= Variational regularization
{wiki}
<Variational regularization> reconstructs an unknown by balancing data fit with a penalty expressing desired structure. A standard model is
$$
u_\alpha\in\operatorname*{argmin}_u\left\{\tfrac12\|Au-f^\delta\|^2+\alpha J(u)\right\},\qquad\alpha>0.
$$
The <linear operator> $A$ predicts data, $f^\delta$ denotes observations, and the extended-real <functional> $J$ can impose smoothness, sparsity or hard constraints. This construction includes nonlinear reconstruction maps even when $A$ itself is linear.<Modern Regularization Methods for Inverse Problems by Martin Benning and Martin Burger>[§5, Definition5.1, equation(47), printed pp19–20]{ref}
For $J(u)=\|u\|^2$, the model is quadratic <Tikhonov regularization>. For $J(u)=\operatorname{TV}(u)$, the penalty controls the <total variation seminorm on a domain> and can retain jumps without demanding a square-integrable classical gradient. Adding the <indicator functional of a constraint set> enforces constraints such as nonnegativity.
Existence, uniqueness and stability are separate conclusions. A <direct method in the calculus of variations> requires suitable compactness of bounded-energy sets and <lower semicontinuity>; convexity alone does not supply compactness. A unique minimizer at fixed $\alpha$ also does not prove convergence to the true object as noise vanishes: the <regularization parameter choice> must balance bias and noise.
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