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Gleipnir: toward practical error analysis for Quantum programs
Practical error analysis is essential for the design, optimization, and evaluation of Noisy Intermediate-Scale Quantum(NISQ) computing. However, bounding errors in quantum programs is a grand challenge, because the effects of quantum errors depend on exponentially large quantum states. In this work, we present Gleipnir, a novel methodology toward practically computing verified error bounds in quantum programs. Gleipnir introduces the (ρ,δ)-diamond norm, an error metric constrained by a quantum predicate consisting of the approximate state ρ and its distance δ to the ideal state ρ. This predicate (ρ,δ) can be computed adaptively using tensor networks based on the Matrix Product States. Gleipnir features a lightweight logic for reasoning about error bounds in noisy quantum programs, based on the (ρ,δ)-diamond norm metric. Furthermore, our experimental results show that Gleipnir is able to efficiently generate tight error bounds for real-world quantum programs with 10 to 100 qubits, and can be used to evaluate the error mitigation performance of quantum compiler transformations.
Scalable Yet Rigorous Floating-Point Error Analysis
Automated techniques for rigorous floating-point round-off error analysis are a prerequisite to placing important activities in HPC such as precision allocation, verification and code optimization on a formal footing. Yet existing techniques cannot provide tight bounds for expressions beyond a few dozen operators; barely enough for HPC. In this work, we offer an approach embedded in a new tool called SATIRE that scales error analysis by four orders of magnitude compared to today’s best-of-class tools. We explain how three key ideas underlying SATIRE help it attain such scale: path strength reduction, bound optimization and abstraction. SATIRE provides tight bounds and rigorous guarantees on significantly larger expressions with well over a hundred thousand operators, covering important examples including FFT, matrix multiplication and PDE stencils.
An Error Analysis Toolkit for Binned Counting Experiments
We introduce the MINERvA Analysis Toolkit (MAT), a utility for centralizing the handling of systematic uncertainties in HEP analyses. The fundamental utilities of the toolkit are the MnvHnD, a powerful histogram container class, and the systematic Universe classes, which provide a modular implementation of the many universe error analysis approach. These products can be used stand-alone or as part of a complete error analysis prescription. They support the propagation of systematic uncertainty through all stages of analysis, and provide flexibility for an arbitrary level of user customization. This extensible solution to error analysis enables the standardization of systematic uncertainty definitions across an experiment and a transparent user interface to lower the barrier to entry for new analyzers.
Numerical coupling of aerosol emissions, dry removal, and turbulent mixing in the E3SM Atmosphere Model version 1 (EAMv1) – Part 2: A semi-discrete error analysis framework for assessing coupling schemes
Abstract. Part 1 (Wan et al., 2024) of this study discusses the motivation and empirical evaluation of a revision to the aerosol-related numerical process coupling in the atmosphere component of the Energy Exascale Earth System Model version 1 (EAMv1) to address the previously reported issue of strong sensitivity of the simulated dust aerosol lifetime and dry removal rate to the model's vertical resolution. This paper complements that empirical justification of the revised scheme with a mathematical justification leveraging a semi-discrete analysis framework for assessing the splitting error of process coupling methods. The framework distinguishes the error due to numerical splitting from the error due to the time integration method(s) used for each individual process. Such a distinction results in a framework that provides an intuitive understanding of the causes of the splitting error. The application of this framework to the dust life cycle in EAMv1 confirms (i) that the original EAMv1 scheme artificially strengthens the effect of dry removal processes and (ii) that the revised splitting reduces that artificial strengthening. While the error analysis framework is presented in the context of the dust life cycle in EAMv1, the framework can be broadly leveraged to evaluate process coupling schemes, both in other physical problems and for any number of processes. This framework will be particularly powerful when the various process implementations support a variety of time integration approaches. Whereas traditional local truncation error approaches require separate consideration of each combination of time integration methods, this framework enables evaluation of coupling schemes independent of particular time integration approaches for each process while still allowing for the incorporation of these specific time integration errors if so desired. The framework also explains how the splitting error terms result from (i) the integration of individual processes in isolation from other processes and (ii) the choices of input state and time step size for the isolated integration of processes. Such a perspective has the potential for the rapid development of alternative coupling approaches that utilize knowledge both about the desired accuracy and about the computational costs of individual processes.
Reading Error Analysis (REA) Software Version 1
This is training presentation for REA (Reading Error Analysis) software. REA is designed for analyzing UGT scope reading errors and for developing reading points optimization algorithms.
Quantum simulation of boson-related Hamiltonians: techniques, effective Hamiltonian construction, and error analysis
Elementary quantum mechanics proposes that a closed physical system consistently evolves in a reversible manner. However, control and readout necessitate the coupling of the quantum system to the external environment, subjecting it to relaxation and decoherence. Consequently, system-environment interactions are indispensable for simulating physically significant theories. A broad spectrum of physical systems in condensed-matter and high-energy physics, vibrational spectroscopy, and circuit and cavity QED necessitates the incorporation of bosonic degrees of freedom, such as phonons, photons, and gluons, into optimized fermion algorithms for near-future quantum simulations. In particular, when a quantum system is surrounded by an external environment, its basic physics can usually be simplified to a spin or fermionic system interacting with bosonic modes. Nevertheless, troublesome factors such as the magnitude of the bosonic degrees of freedom typically complicate the direct quantum simulation of these interacting models, necessitating the consideration of a comprehensive plan. This strategy should specifically include a suitable fermion/boson-to-qubit mapping scheme to encode sufficiently large yet manageable bosonic modes, and a method for truncating and/or downfolding the Hamiltonian to the defined subspace for performing an approximate but highly accurate simulation, guided by rigorous error analysis. In this pedagogical tutorial review, we aim to provide such an exhaustive strategy, focusing on encoding and simulating certain bosonic-related model Hamiltonians, inclusive of their static properties and time evolutions. Specifically, we emphasize two aspects: (1) the discussion of recently developed quantum algorithms for these interacting models and the construction of effective Hamiltonians, and (2) a detailed analysis regarding a tightened error bound for truncating the bosonic modes for a class of fermion-boson interacting Hamiltonians.
Towards sharp error analysis of extended Lagrangian molecular dynamics
The extended Lagrangian molecular dynamics (XLMD) method provides a useful framework for reducing the computational cost of a class of molecular dynamics simulations with constrained latent variables. The XLMD method relaxes the constraints by introducing a fictitious mass ε for the latent variables and solving a set of singularly perturbed ordinary differential equations. While favorable numerical performance of XLMD has been demonstrated in several different contexts in the past decade, mathematical analysis of the method remains scarce. Here, we propose the first error analysis of the XLMD method in the context of a classical polarizable force field model. While the dynamics with respect to the atomic degrees of freedom are general and nonlinear, the key mathematical simplification of the polarizable force field model is that the constraints on the latent variables are given by a linear system of equations. We prove that when the initial value of the latent variables is compatible in a sense that we define, XLMD converges as the fictitious mass ε is made small with $\mathscr{O}$(ε) error for the atomic degrees of freedom and with $\mathscr{O}$($\sqrt{ε}$) error for the latent variables, when the dimension of the latent variable d' is 1. Furthermore, when the initial value of the latent variables is improved to be optimally compatible in a certain sense, we prove that the convergence rate can be improved to $\mathscr{O}$(ε) for the latent variables as well. Numerical results verify that both estimates are sharp not only for d'=1, but also for arbitrary d'. In the setting of general d', we do obtain convergence, but with the non-sharp rate of $\mathscr{O}$($\sqrt{ε}$) for both the atomic and latent variables.
Sources of error in detonation calorimeters and error analysis for neat 2,4,6-triamino-1,3,5- trinitrobenzene (TATB)
Here, a calorimeter for measuring heats of detonation at Lawrence Livermore National Laboratory is described. A calibration precision of 0.2 % at the 95 % confidence interval (CI) is reported. Sources of uncertainty are discussed, including nonequivalent sources, which are those arising from differences between calibration and experimental tests. The systematic error due to nonequivalent sources is bounded to 0.19–0.22 % of the measured heat for a standard detonation test where the confinement material is gold. The recommendation is to correct for the systematic error by adding 0.19 % to the reported value and adding 0.03 % to the uncertainty. It is demonstrated that the precision of a detonation test is variable with testing duration because a source of uncertainty resides in the thermodynamic correction factor k6, which accounts for the contribution to heat by stirring; the generated power is additive and therefore highly impacted by total test time. The relative proportion also varies with the magnitude of heat release and sample size, adding variance to the weight of the error arising from the correction factor. A full error analysis based on the described sources of uncertainty is developed. The methodology is applied to a test series on neat 2,4,6-triamino-1,3,5- trinitrobenzene (TATB), demonstrating an ultimate precision of 0.7 % (single test) for materials testing and a relative standard deviation of 1.8 %.
Analytic error analysis of cross section interpolation methods in nodal diffusion codes - I : Theory
This paper discusses two cross section interpolation methods commonly found in popular nodal codes; the partial derivatives and multiple tables models. The motivation for choosing a model, and thus a case matrix structure, is a trade off between accuracy and computational cost. Due to decades of experience, there are default structures that are sufficient for current light water reactor analysis. However, this is not necessarily the case for advanced reactor designs. Therefore, it is advantageous to understand the sources of error in cross section interpolation models so that the quality of a case matrix may be improved. A mathematical framework for these models is presented in this work that provides a more rigorous connection between the nuclear engineering field's cross section interpolation methods and the broader mathematical field of function approximation. The two cross section models examined in this paper were found to utilize Lagrange interpolation and are a subset of Lagrange tensor products. Classical results of Lagrange polynomial error analysis were then applied to the partial derivative and multiple tables models to derive expressions for the total point-wise error. The analytical results classify the total error into two parts: the model form error and interpolation error. Finally, based on our observations, a better foundation for improving the quality of a case matrix is proposed. (authors)
Implementation and (Inverse Modified) Error Analysis for Implicitly Templated ODE-Nets
We focus on learning unknown dynamics from data using ODE-nets templated on implicit numerical initial value problem solvers. First, we perform inverse modified error analysis of the ODE-nets using unrolled implicit schemes for ease of interpretation. It is shown that training an ODE-net using an unrolled implicit scheme returns a close approximation of an inverse modified differential equation (IMDE). In addition, we establish a theoretical basis for hyperparameter selection when training such ODE-nets, whereas current strategies usually treat numerical integration of ODE-nets as a black box. We thus formulate an adaptive algorithm which monitors the level of error and adapts the number of (unrolled) implicit solution iterations during the training process, so that the error of the unrolled approximation is less than the current learning loss. This helps accelerate training while maintaining accuracy. Several numerical experiments are performed to demonstrate the advantages of the proposed algorithm compared to nonadaptive unrollings and validate the theoretical analysis. Here, we also note that this approach naturally allows for incorporating partially known physical terms in the equations, giving rise to what is termed “gray box” identification.
Global error analysis of the Chebyshev rational approximation method
The Chebyshev rational approximation method (CRAM) has become a widely adopted method for solving nuclear depletion problems. Therefore, understanding CRAM’s accuracy is important for the safe operation of nuclear power plants. This article performs a global error analysis of CRAM and finds that, as the length of the time step approaches zero, the relative error measured between the exact and CRAM solutions at a fixed end time approaches one and infinity for even and odd orders, respectively; for intermediate time step sizes, a minimum in relative error is observed. Finally, we show that the reason for CRAM’s behavior is that the method is inconsistent. Two best practices for using CRAM, derived from these results, are: (1) use CRAM order 16 or higher, (2) if necessary, increase the CRAM order when multiphysics coupling requires smaller time steps.
Rounding Error Analysis of Mixed Precision Block Householder QR Algorithms
Although mixed precision arithmetic has recently garnered interest for training dense neural networks, many other applications could benefit from the speedups and lower storage cost if applied appropriately. The growing interest in employing mixed precision computations motivates the need for rounding error analysis that properly handles behavior from mixed precision arithmetic. We develop mixed precision variants of existing Householder QR algorithms and show error analyses supported by numerical experiments.
Error analysis of numerical methods for thick diffusive neutron transport problems on Shishkin mesh
A thin layer will develop at the boundary if the incoming angular flux is anisotropic in thick diffusive neutron transport problems. Solving such singularly perturbed problems, which have non-smooth solutions with singularity near the boundary, is computationally challenging. Standard finite difference schemes on a uniform mesh cannot yield ε-uniform convergence, where ε is a small parameter, while it can be achieved on a suitable piecewise-uniform Shishkin mesh. We present a formal error analysis of the diamond difference (DD) method and step difference (SD) method for solving the S{sub N} neutron transport equation. The analysis can be extended to other finite difference methods. Numerical results are presented to confirm the error estimates and the advantages of the Shishkin mesh. (author)
Systematic error analysis in the absolute hydrogen gas jet polarimeter at RHIC
The Polarized Atomic Hydrogen Gas Jet Target polarimeter (HJET) is operated at the Relativistic Heavy Ion Collider (RHIC) since 2004 to measure the absolute polarization of each colliding proton beam. Polarimeter detectors and data acquisition were upgraded in 2015 to increase the solid angle, energy range, and to improve the energy and time resolution. These upgrades along with an improved beam intensity and polarization allowed us to greatly reduce the statistical and systematic errors for the proton polarization measurements in RHIC Runs 15 (E beam = 100 GeV) and 17 (255 GeV). For a typical 8 hour RHIC store, the measured proton beam average polarization was about P beam ~ 55 ± 2.0 stat ± 0.3 syst %. The elastic pp analyzing power, AN(t)~0.04, was determined with a precision of about |δA N (t)|~0.0002 in the momentum transfer squared range 0.001<-t<0.020 GeV 2 . In this paper we present a detailed systematic error analysis of the polarization measurements at HJET. Perspectives of using the HJET based absolute polarimeters in the future Electron Ion Collider (EIC) will be also discussed.
Numerical error analysis of SOLPS-ITER simulations of EAST
Abstract Plasma edge simulations with codes like SOLPS-ITER are widely employed to interpret fusion experiments. However, numerical errors appearing in such simulations are rarely investigated, despite their potential large impact on simulation results. These errors consist of the statistical error and the bias, both resulting from the finite number of employed EIRENE Monte Carlo particles and incomplete convergence, and the discretization error due to the finite resolution of the computational grids. In this contribution, the resulting numerical errors on simulations of pure deuterium and neon seeded H-mode EAST discharges are examined. The statistical error can be kept small compared to other numerical error contributions by averaging the plasma profiles. This allows investigating the bias and discretization errors using Richardson extrapolation. It is shown that grid refinement and the number of employed Monte Carlo particles have the largest influence on the result, in agreement with similar studies of an ITER deuterium case. For the first time, numerical error bars on the entire simulated target profiles are determined showing that the largest numerical error is 17.9%, mainly due to the plasma grid discretization. On top, also numerical errors on simulated neutral pressures are investigated in detail, for which the statistical error is dominant. The analysis demonstrates which setup is needed to keep numerical errors limited: the SOLPS-ITER averaging procedure should be employed including enough EIRENE particles, and the involved grids should be sufficiently refined to reduce discretization errors.
A priori error analysis of high-order LL* (FOSLL*) finite element methods
A number of non-standard finite element methods have been proposed in recent years, each of which derives from a specific class of PDE-constrained norm minimization problems. The most notable examples are LL* methods. In this work, we argue that all high-order methods in this class should be expected to deliver substandard uniform h-refinement convergence rates. In fact, one may not even see rates proportional to the polynomial order p > 1 when the exact solution is a constant function. Here, we show that the convergence rate is limited by the regularity of an extraneous Lagrange multiplier variable which naturally appears via a saddle-point analysis. In turn, limited convergence rates appear because the regularity of this Lagrange multiplier is determined, in part, by the geometry of the domain. Numerical experiments support our conclusions.
On the Viability of Video Imaging in Leak Rate Quantification: A Theoretical Error Analysis
Optical gas imaging through multispectral cameras is a promising technique for mitigation of methane emissions through localization and quantification of emissions sources. While more advanced cameras developed in recent years have led to lower uncertainties in measuring gas concentrations, a systematic analysis of the uncertainties associated with leak rate estimation have been overlooked. We present a systematic categorization of the involved uncertainties with a focus on a theoretical analysis of projection uncertainties that are inherent to this technique. The projection uncertainties are then quantified using Large Eddy Simulation experiments of a point source release into the atmosphere. Our results show that while projection uncertainties are typically about 5% of the emission rate, low acquisition times and observation of the gas plume at small distances from the emission source (<10 m) can amount to errors of about 20%. Further, we found that acquisition times on the order of tens of seconds are sufficient to significantly reduce (>50%) the projection uncertainties. These findings suggest robust procedures on how to reduce projection uncertainties, however, a balance between other sources of uncertainty due to operational conditions and the employed instrumentation are required to outline more practical guidelines.