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Regularization is the standard mathematical answer: instead of solving the raw, ill-conditioned system, one solves a modified problem that trades a small, controlled amount of bias for a large reduction in the amplification of noise. Tikhonov regularization is the paradigmatic form, replacing the bare least-squares fit with a penalized objective $\hat{x} = \arg\min_{x} \; \lVert Ax - b \rVert_2^2 + \lambda \lVert Lx \rVert_2^2$. where A is the forward projection operator, b the measured data, L a penalty operator (often favoring smoothness), and $\lambda$ > 0 a regularization parameter controlling the trade-off. As $\lambda$ $\to$ 0 the solution approaches the raw, noise-amplifying least-squares fit; as $\lambda$ grows the solution becomes smoother and more…

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