems with different error models. To improve the error bound we introduce a new extrapolated version of the combination technique and see how the surpluses are affected. To evaluate our theoretical results we perform numerical experiments where we consider integration problems and the gyrokinetic plasma turbulence simulation GENE. The experimental results for the integration problems nicely confirm our derived theoretical results.