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We study the computational scalability of a Gaussian process (GP) frameworkfor solving general nonlinear partial differential equations (PDEs). Thisframework transforms solving PDEs to solving quadratic optimization problemwith nonlinear constraints. Its complexity bottleneck lies in computing withdense kernel matrices obtained from pointwise evaluations of the covariancekernel of the GP and its partial derivatives at collocation points.  We present a sparse Cholesky factorization algorithm for such kernel matricesbased on the near-sparsity of the Cholesky factor under a new ordering ofDiracs and derivative measurements. We rigorously identify the sparsity patternand quantify the exponentially convergent accuracy of the corresponding Vecchiaapproximation of the GP, which is optimal in the Kullback-Leibler divergence.This enables us to compute $\epsilon$-approximate inverse Cholesky factors ofthe kernel matrices with complexity $O(N\log^d(N/\epsilon))$ in space and$O(N\log^{2d}(N/\epsilon))$ in time.  With the sparse factors, gradient-based optimization methods become scalable.Furthermore, we can use the oftentimes more efficient Gauss-Newton method, forwhich we apply the conjugate gradient algorithm with the sparse factor of areduced kernel matrix as a preconditioner to solve the linear system. Wenumerically illustrate our algorithm's near-linear space/time complexity for abroad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, andMonge-Amp\`ere equations. In summary, we provide a fast, scalable, and accuratemethod for solving general PDEs with GPs.

Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes