Adjoint-Based Joint Reconstruction of Heat Source and Initial Condition with Uncertainty Quantification

This paper develops an adjoint-based framework for jointly reconstructing a space–time-dependent internal heat source and an unknown initial temperature field from sparse, noisy measurements. A single adjoint solve supplies the gradients with respect to both fields, so each iteration requires one forward and one adjoint solve. Deterministically, CGLS with discrepancy-principle stopping reaches a comparable regularized solution in about an order of magnitude fewer iterations than Landweber–Fridman. In the Bayesian formulation, Gaussian noise and Matérn priors yield an exact Gaussian posterior; prior-preconditioned conjugate gradients compute the maximum a posteriori estimate, while a low-rank approximation of the prior-preconditioned data-misfit Hessian provides pointwise credible bands. The exact discrete adjoint gives machine-precision gradients, and prior-predictive experiments verify nominal pointwise coverage. Numerical experiments compare the reconstructions and assess sensitivity to noise, discretization, and prior hyperparameters. The Bayesian reconstruction is more accurate and mesh-robust in the reported tests. A calibrated generalized-${\chi }^2$ discrepancy diagnostic detects misspecification caused by an omitted initial-temperature offset and, less strongly, by discontinuous sources outside the prior model. These experiments demonstrate joint reconstruction and scalable uncertainty quantification using only forward and adjoint heat-equation solves.