Sampling logconcave functions arising in statistics and machine learning has\nbeen a subject of intensive study. Recent developments include analyses for\nLangevin dynamics and Hamiltonian Monte Carlo (HMC). While both approaches have\ndimension-independent bounds for the underlying $\\mathit{continuous}$ processes\nunder sufficiently strong smoothness conditions, the resulting discrete\nalgorithms have complexity and number of function evaluations growing with the\ndimension. Motivated by this problem, in this paper, we give a general\nalgorithm for solving multivariate ordinary differential equations whose\nsolution is close to the span of a known basis of functions (e.g., polynomials\nor piecewise polynomials). The resulting algorithm has polylogarithmic depth\nand essentially tight runtime - it is nearly linear in the size of the\nrepresentation of the solution.\n We apply this to the sampling problem to obtain a nearly linear\nimplementation of HMC for a broad class of smooth, strongly logconcave\ndensities, with the number of iterations (parallel depth) and gradient\nevaluations being $\\mathit{polylogarithmic}$ in the dimension (rather than\npolynomial as in previous work). This class includes the widely-used loss\nfunction for logistic regression with incoherent weight matrices and has been\nsubject of much study recently. We also give a faster algorithm with $\n\\mathit{polylogarithmic~depth}$ for the more general and standard class of\nstrongly convex functions with Lipschitz gradient. These results are based on\n(1) an improved contraction bound for the exact HMC process and (2) logarithmic\nbounds on the degree of polynomials that approximate solutions of the\ndifferential equations arising in implementing HMC.\n