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Density estimation for elliptic PDE with random input by preintegration and quasi-Monte Carlo methods

Alexander D. Gilbert, Frances Y. Kuo, Abirami Srikumar

Abstract

In this paper, we apply quasi-Monte Carlo (QMC) methods with an initial preintegration step to estimate cumulative distribution functions and probability density functions in uncertainty quantification (UQ). The distribution and density functions correspond to a quantity of interest involving the solution to an elliptic partial differential equation (PDE) with a lognormally distributed coefficient and a normally distributed source term. There is extensive previous work on using QMC to compute expected values in UQ, which have proven very successful in tackling a range of different PDE problems. However, the use of QMC for density estimation applied to UQ problems will be explored here for the first time. Density estimation presents a more difficult challenge compared to computing the expected value due to discontinuities present in the integral formulations of both the distribution and density. Our strategy is to use preintegration to eliminate the discontinuity by integrating out a carefully selected random parameter, so that QMC can be used to approximate the remaining integral. First, we establish regularity results for the PDE quantity of interest that are required for smoothing by preintegration to be effective. We then show that an $N$-point lattice rule can be constructed for the integrands corresponding to the distribution and density, such that after preintegration the QMC error is of order $\mathcal{O}(N^{-1+ε})$ for arbitrarily small $ε>0$. This is the same rate achieved for computing the expected value of the quantity of interest. Numerical results are presented to reaffirm our theory.

Density estimation for elliptic PDE with random input by preintegration and quasi-Monte Carlo methods

Abstract

In this paper, we apply quasi-Monte Carlo (QMC) methods with an initial preintegration step to estimate cumulative distribution functions and probability density functions in uncertainty quantification (UQ). The distribution and density functions correspond to a quantity of interest involving the solution to an elliptic partial differential equation (PDE) with a lognormally distributed coefficient and a normally distributed source term. There is extensive previous work on using QMC to compute expected values in UQ, which have proven very successful in tackling a range of different PDE problems. However, the use of QMC for density estimation applied to UQ problems will be explored here for the first time. Density estimation presents a more difficult challenge compared to computing the expected value due to discontinuities present in the integral formulations of both the distribution and density. Our strategy is to use preintegration to eliminate the discontinuity by integrating out a carefully selected random parameter, so that QMC can be used to approximate the remaining integral. First, we establish regularity results for the PDE quantity of interest that are required for smoothing by preintegration to be effective. We then show that an -point lattice rule can be constructed for the integrands corresponding to the distribution and density, such that after preintegration the QMC error is of order for arbitrarily small . This is the same rate achieved for computing the expected value of the quantity of interest. Numerical results are presented to reaffirm our theory.
Paper Structure (16 sections, 8 theorems, 75 equations, 4 figures)

This paper contains 16 sections, 8 theorems, 75 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Smoothing by preintegration
  3. Function spaces
  4. Preintegration theory for density estimation
  5. Quasi-Monte Carlo methods
  6. PDE with random input
  7. Parametric variational form
  8. Parametric regularity
  9. Density estimation for PDE with random input
  10. Conditions required for preintegration theory
  11. Verifying the monotonicity condition
  12. Verifying the smoothness of the quantity of interest
  13. Bounding the derivative terms
  14. Error analysis
  15. Numerical results
  16. ...and 1 more sections

Key Result

Theorem 1

For $s, N \in \mathbb{N}$, let $\psi$ be either an exponential or Gaussian weight function and let $g\in\mathcal{H}^{{\boldsymbol{1}}}_{2s}$ with weight parameters ${\boldsymbol{\gamma}} = (\gamma_{{\boldsymbol{\eta}}})_{{\boldsymbol{\eta}}\leq{\boldsymbol{1}}}$. Then a randomly shifted lattice rule where $\phi_\mathrm{tot}$ is the Euler totient function and $\varrho$ also depends on $\psi$. For a

Figures (4)

  • Figure 1: Convergence plots comparing MC and QMC performance (with and without preintegration) for computing the cdf (left) and pdf (right) evaluated at $t=-0.02$ with $\alpha=1$, $\theta=2$ and $16$ random shifts. The numbers in parentheses indicate the empirical rate of convergence.
  • Figure 2: Plots of the computed cdf (left) and pdf (right) for the problem with $\alpha=1$, $\theta=2$ and $N=503$ lattice points. Histograms of 32000 randomly sampled solutions are overlaid.
  • Figure 3: Convergence plots comparing QMC with preintegration performance for computing the cdf at $t=-0.02$ for fixed $\alpha=1$ and varying theta (left) and fixed $\theta=2$ and varying alpha (right) with $16$ random shifts. The magenta lines in both plots represent the same data ($\theta = 2$ and $\alpha=1$).
  • Figure 4: Plots of the cdf (left) and pdf (right) for the problem with $\alpha=30$, $\theta=2$ and $N=503$ lattice points. Histograms of 32000 randomly sampled solutions are overlaid.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Remark 4
  • Lemma 5
  • Corollary 6
  • Remark 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • ...and 1 more