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This dataset accompanies the publication "GenASiS: General Astrophysical Simulation System. II. Self-gravitating Baryonic Matter." This dataset contains simulation outputs from GenASiS for the test problems described in that publication.
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This dataset accompanies the publication "GenASiS: General Astrophysical Simulation System. II. Self-gravitating Baryonic Matter." This dataset contains simulation outputs from GenASiS described in the Appendix of that publication.
The coming massive parallelism of exascale computing presents a pressing challenge for the many DOE simulations of time-dependent partial differential equations (PDEs), which typically use traditional sequential time stepping methods. Since this traditional approach is inherently serial, it presents a sequential bottleneck when moving to exascale computing, because future performance gains will come through greater concurrency, not faster clock speeds. Thus, the goal of this work is to research parallelism in time, i.e., methods that compute multiple time values simultaneously, not sequentially. The focus will be on hyperbolic and chaotic problems of interest to DOE, with the goal of enabling scalable simulations of time-dependent hyperbolic and chaotic problems on future architectures. The chosen methodology for solving these problems parallel-in-time is multigrid, because multigrid (when it works) is a powerful, optimal, and scalable solver for discretized PDEs. Multigrid is already commonly used in many DOE simulations for scalably and optimally solving space-only PDE problems. The areas of hyperbolic and chaotic problems are chosen because of their relevance to problems of programmatic interest to DOE. However, these problems are also well-known to be difficult for parallel-in-time methods, with the most common method, parareal, diverging in many cases. The current state of-the-art for parallel-in-time at LLNL is the multigrid reduction in time (MGRIT) XBraid package, which also struggles for such problems, while still showing some improvement over parareal. In summary, new methods are needed for an efficient parallel-in-time scheme for hyperbolic and chaotic problems, and this work shall research promising new multigrid methods in this area. In particular, this work shall continue researching the directions from the current collaboration with Dr. Falgout, which are laid out in the work Toward Parallel in Time for Chaotic Dynamical Systems and showed the first known results of a parallel-in-time speedup for a chaotic problem. This work outlines two key improvements to XBraid for chaotic problems, the so-called “theta” and “delta-correction” methods. Here, these two improvements will be further researched and improved (including with a new relaxation method inspired by on Least Squares Shadowing (LSS)) and explored for more complicated problems.
The quantum approximate optimization algorithm (QAOA) has the potential to approximately solve complex combinatorial optimization problems in polynomial time. However, current noisy quantum devices cannot solve large problems due to hardware constraints. In this work, we develop an algorithm that decomposes the QAOA input problem graph into a smaller problem and solves MaxCut using QAOA on the reduced graph. The algorithm requires a subroutine that can be classical or quantum—in this work, we implement the algorithm twice on each graph. One implementation uses the classical solver Gurobi in the subroutine and the other uses QAOA. We solve these reduced problems with QAOA. On average, the reduced problems require only approximately 1/10 of the number of vertices than the original MaxCut instances. Furthermore, the average approximation ratio of the original MaxCut problems is 0.75, while the approximation ratios of the decomposed graphs are on average of 0.96 for both Gurobi and QAOA. With this decomposition, we are able to measure optimal solutions for ten 100-vertex graphs by running single-layer QAOA circuits on the Quantinuum trapped-ion quantum computer H1-1, sampling each circuit only 500 times. This approach is best suited for sparse, particularly k-regular graphs, as k-regular graphs on n vertices can be decomposed into a graph with at most $\frac{nk}{k+1}$ vertices in polynomial time. Further reductions can be obtained with a potential trade-off in computational time. In conclusion, while this paper applies the decomposition method to the MaxCut problem, it can be applied to more general classes of combinatorial optimization problems.
This paper reviews current theoretical and numerical approaches to optimization problems governed by partial differential equations (PDEs) that depend on random variables or random fields. Such problems arise in many engineering, science, economics and societal decision-making tasks. This paper focuses on problems in which the governing PDEs are parametrized by the random variables/fields, and the decisions are made at the beginning and are not revised once uncertainty is revealed. Examples of such problems are presented to motivate the topic of this paper, and to illustrate the impact of different ways to model uncertainty in the formulations of the optimization problem and their impact on the solution. A linear–quadratic elliptic optimal control problem is used to provide a detailed discussion of the set-up for the risk-neutral optimization problem formulation, study the existence and characterization of its solution, and survey numerical methods for computing it. Different ways to model uncertainty in the PDE-constrained optimization problem are surveyed in an abstract setting, including risk measures, distributionally robust optimization formulations, probabilistic functions and chance constraints, and stochastic orders. Furthermore, approximation-based optimization approaches and stochastic methods for the solution of the large-scale PDE-constrained optimization problems under uncertainty are described. Some possible future research directions are outlined.
Abstract. We consider robust optimal experimental design (ROED) for nonlinear Bayesian inverse problems governed by partial differential equations (PDEs). An optimal design is one that maximizes some utility quantifying the quality of the solution of an inverse problem. However, the optimal design is dependent on elements of the inverse problem such as the simulation model, the prior, or the measurement error model. ROED aims to produce an optimal design that is aware of the additional uncertainties encoded in the inverse problem and remains optimal even after variations in them. We follow a worst-case scenario approach to develop a new framework for robust optimal design of nonlinear Bayesian inverse problems. The proposed framework (a) is scalable and designed for infinite-dimensional Bayesian nonlinear inverse problems constrained by PDEs; (b) develops efficient approximations of the utility, namely the expected information gain; (c) employs eigenvalue sensitivity techniques to develop analytical forms and efficient evaluation methods of the gradient of the utility with respect to the uncertainties against which we wish to be robust; and (d) employs a probabilistic optimization paradigm that properly defines and efficiently solves the resulting combinatorial max-min optimization problem. The effectiveness of the proposed approach is illustrated for optimal sensor placement problem in an inverse problem governed by an elliptic PDE.
Multiobjective simulation optimization (MOSO) problems are optimization problems with multiple conflicting objectives, where evaluation of at least one of the objectives depends on a black-box numerical code or real-world experiment, which we refer to as a simulation. Whereas an extensive body of research is dedicated to developing new algorithms and methods for solving these and related problems, it is challenging and time-consuming to integrate these techniques into real-world production-ready solvers. This is partly because of the diversity and complexity of modern state-of-the-art MOSO algorithms and methods and partly because of the complexity and specificity of many real-world problems and their corresponding computing environments. The complexity of this problem is only compounded when introducing potentially complex and/or domain-specific surrogate-modeling techniques, problem formulations, design spaces, and data acquisition functions. Here, this paper carefully surveys the current state of the art in MOSO algorithms, techniques, and solvers, as well as problem types and computational environments where MOSO is commonly applied. We then present several key challenges in the design of a parallel multiobjective simulation optimization framework (ParMOO) and how they have been addressed. Finally, we provide two case studies demonstrating how customized ParMOO solvers can be quickly built and deployed to solve real-world MOSO problems.
In previous work, we have introduced a mathematical model for solving a discretized version of the process family design problem. This involves two sets of decision variables. One set selects which unit module designs are included in the process platform out of a candidate set of options; the other set determines which of these unit module designs are assigned to each variant. In this work, we exploit a parallelized Progressive Hedging (PH) algorithm to solve even larger scale design problems. PH is a well-known algorithm traditionally used to solve stochastic programming problems. While our problem is not a two-stage stochastic programming problem, the structure is similar, and it can be directly mapped to the PH approach, which we employ here to solve this deterministic optimization problem. We decompose our problem by process variant. We treat the platform unit module design variables as first-stage and the assignment of unit module designs to variants as second-stage, solving the problem using mpi-sppy. We demonstrate this approach on case studies of CC, water desalination, and refrigeration.
Optimal design of experiments for Bayesian inverse problems has recently gained wide popularity and attracted much attention, especially in the computational science and Bayesian inversion communities. An optimal design maximizes a predefined utility function that is formulated in terms of the elements of an inverse problem, an example being optimal sensor placement for parameter identification. The state-of-the-art algorithmic approaches following this simple formulation generally overlook misspecification of the elements of the inverse problem, such as the prior or the measurement uncertainties. This work presents an efficient algorithmic approach for designing optimal experimental design schemes for Bayesian linear inverse problems such that the optimal design is robust to misspecification of elements of the inverse problem. Specifically, we consider a worst-case scenario approach for the uncertain or misspecified parameters, formulate robust objectives, and propose an algorithmic approach for optimizing such objectives. Furthermore, both relaxation and stochastic solution approaches are discussed with detailed analysis and insight into the interpretation of the problem and the proposed algorithmic approach. Extensive numerical experiments to validate and analyze the proposed approach are carried out for sensor placement in a parameter identification problem.
This project aims to develop efficient and robust computational methods for solving the security-constrained unit commitment and alternating current optimal power flow problem (SC-UC-ACOPF). The SC-UC-ACOPF problem is at the center of the short-term operation of the U.S. Power Grid. It is solved every week, every day, and every 10 minutes to plan for the optimal action of electricity generation and consumption by minimizing the generation cost and maintaining power system reliability against potential disruptions of equipment failures. In mathematical terms, SC-UC-ACOPF is a challenging large-scale mixed-integer nonlinear optimization model. This means that the decisions involve both discrete variables, e.g. the turning on and off of generators and switching of transmission lines and transformers, and continuous decisions, e.g. the amount of energy generated by each generator and the power flows in the power grid. The physics of the power flow is described by nonlinear equations involving real and reactive power and bus voltages. Another key feature is the large number of contingencies, i.e. the system needs to stay reliable in face of failure of any one equipment, such as transmission lines and generators. The U.S. power grids are extremely complicated and large scale with more than 5,000 generators, 50,000 buses, and 100,000 high-voltage transmission lines, making the SC-UC-ACOPF a very large-scale computation challenge. The research developed in this project aims to solve the SC-UC-ACOPF problems in the three timescales, i.e. weekly, daily, and every 10-min. The proposed computational methods are built on a principled algorithmic approach of decomposition and penalization. More specifically, the algorithm develops spatial and temporal decomposition by exploiting the strong temporal coupling and weak spatial coupling of the UC problem and the complementary feature, i.e. weak temporal coupling and strong spatial coupling of the ACOPF problem. The algorithm also leverages recent progresses in strong convex relaxation of ACOPF. A unique feature of the proposed approach is that it generates a valid, global upper bound on the optimal maximum profit. In this way, a global optimality gap is available to measure the quality of the solution. To further speed up computation, the research team has developed a plethora of effective heuristics to strengthen the iterative penalty-based decomposition framework. For instance, a heuristic is developed to construct inner approximations of the time coupling constraints within the time decoupled problems. Contingencies are pre-screened and low-rank matrix computation is exploited to find the almost unique solution to each contingency. A novel heuristic for line switching is proposed and tested with positive impacts on instances where line switching is beneficial. Taking a systematic approach and carefully handling every detail of the problem pays off. The TIM-GO’s performance throughout the trials and the final event was stellar. TIM-GO garnered the second highest total prize money and is ranked in the top three positions across all categories of comparison.
We model transient diffusion in heterogeneous materials using a novel physics-informed neural networks framework (PINNs) termed Adaptive interface physics-informed neural networks or AdaI-PINNs (Roy et al. arXiv preprint arXiv:2406.04626, 2024). AdaI-PINNs utilize different activation functions with trainable slopes tailored to each material region within the computational domain, allowing for a fully automated and adaptive PINNs approach to model interface problems with strongly and weakly discontinuous solutions. To enhance its performance in highly heterogeneous transient diffusion systems, we prescribe a suite of robust practices, including appropriate non-dimensionalization of equations, a biased sampling method, Glorot initialization, and the hard enforcement of boundary and initial conditions. Here we evaluate the efficacy of the proposed method on several benchmark forward and inverse problems. Comparative studies on one-dimensional and two-dimensional benchmark problems reveal that the modified AdaI-PINNs outperform its unmodified counterpart, achieving root-mean-square errors that are at least two orders of magnitude better in forward problems. For inverse problems, the maximum errors in the approximated diffusion coefficients by modified AdaI-PINNs are four orders of magnitude better than those of the unmodified version. Additionally, modified AdaI-PINNs demonstrate improved stability in problems with large material mismatches.
In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors, i.e. the errors between the accurate parameter-to-observable mapping and the surrogate. The statistics of these errors are, however, in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline, i.e. before considering the data, the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to directly and efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.
In this work, we present an efficient and scalable algorithm for performing matrix-vector multiplications (matvecs) for block Toeplitz matrices. Such matrices, which are shift-invariant with respect to their blocks, arise in the context of solving inverse problems governed by autonomous systems, and time-invariant systems in particular. In this article, we consider inverse problems that infer unknown parameters from observational data of a linear time-invariant dynamical system given in the form of partial differential equations (PDEs). Matrix-free Newton-conjugate-gradient methods are often the gold standard for solving these inverse problems, but they require numerous actions of the Hessian on a vector. Matrix-free adjoint-based Hessian matvecs require solution of a pair of linearized forward/adjoint PDE solves per Hessian action, which may be prohibitive for large-scale inverse problems. Time invariance of the forward PDE problem leads to a block Toeplitz structure of the discretized parameter-to-observable (p2o) map defining the mapping from inputs (parameters) to outputs (observables) of the PDEs. This block Toeplitz structure enables us to exploit two key properties: (1) compact storage of the p2o map and its adjoint, and (2) efficient fast Fourier transform–based Hessian matvecs. The proposed algorithm is mapped onto large multi-GPU clusters and achieves more than 80% of peak bandwidth on NVIDIA A100 GPUs. Excellent weak scaling is shown for up to 48 A100 GPUs. For the targeted problems, the implementation executes Hessian matvecs within fractions of a second, which is orders of magnitude faster than can be achieved by conventional matrix-free Hessian matvecs via forward/adjoint PDE solves.
Abstract Quantum annealing (QA) is a novel type of analog computation that aims to use quantum mechanical fluctuations to search for optimal solutions of Ising problems. QA in the transverse Ising model, implemented on D-Wave quantum processing units, are available as cloud computing resources. In this study we report concise benchmarks across three generations of D-Wave quantum annealers, consisting of four different devices, for the NP-hard discrete combinatorial optimization problems unweighted maximum clique and unweighted maximum cut on random graphs. The Ising, or equivalently quadratic unconstrained binary optimization, formulation of these problems do not require auxiliary variables for order reduction, and their overall structure and weights are not highly variable, which makes these problems simple test cases to understand the sampling capability of current D-Wave quantum annealers. All-to-all minor embeddings of size 52, with relatively uniform chain lengths, are used for a direct comparison across the Chimera, Pegasus, and Zephyr device topologies. A grid-search over annealing times and the minor embedding chain strengths is performed in order to determine the level of reasonable performance for each device and problem type. Experiment metrics that are reported are approximation ratios for non-broken chain samples, chain break proportions, and time-to-solution for the maximum clique problem instances. How fairly the quantum annealers sample optimal maximum cliques, for instances which contain multiple maximum cliques, is quantified using entropy of the measured ground state distributions. The newest generation of quantum annealing hardware, which has a Zephyr hardware connectivity, performed the best overall with respect to approximation ratios and chain break frequencies.
The dimer tiling problem asks in how many ways can the edges of a graph be covered by dimers so that each site is covered once. In the special case of a planar graph, this problem has a solution in terms of a free fermionic field theory. We rediscover and explore an expression for the number of coverings of an arbitrary graph by arbitrary objects in terms of an interacting fermionic field theory first proposed by Samuel. Generalizations of the dimer tiling problem, which we call “dimer piling problems,” demand that each site be covered N times by indistinguishable dimers. Our field theory provides a solution of these problems in the large- N limit. We give a similar path integral representation for certain lattice coloring problems. Published by the American Physical Society 2024
By generalizing the unconstrained p-hub approach for the park-and-ride (P&R) facility location problem under the multinomial logit demand function, the capacitated p-hub approach for the problem under the nested logit demand function captures a broader range of real-world cases. To solve this problem optimally, we introduce a mixed-integer linear program and accelerate its solution by enhancing the branch-and-cut procedure. To address the problem at a large scale, we introduce two other polyhedral approaches: variable neighborhood search (VNS) and adaptive randomized rounding (ARR). Downtown areas in Seoul have a high modal share of public transportation and congested road traffic, yet P&R has not been widely implemented. Therefore, we apply the ARR procedure to solve a real-world problem using traffic and geographic data from the Seoul metropolitan area. ARR performs better than VNS and addresses real-world cases. The solutions obtained by ARR present a phased expansion plan that encourages policymakers to start installing a small number of P&Rs immediately.
In this work, we introduce a new iterative quantum algorithm, called Iterative Symphonic Tunneling for Satisfiability problems (IST-SAT), which solves quantum spin glass optimization problems using high-frequency oscillating transverse fields. IST-SAT operates as a sequence of iterations, in which bitstrings returned from one iteration are used to set spin-dependent phases in oscillating transverse fields in the next iteration. Over several iterations, the novel mechanism of the algorithm steers the system toward the problem ground state. We benchmark IST-SAT on sets of hard MAX-3-XORSAT problem instances with exact state vector simulation, and report polynomial speedups over Trotterized adiabatic quantum computation and the best known semi-greedy classical algorithm. When IST-SAT is seeded with a sufficiently good initial approximation, the algorithm converges to exact solution(s) in a polynomial number of iterations. Our numerical results identify a critical Hamming radius, or quality of initial approximation, where the time-to-solution crosses from exponential to polynomial scaling in problem size. This work proposes IST-SAT a new quantum algorithm, which improves upon solutions obtained from initial classical or quantum optimization algorithms. The steering mechanism we introduce through IST-SAT presents a new path toward achieving quantum advantage in optimization.