We consider a nonlinear inverse problem where the unknown is assumed to be piecewise affine, which is motivated by an application in DNA replication analysis. Since traditional algorithmic and theoretical tools from linear inverse problems do not apply, we propose a novel formalism and computational approach to harness it. In the noiseless case, we establish sufficient identifiability conditions, and prove that the solution is the unique minimizer of a non-convex optimization problem. The latter is specially challenging because of its multiple local minima. We propose an optimization algorithm that provably finds the global solution in the noiseless case and is shown to be numerically effective for noisy signals. When instantiated in a DNA replication analysis scenario, where the unknown is a so-called timing profile, the approach is shown to be more computationally effective than the state-of-the-art optimization methods by at least 30 orders of magnitude. Besides, it automatically recovers the full configuration of the DNA replication dynamics, which is crucial for DNA replication analysis and was not possible with previous methods.
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