This paper describes an extension of the Minimum Sobolev Norm interpolation scheme to an approximation scheme. A fast implementation of the MSN interpolation method using the methods for Hierarchical Semiseparable (HSS) matrices is described and experimental results are provided. The approximation scheme is introduced along with a numerically stable solver. Several numerical results are provided comparing the interpolation scheme, the approximation scheme and Thin Plate Splines. A method to decompose images into smooth and rough components is presented. A metric that could be used to distinguish edges and textures in the rough component is also introduced. Suitable examples are provided for both the above.

Prediction of financial time series is a great challenge for statistical models. In general, the stock market times series present high volatility due to its sensitivity to economic and political factors. Furthermore, recently, the covid-19 pandemic has caused a drastic change in the stock exchange times series. In this challenging context, several computational techniques have been proposed to improve the performance of predicting such times series. The main goal of this article is to compare the prediction performance of five neural network architectures in predicting the six most traded stocks of the official Brazilian stock exchange B3 from March 2019 to April 2020. We trained the models to predict the closing price of the next day using as inputs its own previous values. We compared the predictive performance of multiple linear regression, Elman, Jordan, radial basis function, and multilayer perceptron architectures based on the root of the mean square error. We trained all models using the training set while hyper-parameters such as the number of input variables and hidden layers were selected using the testing set. Moreover, we used the trimmed average of 100 bootstrap samples as our prediction. Thus, our approach allows us to measure the uncertainty associate with the predicted values. The results showed that for all times series, considered all architectures, except the radial basis function, the networks tunning provide suitable fit, reasonable predictions, and confidence intervals.

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