In lattice QCD computations a substantial amount of work is spent in solving linear systems arising in Wilson's discretization of the Dirac equations. We show first numerical results of the extension of the two-level DD-αAMG method to a true multilevel method based on our parallel MPI-C implementation. Using additional levels pays off, allowing to cut down the core minutes spent on one system solve by a factor of approximately 700 compared to standard Krylov subspace methods and yielding another speed-up of a factor of 1.7 over the two-level approach.

No abstract

The Adaptive Aggregation-based Domain Decomposition Multigrid method [1] is extended for two degenerate flavors of twisted mass fermions. By fine-tuning the parameters we achieve a speed-up of the order of hundred times compared to the conjugate gradient algorithm for the physical value of the pion mass. A thorough analysis of the aggregation parameters is presented, which provides a novel insight into multigrid methods for lattice QCD independently of the fermion discretization.

No abstract

We study the quantification of uncertainty of Convolutional Neural Networks (CNNs) based on gradient metrics. Unlike the classical softmax entropy, such metrics gather information from all layers of the CNN. We show for the EMNIST digits data set that for several such metrics we achieve the same meta classification accuracy -i.e. the task of classifying predictions as correct or incorrect without knowing the actual label -as for entropy thresholding. We apply meta classification to unknown concepts (out-of-distribution samples) -EMNIST/Omniglot letters, CIFAR10 and noise -and demonstrate that meta classification rates for unknown concepts can be increased when using entropy together with several gradient based metrics as input quantities for a meta classifier. Meta classifiers only trained on the uncertainty metrics of known concepts, i.e. EMNIST digits, usually do not perform equally well for all unknown concepts. If we however allow the meta classifier to be trained on uncertainty metrics for some out-ofdistribution samples, meta classification for concepts remote from EMNIST digits (then termed known unknowns) can be improved considerably.

In the semantic segmentation of street scenes the reliability of the prediction and therefore uncertainty measures are of highest interest. We present a method that generates for each input image a hierarchy of nested crops around the image center and presents these, all re-scaled to the same size, to a neural network for semantic segmentation. The resulting softmax outputs are then post processed such that we can investigate mean and variance over all image crops as well as mean and variance of uncertainty heat maps obtained from pixel-wise uncertainty measures, like the entropy, applied to each crop's softmax output. In our tests, we use the publicly available DeepLabv3+ MobilenetV2 network (trained on the Cityscapes dataset) and demonstrate that the incorporation of crops improves the quality of the prediction and that we obtain more reliable uncertainty measures. These are then aggregated over predicted segments for either classifying between IoU = 0 and IoU > 0 (meta classification) or predicting the IoU via linear regression (meta regression). The latter yields reliable performance estimates for segmentation networks, in particular useful in the absence of ground truth. For the task of meta classification we obtain a classification accuracy of 81.93% and an AUROC of 89.89%. For meta regression we obtain an R 2 value of 84.77%. These results yield significant improvements compared to other approaches.

No abstract

The overlap operator is a lattice discretization of the Dirac operator of quantum chromodynamics, the fundamental physical theory of the strong interaction between the quarks. As opposed to other discretizations it preserves the important physical property of chiral symmetry, at the expense of requiring much more effort when solving systems with this operator. We present a preconditioning technique based on another lattice discretization, the Wilson-Dirac operator. The mathematical analysis precisely describes the effect of this preconditioning in the case that the Wilson-Dirac operator is normal. Although this is not exactly the case in realistic settings, we show that current smearing techniques indeed drive the Wilson-Dirac operator towards normality, thus providing a motivation why our preconditioner works well in computational practice. Results of numerical experiments in physically relevant settings show that our preconditioning yields accelerations of up to one order of magnitude.