# Improved Electrochemical-Mechanical Parameter Estimation Technique for Lithium-Ion Battery Models
## 2 Materials and Methods
### Optimization Algorithm  
Since the optimization problem involves multiple objectives, the derivative of \(f(\mathbf{\theta})\) with respect to the control variables is not easily obtainable: the presence of several objectives complicates the calculation of a unique and continuous gradient, making gradient-based algorithms unsuitable. In addition, changes in geometry often lead to remeshing, which may introduce discontinuities or non-differentiable behavior in the objective function. As a result, the computed gradients may be inaccurate or undefined. Therefore, a derivative-free algorithm is more appropriate for addressing this type of problem. The drawback is the slower convergence of the optimization process.  
Among the derivative-free algorithms implemented in COMSOL Multiphysics, BOBYQA (Bound Optimization BY Quadratic Approximation) is the only one that uses a quadratic interpolation model to approximate the objective function. Unlike gradient-based methods, BOBYQA constructs and updates a local quadratic model \(Q(\mathbf{\theta})\) using only objective function evaluations. In particular, the quadratic model is built to interpolate the objective function at a set of \(m\) interpolation points \(\{\mathbf{\theta}_{1}\ldots,\mathbf{\theta}_{j},\ldots,\mathbf{\theta}_{m}\}\), with \(m\geq n+1\), as shown in Equation (17), where \(\mathbf{\theta}_{j}\) indicates the \(j\)-th parameter set. The interpolation points are strategically chosen and updated, as stated by [PERSON] in his work [21].  
\[Q(\mathbf{\theta}_{j})=f(\mathbf{\theta}_{j}),\quad\forall j=1,\ldots,m \tag{17}\]  
Then, the quadratic approximation of the objective function has the form presented in Equation (18), where \(c\) is a scalar constant, \(\mathbf{g}\) is the estimated gradient, \(\mathcal{H}\) is an approximated Hessian matrix, and \(\mathbf{\theta}_{k}\) is the evaluation point at the k-th iteration.  
\[Q(\mathbf{\theta})=c+\mathbf{g}(\mathbf{\theta}-\mathbf{\theta}_{k})+\frac{1}{2}(\mathbf{\theta} -\mathbf{\theta}_{k})^{T}\mathcal{H}(\mathbf{\theta}-\mathbf{\theta}_{k}) \tag{18}\]  
To determine the next point to evaluate, BOBYQA minimizes the quadratic model \(Q(\mathbf{\theta})\) within a neighborhood of the current point \(\mathbf{\theta}_{k}\), known as the _trust region_. This area is defined by a radius \(\delta_{k}\), which bounds the maximum allowed step from the current iteration. The trust-region subproblem is expressed in Equation (19).  
\[\min_{\mathbf{\theta}\in\mathcal{D}_{k}}Q(\mathbf{\theta})\quad\mathcal{D}_{k}=\mathbf{{\theta}\in\mathbb{R}^{n},\;\|\mathbf{\theta}-\mathbf{\theta}_{k}\|\leq\delta_{k}\} \tag{19}\]After evaluating the objective function at the new candidate point, the radius \(\delta_{k}\) is updated depending on the agreement between the predicted decrease in \(Q(\theta)\) and the actual decrease in \(f(\theta)\). If the model prediction is sufficiently accurate, \(\delta_{k}\) is expanded to allow broader exploration; otherwise, it is reduced to focus on a more local region, enhancing model reliability but risking the possibility of missing the global minimum. In parallel, the quadratic model itself is updated to maintain interpolation accuracy: whenever a new function evaluation is accepted and one of the existing interpolation points is replaced, the curvature of the model--represented by the Hessian approximation \(\mathcal{H}\)--must also be updated. This update ensures that the new model still satisfies the interpolation conditions at the updated set of points.