Browsing by Author "Khoo, Y"

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    Efficient construction of tensor ring representations from sampling
    (2017-11-30) Khoo, Y; Lu, J; Ying, L
    In this note we propose an efficient method to compress a high dimensional function into a tensor ring format, based on alternating least-squares (ALS). Since the function has size exponential in $d$ where $d$ is the number of dimensions, we propose efficient sampling scheme to obtain $O(d)$ important samples in order to learn the tensor ring. Furthermore, we devise an initialization method for ALS that allows fast convergence in practice. Numerical examples show that to approximate a function with similar accuracy, the tensor ring format provided by the proposed method has less parameters than tensor-train format and also better respects the structure of the original function.
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    Solving parametric PDE problems with artificial neural networks
    (2017-11-30) Khoo, Y; Lu, J; Ying, L
    The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of physical quantities derived from PDE can be captured by a few features on the space of random coefficients. Based on such observation, we propose using neural-network, a technique gaining prominence in machine learning tasks, to parameterize the physical quantity of interest as a function of random input coefficients. The simplicity and accuracy of the approach are demonstrated through notable examples of PDEs in engineering and physics.