FFSAS v1.1

Free-Form inversion for Small-Angle Scattering (SAS)

ffsas is a Python library to invert for free-form distributions of model parameters in a polydisperse SAS system. It yields the maximum likelihood estimator of the parameter distributions and the sensitivity and uncertainty of the maximum likelihood estimator. ffsas comes with the following features:

• Generality: ffsas formulates the SAS inverse problem as a Green's-tensor-based multi-linear map, which covers any complex SAS models with multiple parameters. An arbitrary model can be easily implemented by supplying the Green's function for forward modelling, i.e., a function that computes the monodisperse intensity on a structured grid of model parameters (i.e., the Green's tensor). In short, users only need to care about the physics for forward modelling, leaving the inverse problem all to ffsas. Besides, it does not require an initial guess of the parameter distributions, the scaling factor or the source background.

• Efficiency: through theoretical analysis, the inverse problem is simplified as a highly solvable nonlinear programming (NLP) problem with a few equality constraints. It is solved by a trust-region method, implemented in SciPy as scipy.optimize.minimize(method='trust-constr'), officially mentioned as "the most versatile constrained minimization algorithm implemented in SciPy and the most appropriate for large-scale problems". Computation of the Jacobian and Hessian of the χ2 error, as the most expensive step during solution, is accelerated by GPU and mini-batch computation. The idea is borrowed from deep learning and implemented with PyTorch.

• Accuracy: the model parameters and the resulting intensity in a SAS problem can span many orders of magnitude, and a good choice of unit system is essential to avoid an ill-conditioned inverse problem. ffsas automatically analyzes the orders of magnitude of data and parameters so as to determine a proper internal unit system to avoid accuracy loss. Such an internal unit system is hidden from users, who can use an arbitrary unit system for input and output.

• Usability: ffsas can be installed with pip in one line. Its usage only includes four APIs respectively for

  • computing the Green's tensor G given a model and a parameter space
  • define a G-system with this Green's tensor
  • using this G-system to compute intensity given parameter distributions (forward modelling)
  • using this G-system to invert for parameter distributions given intensity data (inversion)

  A built-in logging system produces detailed and readable logs during these processes.

Quick Start

To install ffsas:

pip install ffsas

Follow the User Guide to learn the usage.

Citation

• Full paper (open access): http://doi.org/10.1107/S1600576722006379

• BIBTex:

  @article{Leng:jl5041,
  author = "Leng, Kuangdai and King, Stephen and Snow, Tim and Rogers, Sarah and Markvardsen, Anders and Maheswaran, Satheesh and Thiyagalingam, Jeyan",
  title = "{Parameter inversion of a polydisperse system in small-angle scattering}",
  journal = "Journal of Applied Crystallography",
  year = "2022",
  volume = "55",
  number = "4",
  pages = "966--977",
  month = "Aug",
  doi = {10.1107/S1600576722006379},
  url = {https://doi.org/10.1107/S1600576722006379},
  }

Funding and Support

This work was supported by the ISIS Neutron and Muon Source (ISIS) of the Science and Technology Facilities Council through the ISIS-ML funding, and by Wave I of the UKRI Strategic Priorities Fund under the EPSRC grant (EP/T001569/1), particularly the AI for Science theme in that grant and the Alan Turing Institute. We gratefully acknowledge their support.