Raazesh Sainudiin

Thomas Steinke, Raazesh Sainudiin

Multiplication of n-digit integers by long multiplication requires O(n^2) operations and can be time-consuming. In 1970 A. Schoenhage and V. Strassen published an algorithm capable of performing the task with only O(n log(n)) arithmetic operations over the complex field C; naturally, finite-precision approximations to C are used and rounding errors need to be accounted for. Overall, using variable-precision fixed-point numbers, this results in an O(n(log(n))^(2+Epsilon))-time algorithm. However, to make this algorithm more efficient and practical we need to make use of hardware-based floating-point numbers. How do we deal with rounding errors? and how do we determine the limits of the fixed-precision hardware? Our solution is to use interval arithmetic to guarantee the correctness of results and determine the hardware's limits. We examine the feasibility of this approach and are able to report that 75,000-digit base-256 integers can be handled using double-precision containment sets. This clearly demonstrates that our approach has practical potential; however, at this stage, our implementation does not yet compete with commercial ones, but we are able to demonstrate the feasibility of this technique.

Christian von Koch, William Anzén, Max Fischer et al.

We investigate the ability to detect slag formations in images from inside a Grate-Kiln system furnace with two deep convolutional neural networks. The conditions inside the furnace cause occasional obstructions of the camera view. Our approach suggests dealing with this problem by introducing a convLSTM-layer in the deep convolutional neural network. The results show that it is possible to achieve sufficient performance to automate the decision of timely countermeasures in the industrial operational setting. Furthermore, the addition of the convLSTM-layer results in fewer outlying predictions and a lower running variance of the fraction of detected slag in the image time series.