Linear Operators Research Articles

Linear Operating Range and Extension of Three-Dimensional Active Zero-State PWM for Open-End Winding Drive

Linear Operating Range and Extension of Three-Dimensional Active Zero-State PWM for Open-End Winding Drive

The reconstruction problem for a multivalued linear operator's properties

The reconstruction problem for a multivalued linear operator's properties

Deformation of superintegrability in the Miwa-deformed Gaussian matrix model

We consider an arbitrary deformation of the Gaussian matrix model parametrized by Miwa variables za. One can look at it as a mixture of the Gaussian and logarithmic (Selberg) potentials, which are both superintegrable. The mixture is not, still one can find an explicit expression for an arbitrary Schur average as a linear transform of a polynomial made from the values of skew Schur functions at the Gaussian locus pk=δk,2. This linear operation includes multiplication with an exponential eza2/2 and a kind of Borel transform of the resulting product, which we call multiple and enhanced. The existence of such remarkable formulas appears intimately related to the theory of auxiliary K-polynomials, which appeared in superintegrable correlators at the Gaussian point (strict superintegrability). We also consider in great detail the generating function of correlators ⟨(TrX)k⟩ in this model, and discuss its integrable determinant representation. At last, we describe deformation of all results to the Gaussian β-ensemble. Published by the American Physical Society 2024

Contractivity of Möbius functions of operators

Let T be an injective bounded linear operator on a complex Hilbert space. We characterize the complex numbers λ,μ for which (I+λT)(I+μT)−1 is a contraction, the characterization being expressed in terms of the numerical range of the possibly unbounded operator T−1.When T=V, the Volterra operator on L2[0,1], this leads to a result of Khadkhuu, Zemánek and the second author, characterizing those λ,μ for which (I+λV)(I+μV)−1 is a contraction. Taking T=Vn, we further deduce that (I+λVn)(I+μVn)−1 is never a contraction if n≥2 and λ≠μ.

A fully implicit, asymptotic-preserving, semi-Lagrangian algorithm for the time dependent anisotropic heat transport equation

In this paper, we extend the operator-split asymptotic-preserving, semi-Lagrangian algorithm for time dependent anisotropic heat transport equation proposed in Chacón et al. (2014) [18] to use a fully implicit time integration with backward differentiation formulas. The proposed implicit method can deal with arbitrary heat-transport anisotropy ratios χ∥/χ⊥⋙1 (with χ∥, χ⊥ the parallel and perpendicular heat diffusivities, respectively) in complicated magnetic field topologies in an accurate and efficient manner. The implicit algorithm is second-order accurate temporally and demonstrates an accurate treatment at boundary layers (e.g., island separatrices), which was not ensured by the operator-split implementation. The condition number of the resulting algebraic system is independent of the anisotropy ratio, and is inverted with preconditioned GMRES. We propose a simple preconditioner that renders the finite-dimensional linear operator compact, resulting in mesh-independent convergence rates for topologically simple magnetic fields, and convergence rates scaling as ∼(NΔt)1/4 (with N the total mesh size and Δt the timestep) in topologically complex magnetic-field configurations. We demonstrate the accuracy and performance of the approach with test problems of varying complexity, including an analytically tractable boundary-layer problem in a straight magnetic field, and a topologically complex magnetic field featuring magnetic islands with extreme anisotropy ratios (χ∥/χ⊥=1010).

Linear and nonlinear pseudo-differential operators on p-adic fields

Linear and nonlinear pseudo-differential operators on p-adic fields

An optimally convergent Fictitious Domain method for interface problems

We introduce a novel Fictitious Domain (FD) unfitted method for interface problems associated with a second-order elliptic linear differential operator, that achieves optimal convergence without the need for adaptive mesh refinements nor enrichments of the Finite Element spaces. The key aspect of the proposed method is that it extends the solution into the fictitious domain in a way that ensures high global regularity. Continuity of the solution across the interface is enforced through a boundary Lagrange multiplier. The subdomains coupling, however, is not achieved by means of the duality pairing with the Lagrange multiplier, but through an L2 product with the H1 Riesz representative of the latter, thus avoiding gradient jumps across the interface. Thanks to the enhanced regularity, the proposed method attains an increase, with respect to standard FD methods, of up to one order of convergence in energy norm. The Finite Element formulation of the method is presented, followed by its analysis. Numerical tests on a model problem demonstrate its effectiveness and its superior accuracy compared to standard unfitted methods.

Denoising of imaginary time response functions with Hankel projections

Imaginary-time response functions of finite-temperature quantum systems are often obtained with methods that exhibit stochastic or systematic errors. Reducing these errors comes at a large computational cost—in quantum Monte Carlo simulations, the reduction of noise by a factor of two incurs a simulation cost of a factor of four. In this paper, we relate certain imaginary-time response functions to an inner product on the space of linear operators on Fock space. We then show that data with noise typically does not respect the positive definiteness of its associated Gramian. The Gramian has the structure of a Hankel matrix. As a method for denoising noisy data, we introduce an alternating projection algorithm that finds the closest positive definite Hankel matrix consistent with noisy data. We test our methodology at the example of fermion Green's functions for continuous-time quantum Monte Carlo data and show remarkable improvements of the error, reducing noise by a factor of up to 20 in practical examples. We argue that Hankel projections should be used whenever finite-temperature imaginary-time data of response functions with errors is analyzed, be it in the context of quantum Monte Carlo, quantum computing, or in approximate semianalytic methodologies. Published by the American Physical Society 2024

Theory and Modelling of Isotropic Turbulence: From Incompressible through Increasingly Compressible Flows

Homogeneous isotropic turbulence (HIT) has been a useful theoretical concept for more than fifty years of theory, modelling, and calculations. Some exact results are revisited in incompressible HIT, with special emphasis on the 4/5 Kolmogorov law. The finite Reynolds number effect (FRN), which yields corrections to that law, is investigated, using both Kármán–Howarth-type equations and a statistical spectral closure of the Eddy-Damped Quasi-Normal Markovian (EDQNM)-type. This discussion offers an opportunity to give an extended review of such spectral closures, from weak turbulence, as in wave turbulence theory, to a strong one. Extensions of the 4/5 or 4/3 Kolmogorov/Monin laws to anisotropic cases, such as stably stratified and MHD turbulence, are briefly touched on. Before addressing more recent work on compressible isotropic turbulence, the simplest case of quasi-incompressible turbulence subjected to externally imposed isotropic compression or dilatation is presented. Rapid distortion theory is found to be a poor model in this isotropic case, in contrast with its relevance in strongly anisotropic flow cases. Accordingly, a fully nonlinear approach based on a rescaling of all fluctuating variables is used, in order to show its interplay with the linear operator. This opens the discussion on the cases of homogeneous incompressible turbulence, where RDT and nonlinear models are relevant, provided that anisotropy is accounted for. Finally, isotropic compressible flows of increasing complexity are considered. Recent studies using weak turbulence theory, modelling, and DNS are discussed. A final unpublished study involves interactions between the solenoidal mode, inherited from incompressible turbulence, and the acoustic and entropic modes, which are specific to the compressible problem. An approach to acoustic wave turbulence, with resonant triads, is revisited on this occasion.

Idempotent vector spaces and their linear transformations

Abstract This article extends topics about linear algebra and operator theoretic linear transformations on complex vector spaces to those on bicomplex spaces. For example, Definition 3 for the first time defines algebraically idempotent vector spaces, which generalizes the standard definition of a vector space and which includes bicomplex vector spaces as a special case, along with its dimension and its basis in terms of a corresponding vectorial idempotent representation. The article also shows how an n × n n\times n bicomplex matrix’s idempotent representation leads to a bicomplex Jordan form and a description of its bicomplex invariant subspace lattice diagram. Similarly, in a new way, the article rigorously defines “bicomplex Banach and Hilbert” spaces, and then it expands, for the first time, the theory of compact operators on complex Banach spaces to those on bicomplex Banach spaces. In these ways, the article indicates that the idempotent representation extends complex linear algebra and operator theory in a surprisingly generalized and straightforward way to vector space results with bicomplex and multicomplex scalars.

Adaptivity in Local Kernel Based Methods for Approximating the Action of Linear Operators

Adaptivity in Local Kernel Based Methods for Approximating the Action of Linear Operators

An invitation to resolvent analysis

AbstractResolvent analysis is a powerful tool that can reveal the linear amplification mechanisms between the forcing inputs and the response outputs about a base flow. These mechanisms can be revealed in terms of a pair of forcing and response modes and the associated energy gains (amplification magnitude) at a given frequency. The linear relationship that ties the forcing and the response is represented through the resolvent operator (transfer function), which is constructed through spatially discretizing the linearized Navier–Stokes operator. One of the unique strengths of resolvent analysis is its ability to analyze statistically stationary turbulent flows. In light of the increasing interest in using resolvent analysis to study a variety of flows, we offer this guide in hopes of removing the hurdle for students and researchers to initiate the development of a resolvent analysis code and its applications to their problems of interest. To achieve this goal, we discuss various aspects of resolvent analysis and its role in identifying dominant flow structures about the base flow. The discussion in this paper revolves around the compressible Navier–Stokes equations in the most general manner. We cover essential considerations ranging from selecting the base flow and appropriate energy norms to the intricacies of constructing the linear operator and performing eigenvalue and singular value decompositions. Throughout the paper, we offer details and know-how that may not be available to readers in a collective manner elsewhere. Towards the end of this paper, examples are offered to demonstrate the practical applicability of resolvent analysis, aiming to guide readers through its implementation and inspire further extensions. We invite readers to consider resolvent analysis as a companion for their research endeavors.

Exact hydrodynamic manifolds for the linear Boltzmann BGK equation I: spectral theory

We perform a complete spectral analysis of the linear three-dimensional Boltzmann BGK operator resulting in an explicit transcendental equation for the eigenvalues. Using the theory of finite-rank perturbations, we confirm the existence of a critical wave number kcrit\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_{\ extrm{crit}}$$\\end{document} which limits the number of hydrodynamic modes in the frequency space. This implies that there are only finitely many isolated eigenvalues above the essential spectrum at each wave number, thus showing the existence of a finite-dimensional, well-separated linear hydrodynamic manifold as a combination of invariant eigenspaces. The obtained results can serve as a benchmark for validating approximate theories of hydrodynamic closures and moment methods and provides the basis for the spectral closure operator.

COLAFormer: Communicating local–global features with linear computational complexity

Local and sparse attention effectively reduce the high computational cost of global self-attention. However, they suffer from non-global dependency and coarse feature capturing, respectively. While some subsequent models employ effective interaction techniques for better classification performance, we observe that the computation and memory of these models overgrow as the resolution increases. Consequently, applying them to downstream tasks with large resolutions takes time and effort. In response to this concern, we propose an effective backbone network based on a novel attention mechanism called Concatenating glObal tokens in Local Attention (COLA) with a linear computational complexity. The implementation of COLA is straightforward, as it incorporates global information into the local attention in a concatenating manner. We introduce a learnable condensing feature (LCF) module to capture high-quality global information. LCF possesses the following properties: (1) performing a function similar to clustering, aggregating image patches into a smaller number of tokens based on similarity. (2) a constant number of aggregated tokens regardless of the image size, ensuring that it is a linear complexity operator. Based on COLA, we build COLAFormer, which achieves global dependency and fine-grained feature capturing with linear computational complexity and demonstrates impressive performance across various vision tasks. Specifically, our COLAFormer-S achieves 84.5% classification accuracy, surpassing other advanced models by 0.4% with similar or less resource consumption. Furthermore, our COLAFormer-S can achieve a better object detection performance while consuming only 1/4 of the resources compared to other state-of-the-art models. The code and models will be made publicly available.

Improved Temperature-Scalable DC model for SiC power MOSFET including Quasi-Saturation effect

In this paper, accurate temperature-dependent static model for Silicon-Carbide (SiC) power MOSFET is presented. The proposed model is formed by two equations relating to linear and saturation operating regions. In this model, new formalism of the saturation drain current is introduced to consider the peculiar features observed in the I-V static characteristics of the SiC power MOSFET: a) moderate inversion region, or region of low gate voltages and b) quasi-saturation region, region of high gate voltages at which the drain current becomes less sensitive to the increase of gate voltage. In addition, the model captures with high-precision the transition region between linear and saturation region, pinch-off region, noticed in the output characteristics of the SiC power MOSFETs. It will be shown that the model equations ensure continuity and smooth transition between all operating regions. Temperature scaling of the model is carried out by its temperature scaling parameters. The proposed compact model is simple and efficient using reduced number of technology independent parameters. Simple parameter extraction procedure is described that uses an optimizer algorithm based on good experimental initial guess. Excellent agreement is obtained by comparing model to TCAD simulation and device measurement.

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Yosida distance and existence of invariant manifolds in the infinite-dimensional dynamical systems

We consider the existence of invariant manifolds to evolution equations u ′ ( t ) = A u ( t ) u’(t)=Au(t) , A : D ( A ) ⊂ X → X A:D(A)\subset \mathbb {X}\to \mathbb {X} near its equilibrium A ( 0 ) = 0 A(0)=0 under the assumption that its proto-derivative ∂ A ( x ) \partial A(x) exists and is continuous in x ∈ D ( A ) x\in D(A) in the sense of Yosida distance. Yosida distance between two (unbounded) linear operators U U and V V in a Banach space X \mathbb {X} is defined as d Y ( U , V ) ≔ lim sup μ → + ∞ ‖ U μ − V μ ‖ d_Y(U,V)≔\limsup _{\mu \to +\infty } \| U_\mu -V_\mu \| , where U μ U_\mu and V μ V_\mu are the Yosida approximations of U U and V V , respectively. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of ∂ A \partial A is continuous in the sense of Yosida distance. The Yosida distance approach allows us to generalize the well-known results with possible applications to larger classes of partial differential equations and functional differential equations. The obtained results seem to be new.

A 3D-Var assimilation scheme for vertical velocity with CMA-MESO v5.0

Abstract. Certain vertical motions associated with meso-microscale systems are favorable for convection development and maintenance. Correct initialization of updraft motions is thus significant in convective precipitation forecasts. A three-dimensional variational-based vertical velocity (w) assimilation scheme has been developed within the high-resolution (3 km) CMA-MESO (the Mesoscale Weather Numerical Forecast System of the China Meteorological Administration) model. This scheme utilizes the adiabatic Richardson equation as the observation operator for w, enabling the update of horizontal winds and mass fields of the model's background. The tangent linear and adjoint operators are subsequently developed and undergo an accuracy check. A single-point w observation assimilation experiment reveals that the observational information is effectively spread both horizontally and vertically. Specifically, the assimilation of w contributes to the generation of horizontal wind convergence at lower model levels and divergence at higher model levels, thereby adjusting the locations of convection occurrence. The impact of assimilating w on the forecast is then examined through a series of continuous 10 d runs. Further assimilation of w, in addition to the assimilation of conventional and radial wind data, significantly improves the forecast accuracy of precipitation, resulting in higher FSS (fractions skill score) values and higher ETS (equitable threat score) skills at higher thresholds (5 and 20 mm h−1). However, it should be noted that further assimilation of w can potentially lead to some false precipitation, resulting in slightly lower ETS values at lower thresholds (1 mm h−1) and a neutral impact on BIAS (bias score) skills. An individual case study conducted within the batch experiments reveals that assimilating w has a beneficial impact on the enhancement of vertical motion across different layers of the model, facilitating the transport of moisture from lower to middle–high model levels, thereby leading to an improvement in forecast skills.

Spatially Resolved Temperature Response Functions to CO2 Emissions

AbstractCarbon dioxide (CO2) emissions affect local temperature; quantifying that local response is important for learning about the earth system, the impacts of mitigation, and adaptation needs. We assume the climate system can be represented as a time‐dependent linear system, diagnosing Green's Functions for the spatial temperature response to CO2 emissions based on CMIP6 earth system models. This allows us to emulate the linear component of the temperature response to CO2. This approach is sufficient to capture the spatial temperature response of CMIP6 experiments within one standard deviation of the multimodel spread across most regions, though accuracy is lower in the Southern Ocean and the Arctic. Our approach reveals where nonlinear feedbacks are important in current CMIP6 models, and where the local system response is well represented by a time‐dependent linear differential operator. It incorporates emissions path dependency and may be useful for evaluating large ensembles of emission scenarios.

Nonlinear oscillators dynamics using optimal and modified homotopy perturbation method

This study investigates the several strongly nonlinear oscillators (Van der Pol, Duffing and Rayleigh) for which we have applied the most powerful and advanced optimal semi-analytical technique: optimal and modified homotopy perturbation method (OM-HPM) for the convergent semi-analytical series solution. The numerical simulation demonstrates the high accuracy of the OM-HPM, which is straightforward, does not require any domain decomposition, special transformation, or pade approximations to get the convergent series solution. The key features for the high accuracy of the OM-HPM lies on the best optimal auxiliary linear operator. Therefore, OM-HPM offering a valuable tool for engineers and researcher to analyze the complex nonlinear oscillator.