Most of science runs a forward problem: given the cause, predict the effect. Given the density of the Earth, compute its gravity; given an image, compute its blur. An inverse problem runs it backwards — measure the effect and infer the cause. From X-ray shadows, reconstruct the body (CT scanning); from a blurry photo, recover the sharp one; from seismic echoes, map the rock below.

Written abstractly, a forward model d = G(m) turns a model m into data d; the inverse problem is to recover m from noisy d. The catch that makes it a whole subject: this is almost always ill-posed — tiny noise in the data can blow up into huge errors in the model.

The shape of the journey

It leans on linear algebra — the forward map is a matrix, its SVD diagnoses the trouble — and on statistics, since data is always noisy. Six stages:

• Framing. Forward vs inverse, the linear problem d = Gm, and why it is hard.
• Least squares & the generalized inverse. Best fits, the pseudoinverse, and the SVD picture.
• Ill-posedness & conditioning. The condition number, noise amplification, the Picard condition.
• Regularization. Tikhonov, truncated SVD, choosing the parameter, smoothing, resolution.
• The Bayesian view. Maximum likelihood, priors as regularization, and quantified uncertainty.
• Nonlinear & applications. Linearization, deconvolution, and computed tomography.

The path

Framing the problem.

1. Forward and Inverse Problems
2. The Linear Inverse Problem
3. Why Inverse Problems Are Hard

Least squares & the generalized inverse.

1. The Least-Squares Solution
2. The Generalized Inverse
3. SVD and the Pseudoinverse

Ill-posedness & conditioning.

1. Conditioning
2. Ill-Posedness and Noise
3. The Picard Condition

Regularization.

1. Tikhonov Regularization
2. Truncated SVD
3. Choosing α: the L-Curve
4. Smoothing and General Tikhonov
5. Resolution

The statistical / Bayesian view.

1. The Statistical View
2. Maximum Likelihood = Least Squares
3. The Bayesian Formulation
4. Priors Are Regularization
5. Quantifying Uncertainty

Nonlinear problems & applications.

1. Nonlinear Inverse Problems
2. Deconvolution
3. Computed Tomography

Begin → Forward and Inverse Problems