## 4 Inverse Problems
### Fluid Mechanics and FSI
#### 4.2.2 Optimization Algorithms  
The development of algorithms for numerical optimization is a broad and active field of research. It is not the aim of this article to carry out a comprehensive review of all the available optimization techniques, but rather to list the methods most commonly used in the field of biomechanics, providing a rationale for their selection with specific problems. In the following sections, we loosely follow the classification proposed by [PERSON] and [PERSON] based on the characteristics of the target function [123]. Given the nature and complexity of inverse biomechanics problems, we only consider optimization algorithms that deal with continuous variables and multiple optimization parameters. Updating by Differentiation of Target Function  
In those cases where the target function is continuous and derivable, derivative information can be used to estimate the descent path towards the minimum. First and second-order algorithms refer to optimization methods that incorporate numerical evaluations of the local Jacobian and the Hessian matrix, respectively. To reduce the risk of convergence to local minima, stochastic sampling of these derivatives is incorporated. First-order algorithms can only deal with relatively simple problems and are not suited for inverse biomechanics. However, they are used in other relevant applications, such as the automation of image processing and segmentation for the generation of geometric models [11]. Second-order algorithms have been used in the solution of inverse arterial and myocardial mechanics for the estimation of anisotropic material constants. The most commonly used are the Levenberg-Marquardt [34; 107], Broyden-Fletcher-Goldfarb-Shanno (BFGS), limited BFGS (L-BFGS), and sequential quadratic programming [5]. Updating with No Differentiation of Target Function  
Some optimization algorithms do not require derivative information of the target function to operate. These methods are not as fast as gradient-based counterparts when applied to derivable functions. However, they are advantageous in cases where the functions are not derivable, there are regions with invalid solutions or singularities, the function response is noisy, or the target presents multiple local minima. Since the target functions on simulation-based inverse problems are not analytical functions, but instead, are the simulation outputs, it is prone to some numerical problems, e.g., non-valid solutions due to forward problem divergence. In consequence, most recent inverse method developments have incorporated gradient-free algorithms. The most common algorithms can be classified into two groups direct methods and population methods. Direct methods incorporate deterministic algorithms based on patterns or geometrical constructs for sampling the domain and carry a direct comparison of the target function value. This comparison is then used to define the location of the next sampling point. Powell and Nelder-Mead algorithms have been particularly popular in biomechanics applications and inverse analyses [124; 125]. The main feature of population methods, in contrast to direct methods, is that the initial seed is not a single point in the parameter hyperspace but a pool of candidate optimum solutions. On each iteration, a new pool of candidate solutions is generated by altering the input parameter values following different recombination rules from parent candidates and stochastic variations. Then, each new candidate is evaluated and a new pool is selected to build the next generation. These algorithms have proven to be particularly useful when dealing with noisy target functions, and with multiple local minima. The popular genetic algorithms and particle swarm methods stand out due to their multiple applications, including the solution of inverse problems [117; 126; 127; 128]. These methods require intensive sampling, so they are contraindicated for the solution of inverse problems when the forward problem is computationally expensive [15].