Enhanced alternating energy minimization methods for stochastic galerkin matrix equations
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Computer Science↗
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Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations
In this work, we develop implicit rank-adaptive schemes for time-dependent matrix differential equations. The dynamic low rank approximation (DLRA) is a well-known technique to capture the dynamic low rank structure based on Dirac–Frenkel time-dependent variational principle. In recent years, it has attracted a lot of attention due to its wide applicability. Our schemes are inspired by the three-step procedure used in the rank adaptive version of the unconventional robust integrator (the so called BUG integrator) (Ceruti et al. in BIT Numer Math 62(4):1149–1174, 2022) for DLRA. First, a prediction (basis update) step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using an implicit solves for the small core matrix. Finally, a truncation is made according to a prescribed error threshold. Since the DLRA is evolving the differential equation projected on to the tangent space of the low rank manifold, the error estimate of the BUG integrator contains the tangent projection (modeling) error which cannot be easily controlled by mesh refinement. This can cause convergence issue for equations with cross terms. To address this issue, we propose a simple modification, consisting of merging the row and column spaces from the explicit step truncation method together with the BUG spaces in the prediction step. In addition, we propose an adaptive strategy where the BUG spaces are only computed if the residual for the solution obtained from the prediction space by explicit step truncation method, is too large. Here, we prove stability and estimate the local truncation error of the schemes under assumptions. We benchmark the schemes in several tests, such as anisotropic diffusion, solid body rotation and the combination of the two, to show robust convergence properties.
97 MATHEMATICS AND COMPUTING↗
One-shot omnidirectional pressure integration through matrix inversion
In this work, we present a method to perform 2D and 3D omnidirectional pressure integration from velocity measurements with a single-iteration matrix inversion approach. This work builds upon our previous work, where the rotating parallel ray approach was extended to the limit of infinite rays by taking continuous projection integrals of the ray paths and recasting the problem as an iterative matrix inversion problem. This iterative matrix equation is now 'fast-forwarded' to the 'infinity' iteration, leading to a different matrix equation that can be solved in a single step, thereby presenting the same computational complexity as the Poisson equation. We observe computational speedups of ~10 6 when compared to brute-force omnidirectional integration methods, enabling the treatment of grids of ~10 9 points and potentially even larger in a desktop setup at the time of publication. Further examination of the boundary conditions of our one-shot method shows that omnidirectional pressure integration implements a boundary condition where the boundary points are treated as interior points to the extent that information is available. Finally, we show how the method can be extended from the regular grids typical of particle image velocimetry to the unstructured meshes characteristic of particle tracking velocimetry data.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Composing preconditioners for multiphysics PDE systems with applications to Generalized MHD
New patch smoothers or relaxation techniques are developed for solving linear matrix equations coming from systems of discretized partial differential equations (PDEs). One key linear solver challenge for many PDE systems arises when the resulting discretization matrix has a near null space that has a large dimension, which can occur in generalized magnetohydrodynamic (GMHD) systems. Patch-based relaxation is highly effective for problems when the null space can be spanned by a basis of locally supported vectors. The patch-based relaxation methods that we develop can be used either within an algebraic multigrid (AMG) hierarchy or as stand-alone preconditioners. These patch-based relaxation techniques are a form of well-known overlapping Schwarz methods where the computational domain is covered with a series of overlapping sub-domains (or patches). Patch relaxation then corresponds to solving a set of independent linear systems associated with each patch. In the context of GMHD, we also reformulate the underlying discrete representation used to generate a suitable set of matrix equations. In general, deriving a discretization that accurately approximates the curl operator and the Hall term while also producing linear systems with physically meaningful near null space properties can be challenging. Unfortunately, many natural discretization choices lead to a near null space that includes non-physical oscillatory modes and where it is not possible to span the near null space with a minimal set of locally supported basis vectors. Further discretization research is needed to understand the resulting trade-offs between accuracy, stability, and ease in solving the associated linear systems.
97 MATHEMATICS AND COMPUTING↗
Parametric matrix models
We present a general class of machine learning algorithms called parametric matrix models. In contrast with most existing machine learning models that imitate the biology of neurons, parametric matrix models use matrix equations that emulate physical systems. Similar to how physics problems are usually solved, parametric matrix models learn the governing equations that lead to the desired outputs. Parametric matrix models can be efficiently trained from empirical data, and the equations may use algebraic, differential, or integral relations. While originally designed for scientific computing, we prove that parametric matrix models are universal function approximators that can be applied to general machine learning problems. After introducing the underlying theory, we apply parametric matrix models to a series of different challenges that show their performance for a wide range of problems. For all the challenges tested here, parametric matrix models produce accurate results within an efficient and interpretable computational framework that allows for input feature extrapolation.
Computational science↗
Dynamics of photosynthetic light harvesting systems interacting with N-photon Fock states
Here, we develop a method to simulate the excitonic dynamics of realistic photosynthetic light harvesting systems, including non-Markovian coupling to phonon degrees of freedom, under excitation by N-photon Fock state pulses. This method combines the input–output and the hierarchical equations of motion formalisms into a double hierarchy of density matrix equations. We show analytically that under weak field excitation relevant to natural photosynthesis conditions, an N-photon Fock state input and a corresponding coherent state input give rise to equal density matrices in the excited manifold. However, an N-photon Fock state input induces no off-diagonal coherence between the ground and excited subspaces, in contrast with the coherences created by a coherent state input. We derive expressions for the probability to absorb a single Fock state photon with or without the influence of phonons. For short pulses (or, equivalently, wide bandwidth pulses), we show that the absorption probability has a universal behavior that depends only upon a system-dependent effective energy spread parameter Δ and an exciton–light coupling constant Γ. This holds for a broad range of chromophore systems and for a variety of pulse shapes. We also analyze the absorption probability in the opposite long pulse (narrow bandwidth) regime. We then derive an expression for the long time emission rate in the presence of phonons and use it to study the difference between collective vs independent emission. Finally, we present a numerical simulation for the LHCII monomer (14-mer) system under single photon excitation that illustrates the use of the double hierarchy equations.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Analysis of the neutron matter equation of state and the symmetry energy up to fourth order of chiral effective field theory
We present predictions for the neutron matter equation of state, from leading to fourth order of chiral effective field theory, using recently developed, accurate chiral nucleon-nucleon potentials. For the many-body method, we employ the nonperturbative particle-particle ladder approximation, that is, we solve the G-matrix equation. Furthermore, we find the impact of subleading three-neutron forces to be mild and attractive. We also show order-by-order predictions for the symmetry energy, and discuss its density dependence in relation to empirical constraints. For the nuclear matter equation of state, in this work we adopt an empirical parametrization with good saturation properties. This is to highlight, specifically, the energy and pressure in neutron matter, particularly when comparing with empirical constraints.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Symbol alphabets from plabic graphs II: rational letters
Symbol alphabets of n-particle amplitudes in N = 4 super-Yang-Mills theory are known to contain certain cluster variables of G(4, n) as well as certain algebraic functions of cluster variables. The first paper arXiv:2007.00646 in this series focused on n = 8 algebraic letters. In this paper we show that it is possible to obtain all rational symbol letters (in fact all cluster variables) by solving matrix equations of the form C Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of G + (n-4, n).
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Symbol alphabets from plabic graphs III: n = 9
Symbol alphabets of n-particle amplitudes in N = 4 super-Yang-Mills theory are known to contain certain cluster variables of G(4, n) as well as certain algebraic functions of cluster variables. In this paper we solve the C Z = 0 matrix equations associated to several cells of the totally non-negative Grassmannian, combining methods of arXiv:2012.15812 for rational letters and arXiv:2007.00646 for algebraic letters. We identify sets of parameterizations of the top cell of G + (5, 9) for which the solutions produce all of (and only) the cluster variable letters of the 2-loop nine-particle NMHV amplitude, and identify plabic graphs from which all of its algebraic letters originate.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Symbol alphabets from plabic graphs
Symbol alphabets of n-particle amplitudes in N = 4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4, n) as well as certain algebraic functions of cluster variables. In this paper we suggest an algorithm for computing these symbol alphabets from plabic graphs by solving matrix equations of the form C ∙ Z = 0 to associate functions on Gr(m, n) to parameterizations of certain cells of Gr(k, n) indexed by plabic graphs. For m = 4 and n = 8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-particle amplitude from four plabic graphs.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Incremental Interval Assignment by Integer Linear Algebra with Improvements
Interval Assignment (IA) is the problem of selecting the number of mesh edges (intervals) for each curve for conforming quad and hex meshing. The intervals x is fundamentally integer-valued. Many other approaches perform numerical optimization then convert a floating-point solution into an integer solution, which is slow and error prone. We avoid such steps: we start integer, and stay integer. Incremental Interval Assignment (IIA) uses integer linear algebra (Hermite normal form) to find an initial solution to the meshing constraints, satisfying the integer matrix equation Solving for reduced row echelon form provides integer vectors spanning the nullspace of A. Here we add vectors from the nullspace to improve the initial solution, maintaining Ax = b Heuristics find good integer linear combinations of nullspace vectors that provide strict improvement towards variable bounds or goals. IIA always produces an integer solution if one exists. In practice we usually achieve solutions close to the user goals, but there is no guarantee that the solution is optimal, nor even satisfies variable bounds, e.g. has positive intervals. We describe several algorithmic changes since first publication that tend to improve the final solution. The software is freely available.
97 MATHEMATICS AND COMPUTING↗
Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation
The accuracy of the forward scattering approximation for two-point Green's functions of the Anderson localization model on the Cayley tree is studied. A relationship between the moments of the Green's function and the largest eigenvalue of the linearized transfer-matrix equation is proved in the framework of the supersymmetric functional-integral method. The new large-disorder approximation for this eigenvalue is derived and its accuracy is established. Using this approximation the probability distribution of the two-point Green's function is found and compared with that in the forward scattering approximation (FSA). It is shown that FSA overestimates the role of resonances and thus the probability for the Green's function to be significantly larger than its typical value. The error of FSA increases with increasing the distance between points in a two-point Green's function.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Parallel Implicit Hydrodynamics for High Explosive Burn Calculations
High explosives in hostile environments will require calculational capabilities that model processes, which evolve on timescales from minutes to nanoseconds. Eventually the HE will begin to move metal. To handle this temporal evolution an implicit hydrodynamics coupled to the chemical release of the HE energy is required. In addition, the use of chemical kinetics to model the transition from, the initially, slow heating of a confined high explosive through to deflagration and on to detonation requires many computational zones to model high explosive engineering systems. This requirement means that a fully parallel implicit hydrodynamics is essential. In this paper we present the calculation of a nonlinear matrix equation for the advanced particle pressure that has been made parallel and implemented in our AMR code, BABBO. This new parallel implicit hydrodynamics has been applied to a cookoff problem, as well as, one and two dimensional shock problems. Results are presented and discussed.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Machine Learning–Guided Boolean Matrix Inference for Real-Time O-RAN Conflict Detection
Open Radio Access Networks (O-RAN) are emerging, software-driven cellular architectures that promote flexibility by enabling components from different vendors to interoperate. Multiple control applications called xApps can independently adjust network parameters in near real time, often without awareness of each other's actions. This creates a system highly prone to unintended conflicts and performance degradation due to the inherent complexity of such openness. To model such systems and ultimately prevent or mitigate xApp conflicts, it is essential to understand the dynamic relationships between xApps (A), the control parameters they adjust (P), and the resulting KPI responses (K). While the mappings from A to P and from K to A can often be derived from xApp specifications, the relationship from P to K is typically hidden within the system’s dynamics and must be inferred from observed data. We propose a novel data-driven Boolean inference framework that uncovers the hidden P?K dependencies using machine learning and interpretable rule induction. Continuous parameters and KPIs are first binarized using decision tree classifiers, and a binary influence matrix L is then inferred by solving Boolean matrix equations over time. This compact representation improves interpretability and enables real-time tracking of dynamically evolving parameter-KPI dependencies. We demonstrate the effectiveness of our method in a realistic mobile handover scenario, where it accurately recovers the underlying logic and enables proactive conflict detection.
42 - ENGINEERING↗
Polarization Dependent Excitation and High Harmonic Generation from Intense Mid-IR Laser Pulses in ZnO
The generation of high order harmonics from femtosecond mid-IR laser pulses in ZnO has shown great potential to reveal new insight into the ultrafast electron dynamics on a few femtosecond timescale. In this work we report on the experimental investigation of photoluminescence and high-order harmonic generation (HHG) in a ZnO single crystal and polycrystalline thin film irradiated with intense femtosecond mid-IR laser pulses. The ellipticity dependence of the HHG process is experimentally studied up to the 17th harmonic order for various driving laser wavelengths in the spectral range 3–4 µm. Interband Zener tunneling is found to exhibit a significant excitation efficiency drop for circularly polarized strong-field pump pulses. For higher harmonics with energies larger than the bandgap, the measured ellipticity dependence can be quantitatively described by numerical simulations based on the density matrix equations. The ellipticity dependence of the below and above ZnO band gap harmonics as a function of the laser wavelength provides an efficient method for distinguishing the dominant HHG mechanism for different harmonic orders.
tunneling excitation↗
Matrix Approach to Land Carbon Cycle Modeling
Land ecosystems contribute to climate change mitigation by taking up approximately 30% of anthropogenically emitted carbon. However, estimates of the amount and distribution of carbon uptake across the world's ecosystems or biomes display great uncertainty. The latter hinders a full understanding of the mechanisms and drivers of land carbon uptake, and predictions of the future fate of the land carbon sink. The latter is needed as evidence to inform climate mitigation strategies such as afforestation schemes. To advance land carbon cycle modeling, we have developed a matrix approach. Land carbon cycle models use carbon balance equations to represent carbon exchanges among pools. Our approach organizes this set of equations into a single matrix equation without altering any processes of the original model. The matrix equation enables the development of a theoretical framework for understanding the general, transient behavior of the land carbon cycle. While carbon input and residence time are used to quantify carbon storage capacity at steady state, a third quantity, carbon storage potential, integrates fluxes with time to define dynamic disequilibrium of the carbon cycle under global change. The matrix approach can help address critical contemporary issues in modeling, including pinpointing sources of model uncertainty and accelerating spin-up of land carbon cycle models by tens of times. The accelerated spin-up liberates models from the computational burden that hinders comprehensive parameter sensitivity analysis and assimilation of observational data to improve model accuracy. Such computational efficiency offered by the matrix approach enables substantial improvement of model predictions using ever-increasing data availability. Overall, the matrix approach offers a step change forward for understanding and modeling the land carbon cycle.
54 ENVIRONMENTAL SCIENCES↗
Spin-dependent Seebeck and Nernst effects in an ideal skyrmion gas
In this work, we theoretically and numerically study spin-dependent Seebeck and Nernst effects in 2D ferromagnetic materials with the topological spin texture (skyrmion and vortex) ideal gas. From the numerical solution of the matrix Boltzmann equation for a nonequilibrium distribution function and the Lippmann–Schwinger equation for a T-matrix we find the strong nonlinear behaviors in the thermoelectric coefficients depending on skyrmion/vortex diameters and electron concentrations. In particular, the dramatic dependences in the Seebeck and Nernst coefficients take place at larger magnetic texture sizes where the abrupt sign flip in the vortex Seebeck and Nernst coefficients occurs in the narrow region of electron concentrations. In this case the normalized Nernst coefficient changes from +5 to -7. The spin-dependent thermoelectric coefficients are proportional to T at low temperatures for all skyrmion/vortex sizes.
36 MATERIALS SCIENCE↗
Full Implementation of Matrix Approach to Biogeochemistry Module of CLM5
Abstract Earth system models (ESMs) have been rapidly developed in recent decades to advance our understanding of climate change‐carbon cycle feedback. However, those models are massive in coding, require expensive computational resources, and have difficulty in diagnosing their performance. It is highly desirable to develop ESMs with modularity and effective diagnostics. Toward these goals, we implemented a matrix approach to the Community Land Model version 5 (CLM5) to represent carbon and nitrogen cycles. Specifically, we reorganized 18 balance equations each for carbon and nitrogen cycles among the 18 vegetation pools in the original CLM5 into two matrix equations. Similarly, 140 balance equations each for carbon and nitrogen cycles among the 140 soil pools were reorganized into two additional matrix equations. The vegetation carbon and nitrogen matrix equations are connected to soil matrix equations via litterfall. The matrix equations fully reproduce simulations of carbon and nitrogen dynamics by the original model. The computational cost for forwarding simulation of the CLM5 matrix model was 26% more expensive than the original model, largely due to calculation of additional diagnostic variables, but the spin‐up computational cost was significantly saved. We showed a case study on modeled soil carbon storage under two forcing data sets to illustrate the diagnostic capability that the matrix approach uniquely offers to understand simulation results of global carbon and nitrogen dynamics. The successful implementation of the matrix approach to CLM5, one of the most complex land models, demonstrates that most, if not all, the biogeochemical models can be reorganized into the matrix form to gain high modularity, effective diagnostics, and accelerated spin‐up.
58 GEOSCIENCES↗