# Multilevel Monte Carlo Finite Volume Methods for Shallow Water Equations with Uncertain Topography in Multi-dimensions

@article{Mishra2012MultilevelMC, title={Multilevel Monte Carlo Finite Volume Methods for Shallow Water Equations with Uncertain Topography in Multi-dimensions}, author={Siddhartha Mishra and Christoph Schwab and Jonas Sukys}, journal={SIAM J. Sci. Comput.}, year={2012}, volume={34} }

The initial data and bottom topography, used as inputs in shallow water models, are prone to uncertainty due to measurement errors. We model this uncertainty statistically in terms of random shallow water equations. We extend the multilevel Monte Carlo (MLMC) algorithm to numerically approximate the random shallow water equations efficiently. The MLMC algorithm is suitably modified to deal with uncertain (and possibly uncorrelated) data on each node of the underlying topography grid by the use…

## 42 Citations

### Uncertainty quantification in tsunami modeling using multi-level Monte Carlo finite volume method

- Computer Science
- 2016

This work proposes a paradigm for UQ that combines the recently developed path-conservative spatial discretizations efficiently implemented on cluster of GPUs, with the Recently developed Multi-Level Monte Carlo (MLMC) statistical sampling method and provides a fast, accurate and computationally efficient framework to compute statistical quantities of interest.

### The multi-level Monte Carlo method for simulations of turbulent flows

- PhysicsMonthly Weather Review
- 2018

It is demonstrated that unlike the pointwise solutions, the averaged volume transports are correlated across grid resolutions, and the MLMC method can increase simulation efficiency without losing accuracy in uncertainty assessments.

### Uncertainty quantification in shallow water-sediment flows: a stochastic Galerkin shallow water hydro-sediment-morphodynamic model

- Engineering, Environmental Science
- 2021

### Numerical simulation of the shallow water equations coupled with a precipitation system driven by random forcing

- Environmental Science
- 2018

Quantification of flood risk and flood inundation requires accurate numerical simulations, both in terms of the mathematical theory that underpins the methods used and the manner in which the…

### Multi-level Monte Carlo finite volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium

- Computer Science, MathematicsJ. Comput. Phys.
- 2016

### Optimal fidelity multi-level Monte Carlo for quantification of uncertainty in simulations of cloud cavitation collapse

- PhysicsArXiv
- 2017

This work identifies large uncertainties in the location and magnitude of the peak pressure pulse and presents its statistical correlations and joint probability density functions with the geometrical characteristics of the cloud.

### Multi-level Monte Carlo Finite Difference and Finite Volume Methods for Stochastic Linear Hyperbolic Systems

- Mathematics, Computer Science
- 2013

Novel probabilistic computational complexity analysis is presented which takes into account the sample path dependent complexity of the underlying FDM/FVM solver, due to the random CFL-restricted time step size on account of the wave speed in a random medium.

## References

SHOWING 1-10 OF 27 REFERENCES

### Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions

- Computer Science, MathematicsJ. Comput. Phys.
- 2012

### Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data

- Mathematics, Computer ScienceMath. Comput.
- 2012

A class of numerical schemes of multi-level Monte Carlo Finite Volume (MLMC-FVM) type is presented for the approximation of random entropy solutions as well as of their k-point correlation functions and statistical moments of discontinuous solutions are found to be more regular than pathwise solutions.

### An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

- Computer ScienceJ. Comput. Phys.
- 2009

### Intrusive projection methods with upwinding for uncertain nonlinear hyperbolic systems

- Mathematics
- 2009

This paper deals with stochastic spectral methods for uncertainty propagation and quantiﬁcation in nonlinear hyperbolic systems of conservation laws. We consider problems with parametric uncertainty…

### Static Load Balancing for Multi-level Monte Carlo Finite Volume Solvers

- Computer SciencePPAM
- 2011

A novel load balancing procedure is used to ensure scalability of the MLMC algorithm on massively parallel hardware and is described in great detail.

### Causes and consequences of error in digital elevation models

- Environmental Science
- 2006

The source and nature of errors in digital models of elevation, and in the derivatives of such models, are reviewed, the correction of errors and assessment of fitness for use are examined, and some priorities for future research are identified.

### Multilevel Monte Carlo Path Simulation

- Computer ScienceOper. Res.
- 2008

We show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations. In…

### Uncertainty quantification for systems of conservation laws

- Computer Science, MathematicsJ. Comput. Phys.
- 2009

### An adaptive Stochastic Finite Elements approach based on Newton–Cotes quadrature in simplex elements

- Mathematics
- 2009

### Multilevel Monte Carlo Methods

- MathematicsLSSC
- 2001

Applications to stochastic solution of integral equations are given for the case where an approximation of the full solution function or a family of functionals of the solution depending on a parameter of a certain dimension is sought.