“Grid-free Monte Carlo for PDEs with spatially varying coefficients” by Sawhney, Seyb, Jarosz and Crane

  • ©Rohan Sawhney, Dario Seyb, Wojciech Jarosz, and Keenan Crane




    Grid-free Monte Carlo for PDEs with spatially varying coefficients



    Partial differential equations (PDEs) with spatially varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in nature, since they must first discretize the problem—leading to spatial aliasing, and global meshing/sampling that is costly and error-prone. We describe a method that approximates neither the domain geometry, the problem data, nor the solution space, providing the exact solution (in expectation) even for problems with extremely detailed geometry and intricate coefficients. Our main contribution is to extend the walk on spheres (WoS) algorithm from constant- to variable-coefficient problems, by drawing on techniques from volumetric rendering. In particular, an approach inspired by null-scattering yields unbiased Monte Carlo estimators for a large class of 2nd order elliptic PDEs, which share many attractive features with Monte Carlo rendering: no meshing, trivial parallelism, and the ability to evaluate the solution at any point without solving a global system of equations.


    1. Assyr Abdulle, E Weinan, Björn Engquist, and Eric Vanden-Eijnden. 2012. The heterogeneous multiscale method. Acta Numerica 21 (2012), 1–87.Google ScholarCross Ref
    2. William Abikoff. 1981. The Uniformization Theorem. Amer. Math. Monthly 88, 8 (1981).Google Scholar
    3. Frédéric Alauzet and Adrien Loseille. 2016. A decade of progress on anisotropic mesh adaptation for computational fluid dynamics. Computer-Aided Design 72 (2016).Google Scholar
    4. Robert Anderson, Julian Andrej, Andrew Barker, et al. 2021. MFEM: A modular finite element methods library. Computers & Mathematics with Applications 81 (2021).Google Scholar
    5. Adam Arbree, Bruce Walter, and Kavita Bala. 2011. Heterogeneous Subsurface Scattering Using the Finite Element Method. IEEE TVCG 17, 7 (July 2011), 956–969.Google Scholar
    6. Ivo Babuška and Manil Suri. 1992. On locking and robustness in the finite element method. SIAM J. Numer. Anal. 29, 5 (1992), 1261–1293.Google ScholarDigital Library
    7. Victor Bayona, Natasha Flyer, and Bengt Fornberg. 2019. On the role of polynomials in RBF-FD approximations. J. Comput. Phys. 380 (2019), 378–399.Google ScholarDigital Library
    8. Victor Bayona, Natasha Flyer, Bengt Fornberg, and Gregory A Barnett. 2017. On the role of polynomials in RBF-FD approximations. J. Comp. Phys. 332 (2017).Google Scholar
    9. Ilia Binder and Mark Braverman. 2012. The rate of convergence of the walk on spheres algorithm. Geometric and Functional Analysis 22, 3 (2012), 558–587.Google ScholarCross Ref
    10. Fischer Black and Myron Scholes. 1973. The Pricing of Options and Corporate Liabilities. J. Pol. Econ. 81, 3 (1973), 637–654. http://www.jstor.org/stable/1831029Google ScholarCross Ref
    11. Thomas E Booth. 1985. Monte Carlo variance comparison for expected-value versus sampled splitting. Nucl. Sci. Eng. 89, 4 (1985), 305–309.Google ScholarCross Ref
    12. Thomas E Booth and John S Hendricks. 1984. Importance estimation in forward Monte Carlo calculations. Nuclear Technology-Fusion 5, 1 (1984), 90–100.Google ScholarCross Ref
    13. John C. Bowers, Jonathan Leahey, and Rui Wang. 2011. A Ray Tracing Approach to Diffusion Curves. Proc. EGSR 30, 4 (2011), 1345–1352.Google Scholar
    14. Jeremiah U Brackbill and Hans M Ruppel. 1986. FLIP: A method for adaptively zoned, particle-in-cell calculations of fluid flows. J. Comp. Phys. 65, 2 (1986), 314–343.Google ScholarDigital Library
    15. John Burgess. 2020. RTX on—the NVIDIA Turing GPU. IEEE Micro 40, 2 (2020), 36–44.Google ScholarCross Ref
    16. DM Causon and CG Mingham. 2010. Introductory finite difference methods for PDEs.Google Scholar
    17. Subrahmanyan Chandrasekhar. 1960. Radiative Transfer. Dover Publications, NY.Google Scholar
    18. Per H. Christensen and Wojciech Jarosz. 2016. The Path to Path-Traced Movies. Foundations and Trends in Computer Graphics and Vision 10, 2 (Oct. 2016), 103–175.Google ScholarDigital Library
    19. Michael F. Cohen and John R. Wallace. 1993. Radiosity and Realistic Image Synthesis.Google Scholar
    20. CJ Coleman, DL Tullock, and N Phan-Thien. 1991. An effective boundary element method for inhomogeneous PDEs. J. App. Math. Phys. (ZAMP) 42, 5 (1991).Google Scholar
    21. W. A. Coleman. 1968. Mathematical Verification of a Certain Monte Carlo Sampling Technique and Applications of the Technique to Radiation Transport Problems. Nucl. Sci. Eng. 32, 1 (April 1968), 76–81.Google ScholarCross Ref
    22. Martin Costabel. 1987. Principles of boundary element methods. Computer Physics Reports 6, 1–6 (1987), 243–274.Google ScholarCross Ref
    23. John C. Cox, Jonathan E. Ingersoll, and Stephen A. Ross. 1985. A Theory of the Term Structure of Interest Rates. Econometrica 53, 2 (1985), 385–407.Google ScholarCross Ref
    24. S. N. Cramer. 1978. Application of the Fictitious Scattering Radiation Transport Model for Deep-Penetration Monte Carlo Calculations. Nucl. Sci. Eng. 65, 2 (1978).Google Scholar
    25. Madalina Deaconu and Antoine Lejay. 2006. A random walk on rectangles algorithm. Methodology and Computing in Applied Probability 8, 1 (2006), 135–151.Google ScholarDigital Library
    26. Eugene d’Eon and Geoffrey Irving. 2011. A Quantized-Diffusion Model for Rendering Translucent Materials. Proc. SIGGRAPH 30, 4 (July 2011), 56:1–56:14.Google Scholar
    27. Mathieu Desbrun, Roger D Donaldson, and Houman Owhadi. 2013. Modeling Across Scales: Discrete Geometric Structures in Homogenization and Inverse Homogenization. Multiscale Analysis and Nonlinear Dynamics (2013), 19–64.Google Scholar
    28. Craig Donner, Tim Weyrich, Eugene d’Eon, Ravi Ramamoorthi, and Szymon Rusinkiewicz. 2008. A Layered, Heterogeneous Reflectance Model for Acquiring and Rendering Human Skin. Proc. SIGGRAPH Asia 27, 5 (2008), 140:1–140:12.Google ScholarDigital Library
    29. Dean G Duffy. 2015. Green’s functions with applications. Chapman and Hall/CRC.Google Scholar
    30. Louis J Durlofsky. 1991. Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water resources research 27, 5 (1991).Google Scholar
    31. Bruce B Dykaar and Peter K Kitanidis. 1992. Determination of the Effective Hydraulic Conductivity for Heterogeneous Porous Media. Water Res. R. 28, 4 (1992).Google Scholar
    32. Yalchin Efendiev and Thomas Y Hou. 2009. Multiscale finite element methods: theory and applications. Vol. 4.Google Scholar
    33. Christer Ericson. 2004. Real-time collision detection. Crc Press.Google Scholar
    34. Lawrence C Evans. 1998. Partial differential equations. Vol. 19. Rhode Island, USA.Google Scholar
    35. Florian Fahrenberger, Zhenli Xu, and Christian Holm. 2014. Simulation of electric double layers around charged colloids in aqueous solution of variable permittivity. J. Chem. Phys. 141, 6 (2014).Google ScholarCross Ref
    36. J. A Fleck and E. H Canfield. 1984. A Random Walk Procedure for Improving the Computational Efficiency of the Implicit Monte Carlo Method for Nonlinear Radiation Transport. J. Comput. Phys. 54, 3 (June 1984), 508–523.Google ScholarCross Ref
    37. Natasha Flyer, Bengt Fornberg, Victor Bayona, and Gregory A Barnett. 2016. On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy. J. Comput. Phys. 321 (2016), 21–38.Google ScholarDigital Library
    38. Julian Fong, Magnus Wrenninge, Christopher Kulla, and Ralf Habel. 2017. Production Volume Rendering. In ACM SIGGRAPH Courses. ACM Press, New York, NY, USA.Google Scholar
    39. A. Friedman and K.S. Fu. 1975. Stochastic Differential Equations and Applications.Google Scholar
    40. Thomas-Peter Fries, Hermann Matthies, et al. 2004. Classification and overview of meshfree methods. (2004).Google Scholar
    41. M. Galtier, S. Blanco, C. Caliot, C. Coustet, J. Dauchet, M. El Hafi, V. Eymet, R. Fournier, J. Gautrais, A. Khuong, B. Piaud, and G. Terrée. 2013. Integral Formulation of Null-Collision Monte Carlo Algorithms. 125 (Aug. 2013), 57–68.Google Scholar
    42. Riccardo Gatto. 2013. The von Mises-Fisher distribution of the first exit point from the hypersphere of the drifted Brownian motion and the density of the first exit time. Statistics & Probability Letters 83, 7 (2013), 1669–1676.Google ScholarCross Ref
    43. Iliyan Georgiev, Zackary Misso, Toshiya Hachisuka, Derek Nowrouzezahrai, Jaroslav Křivánek, and Wojciech Jarosz. 2019. Integral Formulations of Volumetric Transmittance. Proc. SIGGRAPH Asia 38, 6 (Nov. 2019), 154:1–154:17.Google Scholar
    44. Frederic Gibou, Ronald Fedkiw, and Stanley Osher. 2018. A review of level-set methods and some recent applications. J. Comput. Phys. 353 (2018), 82–109.Google ScholarCross Ref
    45. Michael B Giles. 2015. Multilevel Monte Carlo methods. Acta Numer. 24 (2015), 259–328.Google ScholarCross Ref
    46. Daniel T Gillespie. 1977. Exact stochastic simulation of coupled chemical reactions. The journal of physical chemistry 81, 25 (1977), 2340–2361.Google Scholar
    47. Denis S Grebenkov. 2007. NMR survey of reflected Brownian motion. Reviews of Modern Physics 79, 3 (2007), 1077.Google ScholarCross Ref
    48. Ralf Habel, Per H. Christensen, and Wojciech Jarosz. 2013. Photon Beam Diffusion: A Hybrid Monte Carlo Method for Subsurface Scattering. Proc. EGSR 32, 4 (2013).Google ScholarDigital Library
    49. Wolfgang Hackbusch. 2015. Hierarchical matrices: algorithms and analysis. Vol. 49.Google Scholar
    50. Francis H Harlow and J Eddie Welch. 1965. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 12 (1965).Google Scholar
    51. John C. Hart. 1996. Sphere Tracing: A Geometric Method for the Antialiased Ray Tracing of Implicit Surfaces. The Visual Computer 12, 10 (Dec. 1996), 527–545.Google ScholarCross Ref
    52. Desmond J Higham. 2001. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM review 43, 3 (2001), 525–546.Google Scholar
    53. J Eduard Hoogenboom and Dávid Légrády. 2005. A critical review of the weight window generator in MCNP. In Proceedings of Monte Carlo Topical Meeting. 17–21.Google Scholar
    54. Yixin Hu, Teseo Schneider, Bolun Wang, Denis Zorin, and Daniele Panozzo. 2020. Fast Tetrahedral Meshing in the Wild. ACM Trans. Graph. 39, 4 (July 2020).Google ScholarDigital Library
    55. Chi-Ok Hwang, Sungpyo Hong, and Jinwoo Kim. 2015. Off-centered Walk-on-Spheres algorithm. J. Comp. Phys. 303 (2015), 331–335.Google ScholarDigital Library
    56. Intel. 2013. Embree: High Performance Ray Tracing Kernels. http://embree.github.io/Google Scholar
    57. Henrik Wann Jensen. 2001. State of the Art in Monte Carlo Ray Tracing for Realistic Image Synthesis. In SIGGRAPH Course Notes.Google Scholar
    58. Henrik W. Jensen, Frank Suykens, and Per H. Christensen. 2001. A Practical Guide to Global Illumination Using Photon Mapping. In ACM SIGGRAPH Courses.Google Scholar
    59. Chenfanfu Jiang, Craig Schroeder, Andrew Selle, Joseph Teran, and Alexey Stomakhin. 2015. The affine particle-in-cell method. ACM TOG 34, 4 (2015), 1–10.Google ScholarDigital Library
    60. Chenfanfu Jiang, Craig Schroeder, Joseph Teran, Alexey Stomakhin, and Andrew Selle. 2016. The material point method for simulating continuum materials. In ACM SIGGRAPH 2016 Courses. 1–52.Google ScholarDigital Library
    61. Daniel Jonsson, Joel Kronander, Jonas Unger, Thomas B. Schon, and Magnus Wrenninge. 2020. Direct Transmittance Estimation in Heterogeneous Participating Media Using Approximated Taylor Expansions. IEEE TVCG (2020), 1–1.Google Scholar
    62. James T. Kajiya. 1986. The Rendering Equation. Proc. SIGGRAPH 20, 4 (Aug. 1986).Google ScholarDigital Library
    63. S. Kakutani. 1944. Two-dimensional Brownian Motion and Harmonic Functions. Proceedings of the Imperial Academy 20, 10 (1944), 706–714.Google ScholarCross Ref
    64. Rudolph Emil Kalman. 1960. A New Approach to Linear Filtering and Prediction Problems. Trans. ASME-Journal of Basic Engineering 82, Series D (1960), 35–45.Google Scholar
    65. Hilbert J Kappen. 2007. An introduction to stochastic control theory, path integrals and reinforcement learning. In AIP conference proceedings, Vol. 887. Amer. Inst. Phys.Google ScholarCross Ref
    66. Markus Kettunen, Eugene d’Eon, Jacopo Pantaleoni, and Jan Novak. 2021. An Unbiased Ray-Marching Transmittance Estimator. (Feb. 2021). arXiv:2102.10294 [cs.GR]Google Scholar
    67. P. Kloeden and E. Platen. 2013. Num. Sol. of Stoc. Diff. Eq. Vol. 23.Google Scholar
    68. David Koerner, Jamie Portsmouth, Filip Sadlo, Thomas Ertl, and Bernd Eberhardt. 2014. Flux-Limited Diffusion for Multiple Scattering in Participating Media. CGF 33, 6 (Sept. 2014), 178–189.Google Scholar
    69. Bastian Krayer and Stefan Müller. 2021. Hierarchical Point Distance Fields. In International Symposium on Visual Computing. Springer, 435–446.Google Scholar
    70. Peter Kutz, Ralf Habel, Yining Karl Li, and Jan Novák. 2017. Spectral and Decomposition Tracking for Rendering Heterogeneous Volumes. Proc. SIGGRAPH 36, 4 (July 2017).Google Scholar
    71. Eric P. Lafortune and Yves D. Willems. 1996. Rendering Participating Media with Bidirectional Path Tracing. In Proc. EGWR. Springer-Verlag, Vienna, 91–100.Google ScholarDigital Library
    72. Antoine Lejay and Sylvain Maire. 2013. New Monte Carlo schemes for simulating diffusions in discontinuous media. J. Comp. and Appl. Mathematics 245 (2013).Google Scholar
    73. Shaofan Li and Wing Kam Liu. 2007. Meshfree particle methods.Google Scholar
    74. Frank Losasso, Ronald Fedkiw, and Stanley Osher. 2006. Spatially adaptive techniques for level set methods and incompressible flow. Computers & Fluids 35, 10 (2006).Google Scholar
    75. Sylvain Maire and Giang Nguyen. 2016. Stochastic finite differences for elliptic diffusion equations in stratified domains. Mathematics & Comp. in Simulation 121 (2016).Google Scholar
    76. Nathan G March, Elliot J Carr, and Ian W Turner. 2021. Numerical investigation into coarse-scale models of diffusion in complex heterogeneous media. Transport in Porous Media 139, 3 (2021), 467–489.Google ScholarCross Ref
    77. Zoë Marschner, Paul Zhang, David Palmer, and Justin Solomon. 2021. Sum-of-squares geometry processing. ACM Transactions on Graphics (TOG) 40, 6 (2021), 1–13.Google ScholarDigital Library
    78. Michael Mascagni and Nikolai A Simonov. 2004a. Monte Carlo methods for calculating some physical properties of large molecules. SIAM J. sc. comp. 26, 1 (2004), 339–357.Google Scholar
    79. Michael Mascagni and Nikolai A Simonov. 2004b. The random walk on the boundary method for calculating capacitance. J. Comput. Phys. 195, 2 (2004), 465–473.Google ScholarDigital Library
    80. Khamron Mekchay and Ricardo H Nochetto. 2005. Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. 43, 5 (2005), 1803–1827.Google ScholarDigital Library
    81. Robert C Merton. 1971. Optimum consumption and portfolio rules in a continuous-time model. In Stochastic optimization models in finance. Elsevier, 621–661.Google Scholar
    82. Robert C Merton and Paul Anthony Samuelson. 1992. Continuous-time finance. (1992).Google Scholar
    83. Bailey Miller, Iliyan Georgiev, and Wojciech Jarosz. 2019. A Null-Scattering Path Integral Formulation of Light Transport. Proc. SIGGRAPH 38, 4 (July 2019), 44:1–44:13.Google Scholar
    84. Jonathan T. Moon, Bruce Walter, and Steve Marschner. 2008. Efficient Multiple Scattering in Hair Using Spherical Harmonics. Proc. SIGGRAPH 27, 3 (Aug. 2008).Google Scholar
    85. Linus Mossberg. 2021. GPU-Accelerated Monte Carlo Geometry Processing for GradientDomain Methods. Ph. D. Dissertation. Linköping University, Linköping, Sweden.Google Scholar
    86. Mervin E. Muller. 1956. Some Continuous Monte Carlo Methods for the Dirichlet Problem. Annals of Mathematical Statistics 27, 3 (Sept. 1956), 569–589.Google ScholarCross Ref
    87. Thomas Müller, Marios Papas, Markus Gross, Wojciech Jarosz, and Jan Novák. 2016. Efficient Rendering of Heterogeneous Polydisperse Granular Media. Proc. SIGGRAPH Asia 35, 6 (Nov. 2016), 168:1–168:14.Google Scholar
    88. Ken Museth. 2013. VDB: High-Resolution Sparse Volumes with Dynamic Topology. ACM TOG 32, 3 (July 2013), 27:1–27:22.Google ScholarDigital Library
    89. Mohammad Sina Nabizadeh, Ravi Ramamoorthi, and Albert Chern. 2021. Kelvin Transformations for Simulations on Infinite Domains. (July 2021).Google Scholar
    90. Vinh Phu Nguyen, Timon Rabczuk, Stéphane Bordas, and Marc Duflot. 2008. Meshless methods: a review and computer implementation aspects. Mathematics and computers in simulation 79, 3 (2008), 763–813.Google Scholar
    91. Jan Novák, Iliyan Georgiev, Johannes Hanika, and Wojciech Jarosz. 2018. Monte Carlo Methods for Volumetric Light Transport Simulation. Computer Graphics Forum (Proc. Eurographics State of the Art Reports) 37, 2 (May 2018), 551–576.Google Scholar
    92. Jan Novák, Andrew Selle, and Wojciech Jarosz. 2014. Residual Ratio Tracking for Estimating Attenuation in Participating Media. ACM TOG 33, 6 (Nov. 2014).Google ScholarDigital Library
    93. Bernt Øksendal. 2003. Stochastic Differential Equations: An Introduction with Applications.Google Scholar
    94. Alexandrina Orzan, Adrien Bousseau, Holger Winnemöller, Pascal Barla, Joëlle Thollot, and David Salesin. 2008. Diffusion Curves: A Vector Representation for Smooth-Shaded Images. Proc. SIGGRAPH 27, 3 (Aug. 2008), 1.Google Scholar
    95. Ozgur Ozdemir. 2005. Variable permittivity dielectric material loaded stepped-horn antenna. (2005).Google Scholar
    96. Paul William Partridge, Carlos Alberto Brebbia, et al. 2012. Dual reciprocity boundary element method.Google Scholar
    97. Mark Pauly, Richard Keiser, Bart Adams, Philip Dutré, Markus Gross, and Leonidas J Guibas. 2005. Meshless animation of fracturing solids. ACM TOG 24, 3 (2005).Google Scholar
    98. Matt Pharr, Wenzel Jakob, and Greg Humphreys. 2016. Physically Based Rendering: From Theory to Implementation (3rd ed.). Morgan Kaufmann, Cambridge, MA.Google ScholarDigital Library
    99. Romain Prévost, Wojciech Jarosz, and Olga Sorkine-Hornung. 2015. A Vectorial Framework for Ray Traced Diffusion Curves. CGF 34, 1 (Feb. 2015), 253–264.Google Scholar
    100. Inigo Quilez. 2020. Monte Carlo PDE Shader. https://www.shadertoy.com/view/WdXfzl.Google Scholar
    101. Matthias Raab, Daniel Seibert, and Alexander Keller. 2008. Unbiased Global Illumination with Participating Media. In Monte Carlo and Quasi-Monte Carlo Methods, Alexander Keller, Stefan Heinrich, and Harald Niederreiter (Eds.). Springer-Verlag, 591–605.Google Scholar
    102. Karl K Sabelfeld. 2018. Application of the von Mises-Fisher distribution to Random Walk on Spheres method for solving high-dimensional diffusion-advection-reaction equations. Statistics & Probability Letters 138 (2018), 137–142.Google ScholarCross Ref
    103. Karl K Sabelfeld and Nikolai A Simonov. 2013. Random walks on boundary for solving PDEs. De Gruyter.Google Scholar
    104. Rohan Sawhney and Keenan Crane. 2020. Monte Carlo Geometry Processing: A Grid-Free Approach to PDE-Based Methods on Volumetric Domains. Proc. SIGGRAPH 39, 4 (2020).Google ScholarDigital Library
    105. Rohan Sawhney, Ruihao Ye, Johann Korndoerfer, and Keenan Crane. 2020. FCPW: Fastest Closest Points in the West. https://github.com/rohan-sawhney/fcpw.Google Scholar
    106. Teseo Schneider, Yixin Hu, Jérémie Dumas, Xifeng Gao, Daniele Panozzo, and Denis Zorin. 2018. Decoupling simulation accuracy from mesh quality. ACM TOG (2018).Google Scholar
    107. Vadim Shapiro and Igor Tsukanov. 1999. Meshfree simulation of deforming domains. Computer-Aided Design 31, 7 (1999), 459–471.Google ScholarCross Ref
    108. Prateek Sharma and Gregory W Hammett. 2007. Preserving monotonicity in anisotropic diffusion. J. Comput. Phys. 227, 1 (2007), 123–142.Google ScholarDigital Library
    109. Jure Slak and Gregor Kosec. 2019. On generation of node distributions for meshless PDE discretizations. SIAM Journal on Scientific Computing 41, 5 (2019), A3202–A3229.Google ScholarDigital Library
    110. Jerome Spanier and Ely Meyer Gelbard. 1969. Monte Carlo Principles and Neutron Transport Problems. Addison-Wesley.Google Scholar
    111. Deborah Sulsky, Shi-Jian Zhou, and Howard L Schreyer. 1995. Application of a particle-in-cell method to solid mechanics. Comp. phys. comms. 87, 1–2 (1995), 236–252.Google Scholar
    112. Xin Sun, Guofu Xie, Yue Dong, Stephen Lin, Weiwei Xu, Wencheng Wang, Xin Tong, and Baining Guo. 2012. Diffusion Curve Textures for Resolution Independent Texture Mapping. Proc. SIGGRAPH 31, 4 (July 2012), 74:1–74:9.Google Scholar
    113. László Szirmay-Kalos, Balázs Tóth, and Milán Magdics. 2011. Free Path Sampling in High Resolution Inhomogeneous Participating Media. CGF 30, 1 (2011), 85–97.Google ScholarCross Ref
    114. Heang K Tuy and Lee Tan Tuy. 1984. Direct 2-D display of 3-D objects. IEEE Computer Graphics and Applications 4, 10 (1984), 29–34.Google ScholarCross Ref
    115. MV Umansky, MS Day, and TD Rognlien. 2005. On numerical solution of strongly anisotropic diffusion equation on misaligned grids. Numerical Heat Transfer, Part B: Fundamentals 47, 6 (2005), 533–554.Google ScholarCross Ref
    116. Eric Veach. 1997. Robust Monte Carlo Methods for Light Transport Simulation. Ph.D. Thesis. Stanford University.Google ScholarDigital Library
    117. Eric Veach and Leonidas J. Guibas. 1995. Optimally Combining Sampling Techniques for Monte Carlo Rendering. In Proc. SIGGRAPH, Vol. 29. ACM Press, 419–428.Google Scholar
    118. Jiří Vorba and Jaroslav Křivánek. 2016. Adjoint-Driven Russian Roulette and Splitting in Light Transport Simulation. Proc. SIGGRAPH 35, 4 (July 2016), 42:1–42:11.Google ScholarDigital Library
    119. John C Wagner and Alireza Haghighat. 1998. Automated variance reduction of Monte Carlo shielding calculations using the discrete ordinates adjoint function. Nucl. Sci. Eng. 128, 2 (1998).Google Scholar
    120. Greg Ward and Rob Shakespeare. 1998. Rendering with Radiance: the art and science of lighting visualization. (1998).Google Scholar
    121. Mathias Willmann, Jesus Carrera, Xavier Sanchez-Vila, O Silva, and Marco Dentz. 2010. Coupling of mass transfer and reactive transport for nonlinear reactions in heterogeneous media. Water resources research 46, 7 (2010).Google Scholar
    122. E. R. Woodcock, T. Murphy, P. J. Hemmings, and T. C. Longworth. 1965. Techniques Used in the GEM Code for Monte Carlo Neutronics Calculations in Reactors and Other Systems of Complex Geometry. In Applications of Computing Methods to Reactor Problems. Argonne National Laboratory.Google Scholar
    123. Yonghao Yue, Kei Iwasaki, Bing-Yu Chen, Yoshinori Dobashi, and Tomoyuki Nishita. 2011. Toward Optimal Space Partitioning for Unbiased, Adaptive Free Path Sampling of Inhomogeneous Participating Media. CGF 30, 7 (2011), 1911–1919.Google ScholarCross Ref
    124. Jovan Zagajac. 1995. A fast method for estimating discrete field values in early engineering design. In Proc. the ACM symp. on Solid modeling and applications. 420–430.Google ScholarDigital Library
    125. Laurent Zalewski, Stéphane Lassue, Daniel Rousse, and Kamel Boukhalfa. 2010. Experimental and numerical characterization of thermal bridges in prefabricated building walls. Energy Conversion and Management 51, 12 (2010), 2869–2877.Google ScholarCross Ref
    126. Qingnan Zhou and Alec Jacobson. 2016. Thingi10K: A Dataset of 10,000 3D-Printing Models. arXiv preprint arXiv:1605.04797 (2016).Google Scholar
    127. Yongning Zhu and Robert Bridson. 2005. Animating sand as a fluid. ACM Transactions on Graphics (TOG) 24, 3 (2005), 965–972.Google ScholarDigital Library
    128. Olgierd Cecil Zienkiewicz and Jian Zhong Zhu. 1992a. The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Internat. J. Numer. Methods Engrg. 33, 7 (1992), 1331–1364.Google ScholarCross Ref
    129. Olgierd Cecil Zienkiewicz and Jian Zhong Zhu. 1992b. The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity. Internat. J. Numer. Methods Engrg. 33, 7 (1992), 1365–1382.Google ScholarCross Ref

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