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Multilevel Monte Carlo front-tracking for random scalar conservation laws. (English) Zbl 1350.65008

BIT 56, No. 1, 263-292 (2016); correction ibid. 58, No. 1, 247-255 (2018).
Many problems in physics and engineering are modeled by systems of hyperbolic conservation or balance laws. The Cauchy problem for such systems has the form \[ U_t + \sum_{j=1}^d {\partial \over \partial x_j} (F_j(U)) = 0, \;\;x = (x_1, \dots, x_d) \in \mathbb{R}^d, \;\;t >0, \]
\[ U(x,0) = U_0(x), \;\;x \in \mathbb{R}^d. \] Here \(U: \mathbb{R}^d \to \mathbb{R}^d\) is the vector of unknowns and \(F_j: \mathbb{R}^d \to \mathbb{R}^d\) is the flux vector for the \(j\)-th direction with \(m\) being a positive integer. It is supposed that the flux functions are Lipschitz continuous with respect to the state variable, for which there exists a unique random entropy solution. A convergence analysis of the multilevel Monte Carlo front-tracking algorithm is presented. An improved complexity estimate in one space dimension is obtained.
In Section, 2 some preliminary notions from probability theory and functional analysis are introduced.
In Section 3, the concept of random entropy solutions is introduced and the well-posedness of the scalar hyperbolic conservation law, i.e. \(m=1\), with random interval data is developed. In Subsection 3.1, the Cauchy problem for the scalar conservation law by setting \(m=1\) is presented. In Subsection 3.2, the distributional solutions in the class of entropy solutions are presented. In Subsection 3.3, the spatially homogeneous random flux functions are considered. Here, the concept of random data for the scalar conservation law is developed. In Subsection 3.4, the random scalar conservation law is rewritten in the terms of random data, introduced in Definition 3.1. Theorem 3.2 states that there exists a random entropy solution of the scalar conservation law.
In Section 4, the multilevel Monte Carlo front-tracking algorithm for numerical solution of hyperbolic conservation law is developed. In Subsection 4.1, the discretization of the scalar conservation law is given. In Theorem 4.1, the convergence of the Monte Carlo estimates \(\mathbb{E}_M[U(\cdot,t)]\) in \(L^2(\Omega;L^1(\mathbb{R}^d))\) to \({\mathcal M}^1(U(\cdot,t))\) is presented. In Subsection 4.2, the concept of the front-tracking in the one-dimensional case is described. Algorithm 1, which is a modification of Graham’s scan, gives the scheme how to calculate all solutions of the Riemann problems. In Subsection 4.2.2, the front-tracking in dimension \(d \geq 2\) is considered. The approximate solutions of the scalar conservation law \(U^{\eta}(x,t)\) is given. In Theorem 4.3 many properties of the quantity \(U^{\eta}(x,t)\) are presented. In Theorem 4.4 the convergence results for the approximation of the random scalar conservation law are presented. In Subsection 4.3, the multiresolution decomposition of the random flux on the phase of the solution is developed. In Subsection 4.4, the convergence analysis of the difference \(\mathbb{E}[U(t)] - \mathbb{E}_L^{MLMC}[U^L(t)]\) of the statistical mean \(\mathbb{E}[U(t)]\) and the expectations of increments for each level are developed. The main result is presented in Theorem 4.5, where an upper bound of the \(L^2\)-norm \(|| \mathbb{E}[U(t)] - \mathbb{E}_L^{MLMC}[U^L(t)] ||^2_{L^2(\Omega;L^1(\mathbb{R}^d))}\) is shown.
In Section 5, a performance of the multilevel Monte Carlo method is tested on several examples with random fluxes in one and two space dimensions.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
35L65 Hyperbolic conservation laws
65C05 Monte Carlo methods
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
65Y20 Complexity and performance of numerical algorithms
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
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References:

[1] Bagrinovskiĭ, K.A., Godunov, S. K.: Difference schemes for multidimensional problems. Dokl. Akad. Nauk SSSR (N.S.) 115, 431-433 (1957) · Zbl 0087.12201
[2] Bressan, A., LeFloch, P.: Uniqueness of weak solutions to systems of conservation laws. Arch. Ration. Mech. Anal. 140(4), 301-317 (1997) · Zbl 0903.35039 · doi:10.1007/s002050050068
[3] Bressan, A., Liu, T.-P., Yang, T.: \[L^1\] L1 stability estimates for \[n\times n\] n×n conservation laws. Arch. Ration. Mech. Anal. 149(1), 1-22 (1999) · Zbl 0938.35093 · doi:10.1007/s002050050165
[4] Dafermos, C.M.: Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38, 33-41 (1972) · Zbl 0233.35014 · doi:10.1016/0022-247X(72)90114-X
[5] Dafermos, C.M.: Hyperbolic conservation laws in continuum physics. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, 3rd edn. Springer, Berlin (2010) · Zbl 1196.35001
[6] E, Weinan, Khanin, K., Mazel, A., Sinai, Y.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math. (2) 151(3), 877-960 (2000) · Zbl 0972.35196
[7] Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1992) · Zbl 0804.28001
[8] Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Analysis, vol. VII, pp. 713-1020. North-Holland, Amsterdam (2000) · Zbl 0981.65095
[9] Giles, M.: Improved multilevel Monte Carlo convergence using the Milstein scheme. In: Monte Carlo and quasi-Monte Carlo methods 2006, pp. 343-358. Springer, Berlin (2008) · Zbl 1141.65321
[10] Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607-617 (2008) · Zbl 1167.65316 · doi:10.1287/opre.1070.0496
[11] Giusti, E.: Minimal surfaces and functions of bounded variation. In: Monographs in Mathematics, vol. 80. Birkhäuser, Basel (1984) · Zbl 0545.49018
[12] Godlewski, E., Raviart, P.-A.: Hyperbolic systems of conservation laws. In: Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 3/4. Ellipses, Paris (1991) · Zbl 0768.35059
[13] Godlewski, E., Raviart, P.-A.: Numerical approximation of hyperbolic systems of conservation laws. In: Applied Mathematical Sciences, vol. 118. Springer, New York (1996) · Zbl 0860.65075
[14] Graham, R.L., Yao, F.F.: Finding the convex hull of a simple polygon. J. Algorithms 4(4), 324-331 (1983) · Zbl 0532.68072 · doi:10.1016/0196-6774(83)90013-5
[15] Heinrich, S.: Multilevel Monte Carlo methods. In: Large-Scale Scientific Computing, pp. 58-67. Springer, Berlin (2001) · Zbl 1031.65005
[16] Holden, H., Holden, L.: On scalar conservation laws in one dimension. In: Ideas and Methods in Mathematical Analysis. Stochastics, and Applications (Oslo, 1988), pp. 480-509. Cambridge Univ. Press, Cambridge (1992) · Zbl 0851.65064
[17] Holden, H., Holden, L., Høegh-Krohn, R.: A numerical method for first order nonlinear scalar conservation laws in one dimension. Comput. Math. Appl., 15(6-8), 595-602 (1988) · Zbl 0658.65085
[18] Holden, H., Karlsen, K.H., Lie, K.-A., Risebro, N.H.: Splitting methods for partial differential equations with rough solutions. In: EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2010) · Zbl 1191.35005
[19] Holden, H., Lindstrøm, T., Øksendal, B., Ubøe, J., Zhang, T.-S.: The Burgers equation with a noisy force and the stochastic heat equation. Commun. Partial Differ. Equ. 19(1-2), 119-141 (1994) · Zbl 0804.35158
[20] Holden, H., Risebro, N.H.: Conservation laws with a random source. Appl. Math. Optim. 36(2), 229-241 (1997) · Zbl 0885.35069 · doi:10.1007/BF02683344
[21] Holden, H., Risebro, N.H.: Front tracking for hyperbolic conservation laws. In: Applied Mathematical Sciences, vol. 152. Springer, New York (2011). (First softcover corrected printing of the 2002 original) · Zbl 1006.35002
[22] Kružkov, S.N.: First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81(123), 228-255 (1970) · Zbl 0202.11203
[23] LeVeque, R.J.: Finite volume methods for hyperbolic problems. In: Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002) · Zbl 1010.65040
[24] Mishra, S.. Risebro, N.H., Schwab, C., Tokareva, S.: Numerical solution of scalar conservation laws with random flux functions. Technical Report 2012-35. Seminar for Applied Mathematics, ETH Zürich, Switzerland (2012) · Zbl 1343.65007
[25] Mishra, S., Schwab, C.: Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data. Math. Comput. 81(280), 1979-2018 (2012) · Zbl 1271.65018 · doi:10.1090/S0025-5718-2012-02574-9
[26] Mishra, S., Schwab, C., Šukys, J.: Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions. J. Comput. Phys. 231(8), 3365-3388 (2012) · Zbl 1402.76083 · doi:10.1016/j.jcp.2012.01.011
[27] Mishra, S., Schwab, C., Šukys, J.: Multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws. In: Uncertainty Quantification in Computational Fluid Dynamics, pp. 225-294. Springer, New York (2013) · Zbl 1276.76066
[28] Perthame, B.: Kinetic formulation of conservation laws. In: Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, New York (2002) · Zbl 1030.35002
[29] van Neerven, J.: Stochastic evolution equations. In: Lecture Notes, ISEM (2007/2008) · Zbl 1132.34062
[30] Wehr, J., Xin, J.: White noise perturbation of the viscous shock fronts of the Burgers equation. Commun. Math. Phys. 181(1), 183-203 (1996) · Zbl 0858.60059 · doi:10.1007/BF02101677
[31] Wehr, J., Xin, J.: Front speed in the Burgers equation with a random flux. J. Stat. Phys. 88(3-4), 843-871 (1997) · Zbl 0920.35141 · doi:10.1023/B:JOSS.0000015175.70862.77
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