Parameter Estimation and Inverse Problems

Parameter estimation and inverse problems represent fundamental topics in engineering and scientific computing, involving the inference of unknown model parameters or system characteristics from observed data. These problems find extensive applications in fields such as signal processing, geophysics, and medical imaging.

Parameter estimation typically refers to determining parameter values that best fit observed data through optimization algorithms when the model structure is known. The least squares method serves as the most classical estimation approach, solving problems by minimizing discrepancies between predicted values and actual observations. For linear models, analytical solutions can be computed efficiently using matrix operations. Nonlinear models require iterative optimization techniques such as gradient descent or the Levenberg-Marquardt algorithm, which can be implemented using MATLAB's fminunc or lsqnonlin functions with proper Jacobian matrix specifications.

Inverse problems present greater challenges, aiming to reconstruct original information from indirect observations. These problems are often ill-posed, meaning solutions may lack uniqueness or stability. Regularization techniques address this issue effectively. Tikhonov regularization remains the most commonly used method, incorporating parameter norm penalties into objective functions to yield more stable solutions through trade-off parameters like lambda. Alternative approaches employ Bayesian frameworks that integrate prior information into estimation processes using probability distributions, which can be implemented through Markov Chain Monte Carlo (MCMC) sampling techniques.

When implementing these algorithms in MATLAB, engineers can leverage built-in optimization toolbox functions and efficient matrix computation capabilities. For linear inverse problems, the backslash operator (\) provides efficient solutions to least squares problems via QR decomposition. More complex scenarios may require custom objective function definitions combined with optimization solvers like fmincon for constrained problems or implementing regularization terms directly in the cost function formulation for numerical computation stability.