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Results for “SINGULAR INTEGRAL EQUATION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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8 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Electroweak three-body decays in the presence of two- and three-body bound states

Recently, formalism has been derived for studying electroweak transition amplitudes for three-body systems both in infinite and finite volumes. The formalism provides exact relations that the infinite-volume amplitudes must satisfy, as well as a relationship between physical amplitudes and finite-volume matrix elements, which can be constrained from lattice QCD calculations. This formalism poses additional challenges when compared with the analogous well-studied two-body equivalent one, including the necessary step of solving integral equations of singular functions. In this work, we provide some non-trivial analytical and numerical tests on the aforementioned formalism. In particular, we consider a case where the three-particle system can have three-body bound states as well as bound states in the two-body subsystem. For kinematics below the three-body threshold, we demonstrate that the scattering amplitudes satisfy unitarity. We also check that for these kinematics the finite-volume matrix elements are accurately described by the formalism for two-body systems up to exponentially suppressed corrections. Finally, we verify that in the case of the three-body bound state, the finite-volume matrix element is equal to the infinite-volume coupling of the bound state, up to exponentially suppressed errors.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Kinematic flow for cosmological loop integrands

Recently, an interesting pattern was found in the differential equations satisfied by the Feynman integrals describing tree-level correlators of conformally coupled scalars in a power-law FRW cosmology [1, 2]. It was proven that simple and universal graphical rules predict the equations for arbitrary graphs as a flow in kinematic space. In this note, we show that the same rules — with one small addition — also determine the differential equations for loop integrands. We explain that both the basis of master integrals and the singularities of the differential equations can be represented by tubings of marked graphs. An important novelty in the case of loops is that some basis functions can vanish, and we present a graphical rule to identify these vanishing functions. Taking this into account, we then demonstrate that the kinematic flow correctly predicts the differential equations for all loop integrands.

Cosmological models↗
DOE OSTI2025

A Polar Scaling Technique for the Regularization of Strongly Singular and Strongly Near-Singular Helmholtz Surface Integrals Evaluated Over 2-D Domains

The numerical integration of expressions containing strong singularities or strong near-singularities has long been a challenging problem in the electromagnetics community. Much attention has been paid to this problem, as strong $1/R^{{2}}$ singularities routinely appear when implementing electromagnetic simulation techniques like the method of moments (MoM). To date, several techniques, from singularity extraction to singularity cancellation (SC), have been employed to deal with problems that require the evaluation of 2-D strongly singular integrals. However, no single technique has been proposed that can deal with both strong singularities and strong near-singularities in a fully numerical manner for arbitrary 2-D domains. Moreover, it has been claimed that the Helmholtz-type strongly singular integral found in the MoM is convergent in a principal value sense, but this convergence value has yet to be proven mathematically. In this work, we will conduct the convergence proof and introduce a “polar scaling” change of variables method that may be used to evaluate Helmholtz integrals with both strong and weak singularities/near-singularities. The technique is fully numerical and can in principle be applied to any planar or curved polygon and any nonsingular basis function. We will also provide numerical results showing useful convergence behavior for integrals involving both exact and near-singularities.

47 OTHER INSTRUMENTATION↗
DOE OSTI2025

Jacobian-based model diagnostics and application to equation oriented modeling of a carbon capture system

It can be difficult to identify the specific variables or equations responsible for convergence issues in large mathematical programming models. The Institute for the Design of Advanced Energy Systems Integrated Platform (IDAES-IP) contains a tool to identify poorly scaled constraints and variables by searching for rows and columns of the Jacobian matrix with small L2-norms. A singular value decomposition is then performed to identify degenerate sets of equations and remaining scaling issues. Here, this work presents a flowsheet developed for post-combustion carbon capture using a monoethanolamine (MEA) solvent system as a case study. This work takes the reader through the entire process of model diagnostics and reformulation, from a basic introduction to the mathematics behind these model diagnostics to the reformulations necessary to make the model numerically robust, including a significantly modified enhancement factor model.

IDAES↗
DOE OSTI2024

Jacobian-based Model Diagnostics and Application to Equation Oriented Modeling of a Carbon Capture System

Equation-oriented (EO) modeling has the potential to enable the effective design and optimization of the operation of advanced energy systems. However, advanced modeling of energy systems results in a large number of variables and non-linear equations, and it can be difficult to search through these to identify the culprit(s) responsible for convergence issues. The Institute for the Design of Advanced Energy Systems Integrated Platform (IDAES-IP) contains a tool to identify poorly scaled constraints and variables by searching for rows and columns of the Jacobian matrix with small L2-norms so they can be rescaled. A further singular value decomposition can be per-formed to identify degenerate sets of equations and remaining scaling issues. This work presents an EO model of a flowsheet developed for post-combustion carbon capture using a monoethanolamine (MEA) solvent system as a case study. The IDAES diagnostics tools were successfully applied to this flowsheet to identify problems to improve model robustness and enable the optimization of process design and operating conditions of a carbon capture system.

Allan, Douglas↗
DOE OSTI2025

Differential equations for cosmological correlators

Cosmological fluctuations retain a memory of the physics that generated them in their spatial correlations. The strength of correlations varies smoothly as a function of external kinematics, which is encoded in differential equations satisfied by cosmological correlation functions. In this work, we provide a broader perspective on the origin and structure of these differential equations. As a concrete example, we study conformally coupled scalar fields in a power-law cosmology. The wavefunction coefficients in this model have integral representations, with the integrands being the product of the corresponding flat-space results and “twist factors” that depend on the cosmological evolution. Similar twisted integrals arise for loop amplitudes in dimensional regularization, and their recent study has led to the discovery of rich mathematical structures and powerful new tools for computing multi-loop Feynman integrals in quantum field theory. The integrals of interest in cosmology are also part of a finite-dimensional basis of master integrals, which satisfy a system of first-order differential equations. We develop a formalism to derive these differential equations for arbitrary tree graphs. The results can be represented in graphical form by associating the singularities of the differential equations with a set of graph tubings. Upon differentiation, these tubings grow in a local and predictive fashion. In fact, a few remarkably simple rules allow us to predict — by hand — the equations for all tree graphs. While the rules of this “kinematic flow” are defined purely in terms of data on the boundary of the spacetime, they reflect the physics of bulk time evolution. We also study the analogous structures in tr ϕ 3 theory, and see some glimpses of hidden structure in the sum over planar graphs. This suggests that there is an autonomous combinatorial or geometric construction from which cosmological correlations, and the associated spacetime, emerge.

Cosmological models↗
DOE OSTI2026

Precise 2D electric field density simulations for superconducting quantum devices

Dielectric loss due to two-level systems is a limiting factor for superconducting qubit relaxation times. These losses arise mostly from nanometer-scale interfacial defect regions in superconducting devices with planar dimensions of microns to millimeters, thus making it resource intensive to accurately simulate the electric field density in these regions with traditional electromagnetic solvers. In this work, we demonstrate a fast boundary integral equation solver that allows precise simulation of electric field density in these thin regions, showing a speedup of around two orders of magnitude over traditional solvers, with relative errors around $10^{-7}$ for a ten-minute solution runtime. By computing participation ratios through Green's first identity without squaring the electric field, our approach is less susceptible to the field singularities near conductor corners. We apply this solver to a basic untrenched coplanar waveguide cross-section, showing that the common assumption of participation ratio linearity with dielectric constant holds well for some interfaces and not others; in particular, while the metal-air (MA) top and corner follow this linear relationship strongly, the MA sidewall does not. We then compare isotropic and anisotropic etching, showing that the MA sidewall and the metal-air-substrate triple junction are the most strongly affected. We are currently leveraging this solver to explore geometries that will uniquely isolate the participation ratios of the different dielectrics. Finally, we are working to combine this solver framework with a full 3D microwave solver to accurately calculate participation ratios for the thin dielectrics that are known sources of loss in superconducting qubits.

Gimbutas, Z. [NIST, Boulder] (ORCID:00000003320982↗
DOE OSTI2026

SODAs: sparse optimization for the discovery of differential and algebraic equations

Differential-algebraic equations (DAEs) integrate ordinary differential equations (ODEs) with algebraic constraints, providing a fundamental framework for developing models of dynamical systems characterized by time-scale separation, conservation laws and physical constraints. While sparse optimization has revolutionized model development by allowing data-driven discovery of parsimonious models from a library of possible equations, existing approaches for dynamical systems assume DAEs can be reduced to ODEs by eliminating variables before model discovery. This assumption limits the applicability of such methods for DAE systems with unknown constraints and time scales. We introduce sparse optimization for differential-algebraic systems (SODAs), a data-driven method for the identification of DAEs in their explicit form. By discovering the algebraic and dynamic components sequentially without prior identification of the algebraic variables, this approach leads to a sequence of convex optimization problems. It has the advantage of discovering interpretable models that preserve the structure of the underlying physical system. To this end, SODAs improves since SODAs is singular numerical stability when handling high correlations between library terms, caused by near-perfect algebraic relationships, by iteratively refining the conditioning of the candidate library. We demonstrate the performance of our method on biological, mechanical and electrical systems, showcasing its robustness to noise in both simulated time series and real-time experimental data.

DAE↗