Optimal Collocation Nodes for Fractional Derivative Operators

Spectral discretizations of fractional derivative operators are examined, where the approximation basis is related to the set of Jacobi polynomials. The pseudo-spectral method is implemented by assuming that the grid, used to represent the function to be differentiated, may not be coincident with the collocation grid. The new option opens the way to the analysis of alternative techniques and the search of optimal distributions of collocation nodes, based on the operator to be approximated. Once the initial representation grid has been chosen, indications on how to recover the collocation grid are provided, with the aim of enlarging the dimension of the approximation space. As a results of this process, performances are improved. Applications to fractional type advection-diffusion equations, and comparisons in terms of accuracy and efficiency are made. As shown in the analysis, special choices of the nodes can also suggest tricks to speed up computations

Systemic risk governance in a dynamical model of a banking system

We consider the problem of governing systemic risk in a banking system model. The banking system model consists in an initial value problem for a system of stochastic differential equations whose dependent variables are the log-monetary reserves of the banks as functions of time. The banking system model considered generalizes previous models studied in [5], [4], [7] and describes an homogeneous population of banks. Two distinct mechanisms are used to model the cooperation among banks and the cooperation between banks and monetary authority. These mechanisms are regulated respectively by the parameters $\alpha$ and $\gamma$. A bank fails when its log-monetary reserves go below an assigned default level. We call systemic risk or systemic event in a bounded time interval the fact that in that time interval at least a given fraction of the banks fails. The probability of systemic risk in a bounded time interval is evaluated using statistical simulation. A method to govern the probability of systemic risk in a bounded time interval is presented. The goal of the governance is to keep the probability of systemic risk in a bounded time interval between two given thresholds. The governance is based on the choice of the log-monetary reserves of a kind of "ideal bank" as a function of time and on the solution of an optimal control problem for the mean field approximation of the banking system model. The solution of the optimal control problem determines the parameters $\alpha$ and $\gamma$ as functions of time, that is defines the rules of the borrowing and lending activity among banks and between banks and monetary authority. Some numerical examples are discussed. The systemic risk governance is tested in absence and in presence of positive and negative shocks acting on the banking system

Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models

Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional ROMs - built, e.g., exclusively through proper orthogonal decomposition (POD) - when applied to nonlinear time-dependent parametrized PDEs. In particular, POD-DL-ROMs can achieve extreme efficiency in the training stage and faster than real-time performances at testing, thanks to a prior dimensionality reduction through POD and a DL-based prediction framework. Nonetheless, they share with conventional ROMs poor performances regarding time extrapolation tasks. This work aims at taking a further step towards the use of DL algorithms for the efficient numerical approximation of parametrized PDEs by introducing the $\mu t$-POD-LSTM-ROM framework. This novel technique extends the POD-DL-ROM framework by adding a two-fold architecture taking advantage of long short-term memory (LSTM) cells, ultimately allowing long-term prediction of complex systems' evolution, with respect to the training window, for unseen input parameter values. Numerical results show that this recurrent architecture enables the extrapolation for time windows up to 15 times larger than the training time domain, and achieves better testing time performances with respect to the already lightning-fast POD-DL-ROMs.Comment: 28 page

A decision-making machine learning approach in Hermite spectral approximations of partial differential equations

The accuracy and effectiveness of Hermite spectral methods for the numerical discretization of partial differential equations on unbounded domains, are strongly affected by the amplitude of the Gaussian weight function employed to describe the approximation space. This is particularly true if the problem is under-resolved, i.e., there are no enough degrees of freedom. The issue becomes even more crucial when the equation under study is time-dependent, forcing in this way the choice of Hermite functions where the corresponding weight depends on time. In order to adapt dynamically the approximation space, it is here proposed an automatic decision-making process that relies on machine learning techniques, such as deep neural networks and support vector machines. The algorithm is numerically tested with success on a simple 1D problem, but the main goal is its exportability in the context of more serious applications.Comment: 22 pages, 4 figure