inverseproblem

An inverse problem seeks to determine unknown causes from observed effects. Given a forward model that maps inputs to observable outputs, an inverse problem attempts to recover the inputs that produced measured data. In contrast, a forward problem computes data from known inputs.

Often the inverse problem is ill-posed: existence, uniqueness, or stability may fail. According to Hadamard, a well-posed problem has a solution, a unique solution, and continuous dependence on data. Many inverse problems lack one of these properties, making reconstructions sensitive to noise.

Common approaches include regularization methods (for example Tikhonov regularization) that incorporate prior information to stabilize the solution; discretization and optimization to minimize data misfit plus penalties; Bayesian formulations that treat unknowns probabilistically and yield posterior distributions; and adjoint-state techniques for efficient computation in large-scale problems.

Applications span medical imaging (computed tomography, deconvolution in imaging), geophysics (seismic inversion and subsurface property estimation), remote sensing, spectroscopy, and deblurring of images.

Mathematically, many inverse problems are framed as F(x) = y, where F is a forward operator, y are observed data, and x are the unknowns to be recovered.

Key challenges include noise, incomplete data, model error, and nonlinearity. Consequently, results are often probabilistic or rely on regularization to express uncertainty and incorporate prior knowledge. Inverse problems are foundational across science and engineering, guiding both theory and practical data analysis.