Procrustes Analysis of Truncated Least Squares Multidimensional Scaling

Multidimensional Scaling (MDS) is an important class of techniques for embedding sets of patterns in Euclidean space. Most often it is used to visualize in mathbbR3 multidimensional data sets or data sets given by dissimilarity measures that are not distance metrics. Unfortunately, embedding n patterns with MDS involves processing O(n2) pairwise pattern dissimilarities, making MDS computationally demanding for large data sets. Especially in Least Squares MDS (LS-MDS) methods, that proceed by finding a minimum of a multimodal stress function, computational cost is a limiting factor. Several works therefore explored approximate MDS techniques that are less computationally expensive. These approximate methods were evaluated in terms of correlation between Euclidean distances in the embedding and the pattern dissimilarities or value of the stress function. We employ Procrustes Analysis to directly quantify differences between embeddings constructed with an approximate LS-MDS method and embeddings constructed with exact LS-MDS. We then compare our findings to the results of classical analysis, i.e. that based on stress value and correlation between Euclidean distances and pattern dissimilarities. Our results demonstrate that small changes in stress value or correlation coefficient can translate to large differences between embeddings. The differences can be attributed not only to the inevitable variability resulting from the multimodality of the stress function but also to the approximation errors. These results show that approximation may have larger impact on MDS than what was thus far revealed by analyses of stress value and correlation between Euclidean distances and pattern dissimilarities

Deep Neural Network-driven hp-adaptive Finite Element Method in three dimensions

Numerical solutions of Partial Differential Equations with Finite Element Method have multiple applications in science and engineering. Several challenging problems require special stabilization methods to deliver accurate results of the numerical simulations. The advection-dominated diffusion problem is an example of such problems. They are employed to model pollution propagation in the atmosphere. Unstable numerical methods generate unphysical oscillations, and they make no physical sense. Obtaining accurate and stable numerical simulations is difficult, and the method of stabilization depends on the parameters of the partial differential equations. They require a deep knowledge of an expert in the field of numerical analysis. We propose a method to construct and train an artificial expert in stabilizing numerical simulations based on partial differential equations. We create a neural network-driven artificial intelligence that makes decisions about the method of stabilizing computer simulations. It will automatically stabilize difficult numerical simulations in a linear computational cost by generating the optimal test functions. These test functions can be utilized for building an unconditionally stable system of linear equations. The optimal test functions proposed by artificial intelligence will not depend on the right-hand side, and thus they may be utilized in a large class of PDE-based simulations with different forcing and boundary conditions. We test our method on the model one-dimensional advection-dominated diffusion problem