## 1 Introduction  
The North Atlantic Treaty Organization (NATO) Science and Technology Organization (STO) working group AVT-331 \"Goal-driven, multi-fidelity approaches for military vehicle system-level design\" is currently investigating computing frameworks that could significantly impact the design of next-generation military vehicles [1]. In order to save computational time for uncertainty quantification, optimization, or design space exploration, the use of surrogate models can be a very good strategy. In a surrogate model, expensive function evaluations are replaced by an approximation, which is computationally inexpensive and can thus be evaluated extensively. Especially polynomial chaos expansions (PCEs) [2, 3, 4] and kriging [5, 6, 7, 8] have been widely used in the community. Kriging accuracy can be improved by a dynamic training point selection process rather than picking all training points randomly at the beginning [9]. This is analogous to the concept of expected improvement (EI) which tries to keep a balance between exploration (space filling) and exploitation (resolve interesting areas in the design space).  
PCE [10, 11, 12, 13] and kriging [14, 15, 16, 17, 18] models support the usage of both high- and lower fidelity training points. A standard multi-fidelity strategy is to combine fittings of the high-fidelity (HF) data (e.g., better models, experimental data, finer meshes) with trends from cheaper, low-fidelity (LF) data (e.g., less sophisticated models, coarser meshes). Thus, the usually less intensively sampled HF data needs to correct the trends given by the intensively sampled LF data, for instance, via the popular correction-based method [18]. The correction function is known as scaling function, calibration, or bridge function, and can be additive [19, 20], multiplicative [21], or hybrid multiplicative/additive [10, 11, 13].  
Additionally, the compressive sensing (CS) theory [22; 23; 24; 25; 26; 27; 28; 29] has shown great potential to reduce the curse of dimensionality for PCEs depending on the decay rate of the PCE coefficients and thus the underlying sparsity of the solution. CS offers a framework to use fewer training points than basis functions and to still recover a good approximation to the sparse solution [27; 28; 29; 30]. CS also increases the robustness to noise thanks to the introduced sparsity constraint [28; 29]. In summary, CS aims at selecting a small number of basis polynomials with great impact on the model response [31] and the resulting surrogate model is called sparse PCE (SPCE).  
Thus far, these two distinct surrogate modeling approaches (SPCE and kriging) have been applied in various fields more or less standalone. There have been a few attempts to combine kriging and SPCE in a systematic way for single-fidelity applications [32; 33], but none for multi-fidelity. SPCE is well known for capturing the overall trends of the solution, whereas kriging handles the values around training points well [32; 33]. In this work, a proposed MF SPCE-Kriging method aims at combining the advantages of both meta-modeling methods, thus expecting fewer training points to be required for constructing an accurate model in lieu of computationally expensive HF simulations.  
The general goal of this work is to construct very accurate surrogate models at small overall computational cost assuming that the dominant cost factor is obtaining high-fidelity training point information. The strategy to accomplish this is to enhance surrogate modeling techniques by employing lower fidelity information, adaptively selecting training points, as well as compressed sensing and to use a combined SPCE-Kriging meta-modeling method.