Mathematics professor named VILLUM Investigator

He is a Dr. Techn., has written more than 100 published articles, four books, is a SIAM Fellow, and a recipient of the most prestigious grant from the European Research Council—an ERC Advanced Grant. He can now add to his CV a VILLUM Investigator grant of DKK 35 million, which he will use—over the next six years—to develop new algorithms that can strengthen calculations of inverse problems. (Read more about the VILLUM Investigator project).

Landmines under strawberry bed

To illustrate his research, he takes out a small box with motifs of strawberries. He has glued some magnets inside the box—like landmines under a strawberry bed.

“We know that there’s something down there, but we can't dig down in the middle of the strawberry bed. That’s out of the question. But we can instead take another magnet and measure on the surface of the soil, and thus make some external measurements. Based on these measurements, we can create some formulas that can calculate what is underground,” says Per Christian Hansen.

Another example—but with the same principle—is to monitor a production that takes place inside a closed tank. If you want to avoid stopping the production to look inside the tank, you can send an electrical current into the tank and measure the resulting output voltage. Based on these measurements, you can say something about what is happening inside the tank.

“You achieve results which—with a certain probability—can tell you what’s happening inside the tank,” says Per Christian Hansen, and continues:

“The only problem is that—right now—we don’t have the necessary formulas and methods for doing this with sufficient information. Or, put differently: The tools we have today do not allow us to enter uncertainties. Specific methods for this have been developed in geophysics and the oil industry, but—for general inverse methods—it’s a problem that we don’t know the uncertainties.”

Error after error

The purpose of Per Christian Hansen’s VILLUM Investigator project is therefore first and foremost to develop algorithms which can be used in all areas with a few adjustments. This is a large mathematical challenge.

Firstly, reservations must be made for the noise which will invariably be contained in the data being measured. Secondly, the model used will also contain errors or things that are not described with sufficient accuracy. Thirdly, a priori knowledge about what the reality looks like may also contain errors. And, finally, there may often also be faults in the program used to make the calculations.

“In fact, all these errors contribute to making your final result inaccurate. And you may not know where the important errors have been introduced in the equation. And that’s my job: Entering these uncertainties in the formula,” says Per Christian Hansen, and adds that the knowledge you may have in specific fields need to come into play to the benefit of all researchers studying inverse problems.

It is therefore also an integral part of the research project that it is precisely to contribute to gathering the research that is being conducted in this field worldwide, explains Per Christian Hansen:

“I find it fascinating that you can describe a variety of different problems with the same equations, so right now I will be concentrating on the mathematics and theory development. But when we’re a bit further along in the project, I will create some software which researchers can use to calculate probabilities, which engineers can use to see whether they have made the right design, and which may—in the long term—also be used as a support tool in, for example, tomography, image analysis, and industrial inspection.”