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Sensitivity analysis of fractional linear systems based on random walks with negligible memory usage

Andrés Centeno, Juan A. Acebrón, José Monteiro

Abstract

A random walk-based method is proposed to efficiently compute the solution of a large class of fractional in time linear systems of differential equations (linear F-ODE systems), along with the derivatives with respect to the system parameters. Such a method is unbiased and unconditionally stable, and can therefore be used to provide an unbiased estimation of individual entries of the solution, or the full solution. By using stochastic differentiation techniques, it can be used as well to provide unbiased estimators of the sensitivities of the solution with respect to the problem parameters without any additional computational cost. The time complexity of the algorithm is discussed here, along with suitable variance bounds, which prove in practice the convergence of the algorithm. Finally, several test cases were run to assess the validity of the algorithm.

Sensitivity analysis of fractional linear systems based on random walks with negligible memory usage

Abstract

A random walk-based method is proposed to efficiently compute the solution of a large class of fractional in time linear systems of differential equations (linear F-ODE systems), along with the derivatives with respect to the system parameters. Such a method is unbiased and unconditionally stable, and can therefore be used to provide an unbiased estimation of individual entries of the solution, or the full solution. By using stochastic differentiation techniques, it can be used as well to provide unbiased estimators of the sensitivities of the solution with respect to the problem parameters without any additional computational cost. The time complexity of the algorithm is discussed here, along with suitable variance bounds, which prove in practice the convergence of the algorithm. Finally, several test cases were run to assess the validity of the algorithm.
Paper Structure (20 sections, 3 theorems, 87 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 20 sections, 3 theorems, 87 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Stochastic methods for computing the solution of differential equations
  4. Stochastic methods for the sensitivity analysis of differential equations
  5. Mathematical preliminaries
  6. Description of the stochastic method
  7. Stochastic representation of the solution
  8. Stochastic representation for the derivatives
  9. Matrix and fractional exponent sensitivities
  10. Time sensitivities
  11. Variance bounds
  12. Implementation
  13. Time complexity of the algorithm
  14. Numerical results
  15. Numerical validation
  16. ...and 5 more sections

Key Result

Theorem 1

A stochastic representation of the system of ODEs with $\mathbf{D}=(d_{ij})_{i,j=1,\ldots,n}$ an arbitrary diagonal matrix, is given by

Figures (4)

  • Figure 1: Flowchart for fractional inverse problems. The F-PDE is discretized onto an F-ODE, then a loss is computed and parameters are updated to reduce that loss until it falls bellow a threshold or a maximum number of iterations is reached. Looping through the last step fast enough is critical for the algorithm to converge in a reasonable time.
  • Figure 2: Power law behaviour of average execution time of a random walk with respect to final time $T$ for different values of the $\alpha$ exponent, being $\alpha_i=\alpha, \forall i$.
  • Figure 3: Deterministic and stochastic results for the loss and its sensitivities with respect to the $\alpha$ value and the first entry of the matrix $\mathbf{A}$.
  • Figure :

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 3
  • ...and 3 more