## 1 Introduction: Problem Setting  
It is well known that environmental security is a very important topic for modern society. The development of reliable and sustainable mathematical models has a significant role at this area. Identifying the most influential factors, such as chemical reaction rates, boundary conditions, and emission levels, through sensitivity analysis techniques provides invaluable insights for enhancing the model's performance. This process not only aids in pinpointing areas requiring more detailed scrutiny but also guides improvements in the overall model structure and parametrization. By understanding which factors have the greatest impact on model outputs, researchers can prioritize efforts to refine these critical components, leading to more accurate and reliable predictions. This will lead to increased reliability and robustness of predictions obtained from large-scale environmental and climate models. The input data for sensitivity analyses were obtained during runs of a large-scale mathematical model for the remote transport of air pollutants. The **Un**ified **D**anish **E**u**elian **M**odel (UNI-DEM) was developed at the Danish National Environmental Research Institute ([[http://www2.dmu.dk/AtmosphericEnvironment/DEM/](http://www2.dmu.dk/AtmosphericEnvironment/DEM/)]([http://www2.dmu.dk/AtmosphericEnvironment/DEM/](http://www2.dmu.dk/AtmosphericEnvironment/DEM/)), 1 April 2015) [1]. A _mathematical model_ depicts an initial problem, frequently expressed through partial differential equations (PDEs) or analogous mathematical structures. After establishment, this model undergoes discretization via methods like finite elements, finite differences, or boundary elements. Significantly, post-discretization, some extensive mathematical models can comprise billions or even trillions of algebraic equations. Resolving these models can require multiple days, even with the use of high-performance computing systems. Hence, the creation of efficient algorithms becomes essential. Comprehensive models depict both critical and less significant phenomena concerning the primary output of the model. Despite equal treatment, it can transpire that addressing a minor phenomenon demands substantial computational resources yet contributes marginally to the solution. An illustration of this situation is provided in this study. Moreover, the model's output might exhibit high sensitivity to a singular measurable input parameter. Possessing prior knowledge regarding the importance of certain parameters can prove immensely advantageous. For example, understanding that a particular parameter substantially influences the model's result implies that it should be measured with increased accuracy. This might necessitate resource allocation for acquiring more precise measuring devices, consequently elevating the overall quality and dependability of the model's forecasts. Let us assume the mathematical model at hand can be described using a function  
\[\mathrm{u}=f(\mathrm{x}),\quad\mathrm{where}\quad\mathrm{x}=(x_{1},x_{2},\ldots,x_{d})\in U^{d}\equiv[0;1]^{d} \tag{1}\]  
which is the vector of input parameters with a joint probability density function (PDF) \(p(\mathrm{x})=p(x_{1},\ldots,x_{d})  
crucial for ensuring the existence of a unique solution to the system. When dealing with large-scale problems involving multiple parameters in differential equations, demonstrating the existence of a unique solution becomes more difficult. In such cases, it is common to assume the existence of a unique solution and proceed by discretizing the problem using a suitable numerical method. Global sensitivity analysis (GSA) offers significant benefits for both model developers and users, enabling them to quantify uncertainties in model outputs, assess the relative contributions of various input factors to these uncertainties, and prioritize efforts for their reduction. There are several critical stages involved in conducting a reliable and efficient sensitivity analysis.