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Abstract Recent advances in optical atomic clocks and optical time transfer have enabled new possibilities in precision metrology for both tests of fundamental physics and timing applications. Here we describe a space mission concept that would place a state-of-the-art optical atomic clock in an eccentric orbit around Earth. A high stability laser link would connect the relative time, range, and velocity of the orbiting spacecraft to earthbound stations. The primary goal for this mission would be to test the gravitational redshift, a classical test of general relativity, with a sensitivity 30 000 times beyond current limits. Additional science objectives include other tests of relativity, enhanced searches for dark matter and drifts in fundamental constants, and establishing a high accuracy international time/geodesic reference. 

This paper presents a probabilistic perspective on iterative methods for approximating the solution x in R^d of a nonsingular linear system Ax = b. Classically, an iterative method produces a sequence x_m of approximations that converge to x in R^d. Our approach, instead, lifts a standard iterative method to act on the set of probability distributions, P(Rd), outputting a sequence of probability distributions  mu_m in P(Rd). The output of a probabilistic iterative method can provide both a “best guess” for x, for example by taking the mean of  mu_m, and also probabilistic uncertainty quantification for the value of x when it has not been exactly determined. A comprehensive theoretical treatment is presented in the case of a stationary linear iterative method, where we characterise both the rate of contraction of  mu_m to an atomic measure on x and the nature of the uncertainty quantification being provided. We conclude with an empirical illustration that highlights the potential for probabilistic iterative methods to provide insight into solution uncertainty. 

This paper presents a probabilistic perspective on iterative methods for approximating the solution x in R^d of a nonsingular linear system Ax = b. Classically, an iterative method produces a sequence x_m of approximations that converge to x in R^d. Our approach, instead, lifts a standard iterative method to act on the set of probability distributions, P(Rd), outputting a sequence of probability distributions  mu_m in P(Rd). The output of a probabilistic iterative method can provide both a “best guess” for x, for example by taking the mean of mu_m, and also probabilistic uncertainty quantification for the value of x when it has not been exactly determined. A comprehensive theoretical treatment is presented in the case of a stationary linear iterative method, where we characterise both the rate of contraction of  mu_m to an atomic measure on x and the nature of the uncertainty quantification being provided. We conclude with an empirical illustration that highlights the potential for probabilistic iterative methods to provide insight into solution uncertainty.