This thesis investigates the potential of solving large-scale linear systems of equations (LSEs) in the Tensor Train (TT) format. First, we study when this approach can outperform traditional solvers like Conjugate Gradient (CG) and Gaussian Elimination (GE). In turn, we examine the time complexity of the TT-solve technique when applied to ill-conditioned, sparse, and symmetric positive definite (SPD) matrices. This enables us to assess the scenarios where TT-solve may offer significant advantages.
During this exploration, we have also identified and proposed solutions for a handful of research gaps that could further optimize the TT-solve process. These include challenges related to poorly factorizing matrix sizes, as well as the need for a deeper understanding of the trade-offs between TT-matrix (TTM) rank and maximal mode size during the decomposition process. Additionally, the study examined the potential for altering the coefficient matrix through techniques like variable reordering and partial Gaussian elimination (PG) to achieve lower TTM-ranks without affecting the solution of the LSE.

Fluvial flooding poses a major threat to mankind and annually leads to major economic losses with many casualties worldwide. The consequences of this can be mitigated when accurate and rapid predictions are available when the water will arrive at which location. Current numerical simulations take a significant amount of time due to their computational demand, which is not affordable during a calamity where each second can make the difference between life and death. Literature has shown promising results in fluvial flood forecasting by using a deep learning model, which generally takes only a fraction of the computation time compared to conventional models. However, these models have thus far only been trained on static landscapes with changing boundary conditions, which prevents the model’s application elsewhere.

This research explores therefore the possibilities of creating and training a generic non-location bound deep learning model which can predict the spatial distribution of fluvial flood arrival times per grid cell. The architecture of the created deep learning model consists of five parallel encoder-decoders, which takes the elevation, slopes, elevated elements, land roughness, and initial water levels into consideration, depending on the dataset. The model is trained, validated and tested on four unique datasets, which consists of 30,000 flooding samples. The degree of complexity within a sample increases with each dataset number.

The average error for the four consecutive test datasets were 0.91, 1.41, 1.25, and 1.76 hours per cell. The differences in the predicted and the groundtruth are relatively small although the deviation tends to become larger at the end of the simulation, in regions with a strong gradient in arrival time, and in hilly and complex landscapes.

In addition, the model has been tested on various benchmark landscapes to examine specific flow phenomena, as well as a realistic test scenario in the Dutch dike ring 48. The model shows satisfactory performances for landscapes other than those present in the dataset, untill it encounters a feature on which the model was not trained for, such as undershots or irregularly shaped waterways.

This research has shown the potential of deep learning in predicting fluvial flood arrival times on unseen before landscapes. Recommendations for further studies include the use of an active dike breach and a variable flood location.