# A numerical study of heat source reconstruction for the advection-diffusion operator: A conjugate gradient method stabilized with SVD

## TL;DR

This paper presents a conjugate gradient method stabilized with SVD for reconstructing heat sources in advection-diffusion models, effectively handling heterogeneity and noise in thermomechanical heat source identification.

## Contribution

It introduces a novel inversion algorithm combining conjugate gradient with SVD-based regularization for heat source reconstruction in advection-diffusion problems.

## Key findings

- The SVD-stabilized CG method outperforms Tikhonov regularization in noisy data scenarios.
- Spectral discretization and modal truncation improve the stability and accuracy of the inversion.
- The approach effectively captures heterogeneous heat sources influenced by advection.

## Abstract

In order to better understand micromechanical phenomena such as viscoelasticity and plasticity, the thermomechanical viewpoint is of prime importance but requires calorimetric measurements to be performed during a deformation process. Infrared imaging is commonly used to this aim but does not provide direct access to the intrinsic volumetric Thermomechanical Heat Sources (THS). An inversemethod is needed to convert temperature fields in the former quantity. The one proposed here relies on adiffusion-advection heat transfer model. Advection is generally not considered in such problems but due to plastic instabilities, a heterogeneous and non-negligible velocity field can play a role in the local heat transfer balance. Discretization of the governing equation is made through appropriate spectral approach. Spatial regularization is then achieved through regular modal truncation. The objective of the inversion process lies in a proper identification of the decomposition coefficients (states) which minimize the residuals. When a Conjugate Gradient Method (CGM) is applied to this nonlinear least square optimization, the use of Karhunen-Loeve Decomposition (KLD) or Singular Value Decomposition (SVD) on gradient vectors is shown to produce very good temporal regularization. Two test-cases were explored for noisy data which show that this algorithm performs very well when compared to the Tikhonov penalized conjugate gradient method.

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Source: https://tomesphere.com/paper/1704.06454