New preprint on multilevel Monte Carlo with non-nested meshes available!

Our new paper on Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes is now available on arXiv!


When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. In this paper, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in a MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In a MLMC setting, a good coupling is enforced and the telescoping sum is respected.

Hjernens vannveier in Bergen

This week I’ve been in Odense for a PhD defense (Congratulations to Dr. Christian Valdemar Hansen!), while today I am giving at a talk at Nansensenteret i Bergen for a meeting organized by Norges Tekniske Vitenskapsakademi and Tekna. I will talk about the brain’s waterscape/Hjernens vannveier, of course. This presentation targets a semi-academic, semi-technical audience and the slides are publicly available.

Waterscapes at ENUMATH 2017

The European Conference on Numerical Mathematics and Advanced Applications CG2_3d(ENUMATH) was organized by the University of Bergen at Voss this week, Sept 25-29 2017. The Waterscapes research teams were well represented at the conference with the plenary open lecture on “The operator preconditioning framework with various applications to interstitial fluid flow and the aging human brain” given by Waterscape collaborator Kent-Andre Mardal, and I presenting a plenary lecture on “Compatible discretizations in our hearts and mind”. Slides available on SlideShare:

Waterscapes at Cutting Edge 2017

colin3000The Cutting Edge festival is “Norway’s largest interdisciplinary business jam session for scientists, technologists, entrepreneurs, investors and policy makers”. The 2017 edition had a session on “Brain and Aging” where I presented the core vision of the Waterscapes projects: to improve our fundamental understanding of fluid flow and solute transport in brain tissue using mathematical modelling. Here are the Cutting Edge 2017 slides (all simulations were produced using the Waterscapes solver collection), and the talk:

Every ballet ensemble or opera has a primadonna.  Most football teams have a primadonna.  Hey, probably every work place has at least one primadonna.  You know, the one that gets and takes all of the attention. In the brain, the neuron is the primadonna, the one that everyone cares about. But, I think we all know that in a ensemble or a football team or a work place, it is all the different pieces playing together that really matters. So, in the brain, we should also care about the other types of brain cells those called glial cells, which very rarely anyone has heard of, we should care about the water-filled spaces between cells, we should care about the brain’s blood vessels. And this is precisely I call the brain’s waterscape: the fluid-filled pathways, the canals and the water channels within the brain.

We should care about these because they are all crucial for the well-being of the brain, for the well-being of the primadonna neurons. And we know that a number of brain disorders, like Alzheimer’s, like Parkinson’s, like oedema, like Stroke are associated with abnormalities in the brain’s waterscape. But at the same time, we know surprisingly little about how the fundamental mechanisms underlying the waterscape, about the fundamental mechanisms for water balance and water-enhanced clearance of metabolic waste in the brain. One reason for this is that experimental investigations into the mechanics of the human brain are extremely challenging: both from a practical and an ethical viewpoint.

So in view of this challenge, I and my research team, we are spear-heading a different approach into investigating the brain’s waterscape, a mathematical approach. We are developing and using mathematical and computational models to describe fluid flow and transport within brain tissue. Our mathematical models are physics-based, so what we do is to combine basic laws of physics for how fluids move and how fluid can transport solutes and how fluids interact with structures like biological cells, and we express these as mathematical equations. These equations can then predict how the modelled system will evolve over time.

This is the simplest mathematical model that I know of. It is called the diffusion equation, and it describes how a solute could spread in brain tissue. The solute could be like injected dye or injected drugs or natural by-products like protein fragments from the brain’s metabolism. These models consist of three main components: an unknown, something known and the link between the known and the unknown. The unknown, or the output,  is what we want to predict or compute, in this case the concentration of the solute over time and space. The known, or the input parameters, could be the geometry of the brain, the initial amount of the solute, and some hopefully quantifyiable properties of the brain tissue in question. And the physical laws that we choose to include provides the link between the two.

Now, this looks like a very innocent equation, but it actually impossible to solve using pen and paper only in general. So, we actively develop and use high performance numerical simulation technology to compute the output  of this model.  These illustrations on your far right show how this model predicts that dye injected inside a  brain would spread over time.  Now, we can add complexity to this model … we can account for additional solute transport … we can account for more complex physics, but the core idea remains the same.

Now, the remarkable thing about computational models is that we can experiment and make predictions in a way that would never be possible in physical experiments. The first aspect we aim at is a better understanding of the fundamental mechanisms for how metabolic waste and other solutes can be cleared from the brain, which is essential to understand how normal clearance and, to understand what fails and leads to accumulation of waste in Alzheimer’s patients for instance, and also to understand how treatment drugs can be delivered into the brain.  Using these computational models we can test clinical hypotheses, that may be impossible to test in a lab or in a patient, to see if existing hypotheses make physical sense. In this way, we envision using mathematical models to drive basic medical science forward.

But we can go much further than that. We can make predictions under changing scenarios. For instance, we know that arteries stiffen and that brains shrink as we age. And we can represent this behaviour in the model by representing it as a change in the input parameters of our equations. We can thus make predictions for aging brains, and link these predictions with features of the disease. And if we can understand the underlying cause of a disease, that will be a crucial step on the way to cure or prevent diseases, like Alzheimer’s, where we have no cure today.

And going even further, another possibility is to create personalized models. By including person-specific information in the input parameters of our models, like person-specific geometries, we can create simulations and make predictions specific to an individual. This means that we can evaluate therapies and even design optimal therapies tailored to highly stratified patient groups or even tailored to each and every one of us.

So, in the next years, I envision that mathematical models of the brain’s waterscape can provide a more practical, more ethical and less expensive avenue to understand the brain’s mechanisms, how these vary from individual to individual, and ultimately towards designing, improving and tailoring medical treatments.

And as such find a way to keep the brain’s primadonnas, the neurons, happy and healthy. Thank you.