New generation inversion codes

A. Asensio Ramos Instituto de Astrofísica de Canarias (Spain)

J. de la Cruz Rodríguez

Institute for Solar Physics (Sweden)

github.com/aasensio

@aasensior

aasensio.github.io/blog

SIR, Nicole, Helix+, Hazel, MILOS, PCA, …

Old generation – From 70s and still R&R

Teenage generation – 21st century

• PCA regularized deconvolution (IAC) • Spatially coupled inversions [van Noort’s talk]

EST

DKIST GREGOR

The new generation

Regularization

Why regularization?

• Pixel-by-pixel inversion

• No spatial correlation is used at all

• Noise induces degeneracies

• Your inversion code is surely biased (and you don’t know it)

• We know many things from our solution

Include a-priori knowledge Maximum-likelihood maximum a-posteriori

A simple observation

1 MB (RAW) 129 kB (JPEG) - lossy

286 kB (PNG) - lossless

Do we need information for all pixels?

Nunknowns=Npixel x Nparameters

Do we need information for all pixels?

Nunknowns=Npixel x Nparameters

1000 x 1000

Do we need information for all pixels?

Nunknowns=Npixel x Nparameters

1000 x 1000 ~10-20

Do we need information for all pixels?

Nunknowns=Npixel x Nparameters

1000 x 1000 ~10-20 ~107

Do we need information for all pixels?

Nunknowns=Npixel x Nparameters

1000 x 1000 ~10-20 ~107

Redundant information!

The importance of spatial correlation

Rempel et al. (2009)

The Trick™

Use a sparsifying (compressive) transformation

Use a sparsifying (compressive) transformation

• Solution is sparse • Reduction in the number of unknowns

• Transformation is global

• A pixel contributes to all modes

Compression – Discrete Cosine Transform

Compression – Daubechies Wavelets

Compression – Discrete Cosine Transform

Compression – Daubechies Wavelet

Compression – Physical parameters DCT

Compression – Physical parameters Wavelet

Compression – Physical parameters DCT

Compression – Physical parameters Wavelet

The problem to solve

The problem to solve

How to solve the problem

Levenberg-Marquardt (gradient descent+Newton) cannot be used because the Hessian matrix is huge

• Cannot be built and/or approximated • Contrary to what happens with e.g., van Noort’s method

• Cannot be inverted

• Matrix is not sparse

Need to use accelerated first-order methods

First-order proximal algorithms

First-order proximal algorithms

Optimization of a convex+non-convex function

Proximal gradient descent methods broadly used in ``big data’’

Proximal operators

Proximal operators

For the l0 norm, the proximal operator is the hard-thresholding operator

Proximal operators

For the l0 norm, the proximal operator is the hard-thresholding operator

Proximal operator – Intuitive idea

Admissible solutions Non-admissible

solutions

Computer code

• MPI parallel general inversion code (C++)

• Uses the FISTA algorithm (Beck & Teboulle 2009) with restarting (O’Donoghue & Candès 2012)

• Forward model is modular (+gradient)

• Sparsity: DCT, Wavelet, …

• Spatial deconvolution is added trivially

• We are studying thresholding strategies

Sunspot - Comparison

B

qB

fB

v

Dl

30% compression factor!!

Quiet Sun - Comparison

B

qB

fB

v

Dl

30% compression factor!!

Inversion+deconvolution: straightforward

Convergence

Extensions

• 2D inversions with gradients

• Full 3D inversions invert a cube

• Regularized versions of SIR

• Inclusion of unknown systematics plus the signal

• Inversion + deconvolution