The paper by Thomas Gerstner and Michael Griebel discusses the numerical integration of multivariate functions over \(d\)-dimensional cubes using sparse grid methods, which were first introduced by Smolyak. The authors review existing algorithms and present new ones, focusing on the construction of multivariate quadrature formulas through combinations of tensor products of one-dimensional formulas. They emphasize the use of extended Gauss (Patterson) quadrature formulas as the one-dimensional basis, demonstrating their superiority over previous methods based on trapezoidal, Clenshaw-Curtis, and Gauss rules through numerical experiments. The paper also explores improvements for path integral computations by combining generalized Smolyak quadrature with the Brownian bridge construction. The authors highlight the effectiveness of Smolyak’s construction in overcoming the "curse of dimension" for functions with bounded mixed derivatives, achieving independence of the number of function evaluations and numerical accuracy from the dimension of the problem. The paper covers various applications, including statistical mechanics, financial derivatives valuation, and the numerical computation of path integrals, and provides a comprehensive review of related methods such as Monte Carlo, Quasi-Monte Carlo, lattice rules, and neural network-based approximations.The paper by Thomas Gerstner and Michael Griebel discusses the numerical integration of multivariate functions over \(d\)-dimensional cubes using sparse grid methods, which were first introduced by Smolyak. The authors review existing algorithms and present new ones, focusing on the construction of multivariate quadrature formulas through combinations of tensor products of one-dimensional formulas. They emphasize the use of extended Gauss (Patterson) quadrature formulas as the one-dimensional basis, demonstrating their superiority over previous methods based on trapezoidal, Clenshaw-Curtis, and Gauss rules through numerical experiments. The paper also explores improvements for path integral computations by combining generalized Smolyak quadrature with the Brownian bridge construction. The authors highlight the effectiveness of Smolyak’s construction in overcoming the "curse of dimension" for functions with bounded mixed derivatives, achieving independence of the number of function evaluations and numerical accuracy from the dimension of the problem. The paper covers various applications, including statistical mechanics, financial derivatives valuation, and the numerical computation of path integrals, and provides a comprehensive review of related methods such as Monte Carlo, Quasi-Monte Carlo, lattice rules, and neural network-based approximations.