|
Fred Vermolen
(Hasselt University)
|
Simulation of post-burned skin using principles from morphoelasticity
Each year the lives of hundreds of thousands people are heavily impacted by severe burn injuries. Although nowadays clinical technologies allow most patients to survive heavy burn traumas, these burn injuries often come with hypertrophic scars and contractures, which impair the mobility of patients. In order to minimize the impact to the patients, therapies based on principles such as dressings, ointments, splinting and skin grafting (skin transplantation) are applied. In order to optimize treatment, a quantitative description of the underlying biological mechanisms is needed and for this reason, mathematical models have been constructed. In this talk, we present a continuum-based model that is constructed with principles from morphoelasticity. Morphoelasticity is a mathematical formalism that simultaneously deals with elasticity and microstructural changes in the tissue. We will show some mathematical results regarding stability of the model, as well as neural network simulations that reproduce the simulations at very high computational speed.
|
|
Jan ten Thije Boonkkamp
(Eindhoven University of Technology)
|
Inverse methods for freeform optical design
Inverse methods for freeform optical design compute the shape of an optical surface (reflector/lens) that converts a given light source distribution, typically LED, in a desired target distribution. The surface is referred to as freeform, as it has no symmetry. Applying the principles of geometrical optics, we can derive a geometrical description of an optical system, which for many systems can be formulated in terms of a so-called cost function as known from optimal transport theory. From this description we can subsequently derive a PDE for the optical map, connecting a point in the source domain with a point in the target domain. Combining this equation with energy conservation, we can derive a fully nonlinear PDE for the shape of the optical surface.
We have developed the following classification for optical systems. First, the simplest optical systems, typically a reflector transferring a parallel source distribution in a far-field target distribution, can be described by a quadratic cost function, and the corresponding PDE for the optical surface is the standard Monge-Ampère equation. Second, for more complicated systems, such as a lens converting a point-source distribution in a far-field target distribution, the cost function is non-quadratic and the governing equation for the optical surface is a generalized Monge-Ampère equation. Finally, there are optical systems, typically for near-field target distributions, that no longer allow a description in terms of a cost function. Instead, a generalization in terms of a generating function is possible, and the corresponding PDE for the optical surface is a generated Jacobian equation.
We have developed least-squares solvers for all three classes of optical systems. Our least-squares solvers for the Monge-Ampère equations are two-stage algorithms, i.e., in the first stage the optical map is computed, and in the second stage, the shape of the optical surface is reconstructed from the optical map. Both stages are least-squares solvers. The least-squares solver for the generated Jacobian equation is a slight modification of these methods. Our algorithms can handle quite complicated source and target distributions, and are currently transferred to industrial production code. We will demonstrate the performance of our solvers for several examples.
|
|
Francesc Verdugo Rojano
(VU Amsterdam)
|
PartitionedArrays.jl: A new data-parallel programming model to speedup the development of large-scale scientific computing applications.
In this talk we present PartitionedArrays.jl, a parallel (sparse) linear algebra library implemented in the Julia programming language. PartitionedArrays.jl provides a modern alternative to well-known packages such as Petsc, which consider more traditional HPC languages such as C/C++ and Fortran. PartitionedArrays.jl is not just a translation into Julia of preexisting libraries, but a new parallel programming framework that aims at simplifying the implementation of parallel numerical algorithms and the development of large-scale scientific computing applications. Virtually all large-scale linear solvers are based on the Message Passing Interface (MPI) for parallelization. While MPI provides an efficient communication layer, it makes challenging the development cycle. When debugging MPI code, a portion of the bugs can be tracked with conventional tools in a single MPI rank, but genuine parallel errors need to be debugged in parallel. Fixing an application that requires at least 100 ranks to crash can be very challenging, even with state-of-the-art (often proprietary) MPI debuggers. PartitionedArrays.jl introduces an alternative data-parallel programming model that aims to solve these problems. This interface exposes a set of communication directives, which are defined as simple operations on arrays (hence the name PartitionedArrays.jl). This interface has been proven successful to implement distributed linear algebra data structures (vectors and sparse matrices), and key computational kernels (distributed sparse matrix-vector and matrix-matrix products). With these tools, we were able to solve large Finite Element computations with nearly optimal weak and strong scaling up to tens of thousands of CPU cores [doi.org/10.21105/joss.04157].
|
|
Vanja Nikolić
(Radboud University)
|
Robust error bounds for ultrasonic models in the inviscid limit
A growing number of ultrasound applications in medicine and industry fuels research in nonlinear acoustics. Since sound evolution is quasilinear at high frequencies or intensities, these applications give rise to many interesting mathematical questions involving nonlinear wave equations. The talk will focus on the influence of dissipation on the behavior of discrete ultrasonic models. In the vanishing limit of their strong damping, they are known to undergo a singular behavior change, switching from parabolic-like to hyperbolic evolution. In the talk, we will present recent results concerning robust numerical analysis of such ultrasonic equations in the singular inviscid limit.
|
|
Carolina Urzúa-Torres
(Delft University of Technology)
|
Block Calderón preconditioning for wave scattering at multi-screens
We are interested in the numerical solution of time-harmonic scattering by so-called multi-screens, which are geometries composed of panels meeting at junction lines. This is modelled via first kind integral equations using the framework proposed by Claeys and Hiptmair. The key realisation is that solutions of the related boundary integral equations belong to jump spaces, that can be represented as the quotient-space of a multi-trace space and a single trace space. Hence, the corresponding Galerkin discretization via quotient-space boundary element methods is up to the task. However, it does not address the ill-conditioning of the arising Galerkin matrices and the performance of iterative solvers deteriorates significantly when increasing the mesh refinement.
As a remedy, we introduce a Calderón-type preconditioner and discuss two possible multi-trace discretizations. First, we work with the full multi-trace discrete space, which contains many more degrees of freedom (DoFs) than strictly required. Then, we propose a representation of the quotient-space that reduces significantly the number of degrees of freedom while still allowing for efficient Calderón preconditioning. For this, we exploit the fact that the solution to the scattering problem is determined only up to a function in the single trace space. This implies that if we modify the single trace subspace of the multi-trace discrete space, the solution, as an element of the quotient-space, is unaffected.
|
|
Richard Stevens
(University of Twente)
|
Modeling the fluid physics of wind farms
The performance of large wind farms depends on the development of turbulent wind turbine wakes and the interaction between these wakes. Turbulence also plays a crucial role in transporting kinetic energy from the large-scale geostrophic winds in the atmospheric boundary layer towards heights where wind farms can harvest this energy. High-resolution large-eddy simulations (LES) are ideal for understanding these flow phenomena. Much has been learned from wind farm simulations, which initially focused on 'idealized' situations. The community increasingly focuses on modeling more complex situations, such as the effect of complex terrain and different atmospheric stability conditions. As wind farms become larger, the need to improve their design and develop control strategies to mitigate wake effects increases. However, due to the large separation of length scales and the number of cases, using LES for wind farm design is unfeasible. Therefore, LES is used to develop computationally more tractable modeling approaches ranging from Reynolds Average Navier Stokes (RANS) models to analytical modeling approaches. In this presentation, we will give particular attention to the challenges of modeling wind farm dynamics in large-eddy simulations and emerging challenges to account for the effect of mesoscale flow phenomena in these simulations.
|
|
Francesca Cavallini
(VU Amsterdam)
|
Numerical methods for Uncertainty Quantification in neurobiological models for the cerebral cortex neural activity
In this work, we develop theory and numerical schemes for UQ in Wilson-Cowan models, as prototypes for the cortex activity subject to an external stimulation. More specifically, in the context of forward UQ, we introduce uncertainty in all possible inputs to the model (i.e. synaptic kernel, firing rate, initial state, external stimulus) by defining the abstract problem with random data on a probability space and we demonstrate the existence of an integrable and measurable solution in Bochner spaces. Under realistic assumptions on the data, we parametrize the model's equations in terms of random vectors and pose the problem in the image probability space (bounded or unbounded). Furthermore, we rigorously extend well-posedness and regularity statements to semidiscrete solutions in finite-dimensional approximating subspaces, a necessary step when spatio-temporal projections are used for the numerically approximated solutions.
Importantly, our theoretical analysis is independent of a particular choice of projector and we prove statements that are applicable to the entire range of numerical schemes for time-dependent spatially-extended systems.
Exploiting this theoretical framework, we introduce and study the convergence of one of the classical, non-intrusive methods for UQ, Stochastic Collocation, for multi-parameter neural fields on one-dimensional domains. We test our code on different instances: linear and non-linear equations, with Gaussian and uniform parameters with increasing variance, and different time intervals. The comparison with a standard Monte Carlo scheme just highlights the need of fast and advanced computational methods, even for a simple and not so realistic linear neural field.
In a preliminary work, we integrate experimental data and tackle the Inverse Problem. We take a Bayesian approach and use the interpolation results from stochastic collocation as a surrogate model, and exploit it to build an approximation to the posterior distribution.
|
|
Toon Ingelaere
(KU Leuven)
|
Single-ensemble multilevel Monte Carlo for interacting-particle methods
Interacting-particle methods have found applications in domains such as filtering, optimization and posterior sampling. These parallelizable and often gradient-free algorithms use an ensemble of particles that evolve in time, based on the combination of well-chosen dynamics and interaction between the particles. For computationally expensive dynamics – for example Bayesian inversion with an expensive forward model, or optimization of an expensive objective function – the cost of attaining a high accuracy quickly becomes prohibitive. We exploit a hierarchy of approximations to this forward model and apply multilevel Monte Carlo (MLMC) techniques, improving the asymptotic cost-to-error relation. More specifically, we use MLMC at each time step to estimate the interaction term within a single, globally-coupled ensemble. Numerical experiments corroborate our analysis of the improved asymptotic cost-to-error rate.
|
|
Balint Negyesi
(Delft University of Technology)
|
A Deep BSDE approach for the simultaneous pricing and delta-gamma hedging of large portfolios consisting of high-dimensional multi-asset Bermudan options
Direct financial applications of our recent works on the One Step Malliavin (OSM) scheme (Negyesi et
al. 2024a, 2024b) are presented. Therein a new discretization of (discretely reflected) Markovian backward stochastic differential equations is given which appear naturally in the pricing and hedging of Bermudan options. Our novel discretization schemes exploit a linear BSDE representation for the Z process stemming from Malliavin calculus, which involves a Γ process, corresponding to second-order Greeks of the associated option’s price. The resulting schemes give a robust and efficient way to perform not just delta- but also delta-gamma hedging. The main contribution of this work is to apply these techniques in the context of portfolio risk management. Large portfolios of a mixture of European and Bermudan derivatives are cast into the framework of a system of discretely reflected BSDEs. The resulting system of equations is simultaneously solved by a neural network regression Monte Carlo approach, similarly to the recently emerging class of Deep BSDE methods. Numerical results are presented on large-scale, high-dimensional portfolios, consisting of several European, Bermudan options with possibly different early exercise features. These demonstrate that our hedging strategies significantly outperform benchmark methods, both in the case of standard delta- and delta-gamma hedging.
|
|
|
