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1. CFL Optimized Forward–Backward Runge–Kutta Schemes for the Shallow-Water Equations

   https://doi.org/10.1175/MWR-D-23-0113.1  

   Lilly, Jeremy R. ; Engwirda, Darren ; Capodaglio, Giacomo ; Higdon, Robert L. ; Petersen, Mark R. ( December 2023 , Monthly Weather Review)

   We present the formulation and optimization of a Runge–Kutta-type time-stepping scheme for solving the shallow-water equations, aimed at substantially increasing the effective allowable time step over that of comparable methods. This scheme, called FB-RK(3,2), uses weighted forward–backward averaging of thickness data to advance the momentum equation. The weights for this averaging are chosen with an optimization process that employs a von Neumann–type analysis, ensuring that the weights maximize the admittable Courant number. Through a simplified local truncation error analysis and numerical experiments, we show that the method is at least second-order in time for any choice of weights and exhibits low dispersion and dissipation errors for well-resolved waves. Further, we show that an optimized FB-RK(3,2) can take time steps up to 2.8 times as large as a popular three-stage, third-order strong stability-preserving Runge–Kutta method in a quasi-linear test case. In fully nonlinear shallow-water test cases relevant to oceanic and atmospheric flows, FB-RK(3,2) outperforms SSPRK3 in admittable time step by factors roughly between 1.6 and 2.2, making the scheme approximately twice as computationally efficient with little to no effect on solution quality.

   The purpose of this work is to develop and optimize time-stepping schemes for models relevant to oceanic and atmospheric flows. Specifically, for the shallow-water equations we optimize for schemes that can take time steps as large as possible while retaining solution quality. We find that our optimized schemes can take time steps between 1.6 and 2.2 times larger than schemes that cost the same number of floating point operations, translating directly to a corresponding speedup. Our ultimate goal is to use these schemes in climate-scale simulations.

    

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2. The Communication Complexity of Optimization

   https://doi.org/10.1137/1.9781611975994.106  

   Vempala, Santosh S. ; Wang, Ruosong ; Woodruff, David P. ( January 2020 , ACM-SIAM Symposium on Discrete Algorithms)

   null (Ed.)

   We consider the communication complexity of a number of distributed optimization problems. We start with the problem of solving a linear system. Suppose there is a coordinator together with s servers P1, …, Ps, the i-th of which holds a subset A(i) x = b(i) of ni constraints of a linear system in d variables, and the coordinator would like to output an x ϵ ℝd for which A(i) x = b(i) for i = 1, …, s. We assume each coefficient of each constraint is specified using L bits. We first resolve the randomized and deterministic communication complexity in the point-to-point model of communication, showing it is (d2 L + sd) and (sd2L), respectively. We obtain similar results for the blackboard communication model. As a result of independent interest, we show the probability a random matrix with integer entries in {–2L, …, 2L} is invertible is 1–2−Θ(dL), whereas previously only 1 – 2−Θ(d) was known. When there is no solution to the linear system, a natural alternative is to find the solution minimizing the ℓp loss, which is the ℓp regression problem. While this problem has been studied, we give improved upper or lower bounds for every value of p ≥ 1. One takeaway message is that sampling and sketching techniques, which are commonly used in earlier work on distributed optimization, are neither optimal in the dependence on d nor on the dependence on the approximation ε, thus motivating new techniques from optimization to solve these problems. Towards this end, we consider the communication complexity of optimization tasks which generalize linear systems, such as linear, semi-definite, and convex programming. For linear programming, we first resolve the communication complexity when d is constant, showing it is (sL) in the point-to-point model. For general d and in the point-to-point model, we show an Õ(sd3L) upper bound and an (d2 L + sd) lower bound. In fact, we show if one perturbs the coefficients randomly by numbers as small as 2−Θ(L), then the upper bound is Õ(sd2L) + poly(dL), and so this bound holds for almost all linear programs. Our study motivates understanding the bit complexity of linear programming, which is related to the running time in the unit cost RAM model with words of O(log(nd)) bits, and we give the fastest known algorithms for linear programming in this model. Read More: https://epubs.siam.org/doi/10.1137/1.9781611975994.106 

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3. Improving Column Generation for Vehicle Routing Problems via Random Coloring and Parallelization

   https://doi.org/10.1287/ijoc.2021.1105  

   Yu, Miao ; Nagarajan, Viswanath ; Shen, Siqian ( November 2021 , INFORMS Journal on Computing)

   We consider a variant of the vehicle routing problem (VRP) where each customer has a unit demand and the goal is to minimize the total cost of routing a fleet of capacitated vehicles from one or multiple depots to visit all customers. We propose two parallel algorithms to efficiently solve the column-generation-based linear-programming relaxation for this VRP. Specifically, we focus on algorithms for the “pricing problem,” which corresponds to the resource-constrained elementary shortest path problem. The first algorithm extends the pulse algorithm for which we derive a new bounding scheme on the maximum load of any route. The second algorithm is based on random coloring from parameterized complexity which can be also combined with other techniques in the literature for improving VRPs, including cutting planes and column enumeration. We conduct numerical studies using VRP benchmarks (with 50–957 nodes) and instances of a medical home care delivery problem using census data in Wayne County, Michigan. Using parallel computing, both pulse and random coloring can significantly improve column generation for solving the linear programming relaxations and we can obtain heuristic integer solutions with small optimality gaps. Combining random coloring with column enumeration, we can obtain improved integer solutions having less than 2% optimality gaps for most VRP benchmark instances and less than 1% optimality gaps for the medical home care delivery instances, both under a 30-minute computational time limit. The use of cutting planes (e.g., robust cuts) can further reduce optimality gaps on some hard instances, without much increase in the run time. Summary of Contribution: The vehicle routing problem (VRP) is a fundamental combinatorial problem, and its variants have been studied extensively in the literature of operations research and computer science. In this paper, we consider general-purpose algorithms for solving VRPs, including the column-generation approach for the linear programming relaxations of the integer programs of VRPs and the column-enumeration approach for seeking improved integer solutions. We revise the pulse algorithm and also propose a random-coloring algorithm that can be used for solving the elementary shortest path problem that formulates the pricing problem in the column-generation approach. We show that the parallel implementation of both algorithms can significantly improve the performance of column generation and the random coloring algorithm can improve the solution time and quality of the VRP integer solutions produced by the column-enumeration approach. We focus on algorithmic design for VRPs and conduct extensive computational tests to demonstrate the performance of various approaches. 

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4. The Noise Collector for sparse recovery in high dimensions

   https://doi.org/10.1073/pnas.1913995117  

   Moscoso, Miguel ; Novikov, Alexei ; Papanicolaou, George ; Tsogka, Chrysoula ( May 2020 , Proceedings of the National Academy of Sciences)

   The ability to detect sparse signals from noisy, high-dimensional data is a top priority in modern science and engineering. It is well known that a sparse solution of the linear system A ρ = b 0 can be found efficiently with an ℓ 1 -norm minimization approach if the data are noiseless. However, detection of the signal from data corrupted by noise is still a challenging problem as the solution depends, in general, on a regularization parameter with optimal value that is not easy to choose. We propose an efficient approach that does not require any parameter estimation. We introduce a no-phantom weight τ and the Noise Collector matrix C and solve an augmented system A ρ + C η = b 0 + e , where e is the noise. We show that the ℓ 1 -norm minimal solution of this system has zero false discovery rate for any level of noise, with probability that tends to one as the dimension of b 0 increases to infinity. We obtain exact support recovery if the noise is not too large and develop a fast Noise Collector algorithm, which makes the computational cost of solving the augmented system comparable with that of the original one. We demonstrate the effectiveness of the method in applications to passive array imaging. 

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5. A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Systems

   Henzinger, Monika ; Jin, Billy ; Peng, Richard ; Williamson, David P. ( January 2023 , Leibniz international proceedings in informatics)

   Tauman Kalai, Yael (Ed.)

   Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems of the form 𝐋𝐱 = 𝐛, where 𝐋 is the Laplacian matrix of a weighted graph with weights w(i,j) > 0 on the edges. The solution 𝐱 of the linear system can be interpreted as the potentials of an electrical flow in which the resistance on edge (i,j) is 1/w(i,j). Kelner, Orrechia, Sidford, and Zhu [Kelner et al., 2013] give a combinatorial, near-linear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law by updating flows around cycles (cycle toggling). In this paper, we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law by cut toggling: each iteration updates all potentials on one side of a fundamental cut of a spanning tree by the same amount. We prove that this dual algorithm also runs in a near-linear number of iterations. We show, however, that if we abstract cut toggling as a natural data structure problem, this problem can be reduced to the online vector-matrix-vector problem (OMv), which has been conjectured to be difficult for dynamic algorithms [Henzinger et al., 2015]. The conjecture implies that the data structure does not have an O(n^{1-ε}) time algorithm for any ε > 0, and thus a straightforward implementation of the cut-toggling algorithm requires essentially linear time per iteration. To circumvent the lower bound, we batch update steps, and perform them simultaneously instead of sequentially. An appropriate choice of batching leads to an Õ(m^{1.5}) time cut-toggling algorithm for solving Laplacian systems. Furthermore, we show that if we sparsify the graph and call our algorithm recursively on the Laplacian system implied by batching and sparsifying, we can reduce the running time to O(m^{1 + ε}) for any ε > 0. Thus, the dual cut-toggling algorithm can achieve (almost) the same running time as its primal cycle-toggling counterpart. 

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