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We present a new deterministic algorithm for the sparse Fourier transform problem, in which we seek to identify k N significant Fourier coefficients from a signal of bandwidth N . Previous deterministic algorithms exhibit quadratic runtime scaling, while our algorithm scales linearly with k in the average case. Underlying our algorithm are a few simple observations relating the Fourier coefficients of time-shifted samples to unshifted samples of the input function. This allows us to detect when aliasing between two or more frequencies has occurred, as well as to determine the value of unaliased frequencies. We show that empirically our algorithm is orders of magnitude faster than competing algorithms. 1 See [GST08] for a "user-friendly" description of the improved algorithm. 2 Specifically, the runtime is O(k 2 ·log N ·|S|), where S is the set of samples read by the algorithm. This set takes the form S = log N =1 A−B , where A has ε-discrepancy on rank 2 Bohr sets, B ε-approximates the uniform distribution on [0, 2 − 1] ∩ Z, and A − B is the difference set. Using constructions from [Kat89] one has |A| = O(ε −1 log 4 N ), |B | = O(ε −3 log 4 N ); setting ε = Θ(k −1 ) and noting that | A − B | = O ( |A − B |) and |A − B | = O(|A||B |)

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Abstract. In this work we discuss a class of defect correction methods which is easily adapted to create parallel time integrators for multi-core architectures and is ideally suited for developing methods which can be order adaptive in time. The method is based on Integral Deferred Correction (IDC), which was itself motivated by Spectral Deferred Correction by Dutt, Greengard and Rokhlin (BIT-2000).The method presented here is a revised formulation of explicit IDC, dubbed Revisionist IDC, which can achieve p th -order accuracy in "wall-clock time" comparable to a single forward Euler simulation on problems where the time to evaluate the right-hand side of a system of differential equations is greater than latency costs of inter-processor communication, such as in the case of the N -body problem. The key idea is to re-write the defect correction framework so that, after initial startup costs, each correction loop can be lagged behind the previous correction loop in a manner that facilitates running the predictor and M = p − 1 correctors in parallel on an interval which has K steps, where K p. We prove that given an r th -order Runge-Kutta method in both the prediction and M correction loops of RIDC, then the method is order r × (M + 1).The parallelization in Revisionist IDC uses a small number of cores (the number of processors used is limited by the order one wants to achieve). Using a four-core CPU, it is natural to think about fourth-order RIDC built with forward Euler, or eighth-order RIDC constructed with secondorder Runge-Kutta. Numerical tests on an N -body simulation show that RIDC methods can be significantly faster than popular Runge-Kutta methods such as the classical fourth-order RungeKutta scheme.In a PDE setting, one can imagine coupling RIDC time integrators with parallel spatial evaluators, thereby increasing the parallelization. The ideas behind RIDC extend to implicit and semiimplicit IDC and have high potential in this area.

Abstract. Spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs) were introduced by Dutt, Greengard and Rokhlin (2000). It was shown in that paper that SDC methods can achieve arbitrary high order accuracy and possess nice stability properties. Their SDC methods are constructed with low order integrators, such as forward Euler or backward Euler, and are able to handle stiff and non-stiff terms in the ODEs. In this paper, we use high order Runge-Kutta (RK) integrators to construct a family of related methods, which we refer to as integral deferred correction (IDC) methods. The distribution of quadrature nodes is assumed to be uniform, and the corresponding local error analysis is given. The smoothness of the error vector associated with an IDC method, measured by the discrete Sobolev norm, is a crucial tool in our analysis. The expected order of accuracy is demonstrated through several numerical examples. Superior numerical stability and accuracy regions are observed when high order RK integrators are used to construct IDC methods.