Adjointbasierte

Adjointbasierte methods, often translated as adjoint-based methods, are a class of computational techniques used to efficiently compute gradients or sensitivities of a function with respect to a large number of input parameters. This is particularly useful in fields such as optimization, inverse problems, and uncertainty quantification, where understanding how output changes with small variations in input is crucial.

The core idea behind adjoint-based methods is to reformulate the gradient computation in a way that avoids calculating the gradient for each parameter individually, which can be computationally prohibitive for high-dimensional problems. Instead, an adjoint equation is derived, which is typically a system of equations related to the original forward problem. The solution of this adjoint equation, along with a few inexpensive dot products, allows for the simultaneous computation of the gradient with respect to all input parameters.

These methods are widely employed in computational fluid dynamics for shape optimization, in machine learning for training neural networks (where backpropagation is a specific form of adjoint-based method), and in scientific computing for sensitivity analysis of complex models. The efficiency gain is significant, as the cost of computing the gradient using an adjoint-based approach typically scales linearly with the number of parameters, rather than quadratically or worse as in finite difference methods. This makes adjoint-based methods indispensable for tackling large-scale problems.