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Elementary flare profiles fitted to the STIX light curves using differential evolution algorithm

Marek Stęślicki¹, Karol Kułaga², Anna Kępa¹

Differential evolution algorithms

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Differential evolution (DE) belongs to the class of evolutionary algorithms designed to solve optimization problems with real-valued parameters. It is very powerful method for the black-box optimization, where the function to be optimized is very complex and it is difficult to find the appropriate derivatives necessary to compute its extreme. DE (Storn & Price, 1997) is based on the biological evolutionary processes: mutation, crossover, and selection.

The most important characteristics of DE are the following:

• The mutation is provided by arithmetic combinations of individuals, not as the result of small perturbations to the genes as in genetic algorithm (Feoktistov & Janaqi, 2004).
• DE works with three equally large populations of individuals: the population of parents, trial, and descendants.
• The size of populations (a set of potential solution) - usually does not change during the evolution process.

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A typical working scheme for DE:

This approach is sufficiently efficient for light curves consisting individual well separated EFP, however the calculation time growing exponentially with more complex light curves. Therefore more efficient are DE algorithms. The example of more complex fit using DE is shown below.

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where

After the launch of the Spectrometer Telescope for Imaging X-rays (STIX) onboard Solar Orbiter on 10 February 2020, it is recording solar X-rays almost uninterruptedly since January 2021 covering the rising phase of solar cycle 25. STIX registers many number of flares. We present a methodology is given to determine basic parameters of flares from their X-ray light curves. One is a semiautomatic flare detection procedure that gives start, peak, and end times for single (“elementary”) flare events under the assumption that the light curve is a simple convolution of a Gaussian and exponential decay functions. More complex flares with multiple peaks can generally be described by a sum of such elementary flares. The flare time profiles are hard to fit automatically using standard methods. I will present a method based on differential evolution used to fit elementary flare profiles to the STIX light curves.

1. Solar Physics Division, Space Research Centre Polish Academy of Sciences, Wroclaw, Poland.

2. Astronomical Institute, University of Wroclaw, Wroclaw, Poland.

References

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Aschwanden, Dennis, & Benz, 1998, ApJ 497, 972

Aschwanden & Freeland, 2012, ApJ 754, 112

Feoktistov & Janaqi, 2004, IPDPS’04, 165

Grycuk, M. et al. 2017, Solar Physics, 2017, 292

Storn, R., & Price, K. 1997, Global Optimiz., 11, 341

We acknowledge financial support from the Polish National Science Centre grant number 2020/39/B/ST9/01591

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STIX light curves complexity

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The EFP is a complex function with very steep (exponential) gradients. Additionally, the complex SXR light curves observed by STIX, especially during close approaches to the Sun are very difficult to fit using standard methods (solving the least-squares problem).

One of the easiest methods to solve this problem is to simply guess the solution, by checking multiple random sets of parameters. Additionally the solution (red curve) can be improved by applying least-squares method (green curve), e.g. mpfit. Such simple method on modern PC can gives a solution after ~10 s.

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Jan 14,2022

Sep 20,2021