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We introduce a Monte Carlo method for computing derivatives of the solution to a partial differential equation (PDE) with respect to problem parameters (such as domain geometry or boundary conditions). Derivatives can be evaluated at arbitrary points, without performing a global solve or constructing a volumetric grid or mesh. The method is hence well suited to inverse problems with complex geometry, such as PDE-constrained shape optimization. Like otherwalk on spheres (WoS)algorithms, our method is trivial to parallelize, and is agnostic to boundary representation (meshes, splines, implicit surfaces,etc.), supporting large topological changes. We focus in particular on screened Poisson equations, which model diverse problems from scientific and geometric computing. As in differentiable rendering, we jointly estimate derivatives with respect to all parameters---hence, cost does not grow significantly with parameter count. In practice, even noisy derivative estimates exhibit fast, stable convergence for stochastic gradient-based optimization, as we show through examples from thermal design, shape from diffusion, and computer graphics. 

Numerous scientific and engineering applications require solutions to boundary value problems (BVPs) involving elliptic partial differential equations, such as the Laplace or Poisson equations, on geometrically intricate domains. We develop a Monte Carlo method for solving such BVPs with arbitrary first-order linear boundary conditions---Dirichlet, Neumann, and Robin. Our method directly generalizes thewalk on stars (WoSt)algorithm, which previously tackled only the first two types of boundary conditions, with a few simple modifications. Unlike conventional numerical methods, WoSt does not need finite element meshing or global solves. Similar to Monte Carlo rendering, it instead computes pointwise solution estimates by simulating random walks along star-shaped regions inside the BVP domain, using efficient ray-intersection and distance queries. To ensure WoSt producesbounded-varianceestimates in the presence of Robin boundary conditions, we show that it is sufficient to modify how WoSt selects the size of these star-shaped regions. Our generalized WoSt algorithm reduces estimation error by orders of magnitude relative to alternative grid-free methods such as thewalk on boundaryalgorithm. We also developbidirectionalandboundary value cachingstrategies to further reduce estimation error. Our algorithm is trivial to parallelize, scales sublinearly with increasing geometric detail, and enables progressive and view-dependent evaluation. 

We introduce a suite of path sampling methods for differentiable rendering of scene parameters that do not induce visibility-driven discontinuities, such as BRDF parameters. We begin by deriving a path integral formulation for differentiable rendering of such parameters, which we then use to derive methods that importance sample paths according to this formulation. Our methods are analogous to path tracing and path tracing with next event estimation for primal rendering, have linear complexity, and can be implemented efficiently using path replay backpropagation. Our methods readily benefit from differential BRDF sampling routines, and can be further enhanced using multiple importance sampling and a loss-aware pixel-space adaptive sampling procedure tailored to our path integral formulation. We show experimentally that our methods reduce variance in rendered gradients by potentially orders of magnitude, and thus help accelerate inverse rendering optimization of BRDF parameters. 

Grid-free Monte Carlo methods such aswalk on spherescan be used to solve elliptic partial differential equations without mesh generation or global solves. However, such methods independently estimate the solution at every point, and hence do not take advantage of the high spatial regularity of solutions to elliptic problems. We propose a fast caching strategy which first estimates solution values and derivatives at randomly sampled points along the boundary of the domain (or a local region of interest). These cached values then provide cheap, output-sensitive evaluation of the solution (or its gradient) at interior points, via a boundary integral formulation. Unlike classic boundary integral methods, our caching scheme introduces zero statistical bias and does not require a dense global solve. Moreover we can handle imperfect geometry (e.g., with self-intersections) and detailed boundary/source terms without repairing or resampling the boundary representation. Overall, our scheme is similar in spirit tovirtual point lightmethods from photorealistic rendering: it suppresses the typical salt-and-pepper noise characteristic of independent Monte Carlo estimates, while still retaining the many advantages of Monte Carlo solvers: progressive evaluation, trivial parallelization, geometric robustness,etc.We validate our approach using test problems from visual and geometric computing. 

Grid-free Monte Carlo methods based on thewalk on spheres (WoS)algorithm solve fundamental partial differential equations (PDEs) like the Poisson equation without discretizing the problem domain or approximating functions in a finite basis. Such methods hence avoid aliasing in the solution, and evade the many challenges of mesh generation. Yet for problems with complex geometry, practical grid-free methods have been largely limited to basic Dirichlet boundary conditions. We introduce thewalk on stars (WoSt)algorithm, which solves linear elliptic PDEs with arbitrary mixed Neumann and Dirichlet boundary conditions. The key insight is that one can efficiently simulate reflecting Brownian motion (which models Neumann conditions) by replacing the balls used by WoS withstar-shapeddomains. We identify such domains via the closest point on the visibility silhouette, by simply augmenting a standard bounding volume hierarchy with normal information. Overall, WoSt is an easy modification of WoS, and retains the many attractive features of grid-free Monte Carlo methods such as progressive and view-dependent evaluation, trivial parallelization, and sublinear scaling to increasing geometric detail. 

When viewed under coherent illumination, scattering materials such as tissue exhibit highly varying speckle patterns. Despite their noise-like appearance, the temporal and spatial variations of these speckles, resulting from internal tissue dynamics and/or external perturbation of the illumination, carry strong statistical information that is highly valuable for tissue analysis. The full practical applicability of these statistics is still hindered by the difficulty of simulating the speckles and their statistics. This paper proposes an efficient Monte Carlo framework that can efficiently sample physically correct speckles and estimate their covariances. While Monte Carlo algorithms were originally derived for incoherent illumination, our approach simulates complex-valued speckle fields. We compare the statistics of our speckle fields against those produced by an exact numerical wave solver and show a precise agreement, while our simulator is a few orders of magnitude faster and scales to much larger scenes. We also show that the simulator predictions accurately align with existing analytical models and simulation strategies, which currently address various partial settings of the general problem.