A sketched finite element method for elliptic models

We consider a sketched implementation of the finite element method for elliptic partial differential equations on high-dimensional models. Motivated by applications in real-time simulation and prediction we propose an algorithm that involves projecting the finite element solution onto a low-dimensional subspace and sketching the reduced equations using randomised sampling. We show that a sampling distribution based on the leverage scores of a tall matrix associated with the discrete Laplacian operator, can achieve nearly optimal performance and a significant speedup. We derive an expression of the complexity of the algorithm in terms of the number of samples that are necessary to meet an error tolerance specification with high probability, and an upper bound for the distance between the sketched and the high-dimensional solutions. Our analysis shows that the projection not only reduces the dimension of the problem but also regularises the reduced system against sketching error. Our numerical simulations suggest speed improvements of two orders of magnitude in exchange for a small loss in the accuracy of the prediction

Surrogate models for laser powder bed fusion digital twins

Accurate and fast modeling of the temperature distribution and phase transitions in laser powder bed fusion is a major milestone in achieving its quality assurance. Commonly referred to as digital twin technology, the goal is to fi nd agile, fast-to-compute but also sufficiently accurate simulators that can replicate the 3D printing process while enhancing the quality of its outcomes. While the nonlinear heat equation in the context of laser powder bed fusion is numerically solved by the finite element method, three time-efficient surrogates are proposed as fast alternatives with different trade-offs between model accuracy, robustness, offline preparation, and online execution time. The fi rst one is the reduced Gaussian process surrogate, which is a data-driven model equipped with a nonlinear dimension reduction scheme. It outperforms in real-time execution online managing to predict temperature profi les almost instantly, though it is comparably less accurate, not robust to random anisotropy, and requires o ine preparation of data generation, nonlinear dimension reduction, and training. The second one is the sketched surrogate with data-driven local projection. It projects the accurate but high-dimensional nite element method solution with a low-dimensional basis formed by subsampled training temperatures and then bypasses the majority of costly computations for the temperature-dependent matrices in the projected model by randomized sketching. It is the most accurate surrogate while lacking robustness, necessitating the same offline preparation, and taking more time compared with the rst surrogate. The third one is the sketched surrogate with online local projection. Its projection bases are generated in the process of simulation by combining previous temperature pro les and locally deployed anisotropic Gaussian functions, while the sketching process utilizes efficient sampling within out replacement based on approximate optimal sampling distributions. Both the projection and the sketching are designed to implement alongside the printing process, which makes this surrogate capable of handling different process parameters without requiring prior computations offline. The third surrogate, therefore, is accurate, robust, and requires no offline preparation, although it entails longer online execution time compared to the other two surrogates. A series of numerical experiments are carried out to present and compare the performance of the three surrogates, which assumes a two-layer printing process with a fixed laser beam trajectory using different printing attributes (laser power and scan speed) and arbitrary thermal conductivity anisotropy. All three surrogates are also principally feasible in other thermal-driven additive manufacturing to obtain better quality assurance with techniques like uncertainty management and closed-loop control