Norm Estimates Research Articles

Error Analysis for a Parabolic PDE Model Problem on a Coupled Moving Domain in a Fully Eulerian Framework

We introduce an unfitted finite element method with Lagrange-multipliers to study an Eulerian time stepping scheme for moving domain problems applied to a model problem where the domain motion is implicit to the problem. We consider a parabolic partial differential equation (PDE) in the bulk domain, and the domain motion is described by an ordinary differential equation (ODE), coupled to the bulk partial differential equation through the transfer of forces at the moving interface. The discretisation is based on an unfitted finite element discretisation on a time-independent mesh. The method-of-lines time discretisation is enabled by an implicit extension of the bulk solution through additional stabilisation, as introduced by Lehrenfeld & Olshanskii (ESAIM: M2AN, 53:585-614, 2019). The analysis of the coupled problem relies on the Lagrange-multiplier formulation, the fact that the Lagrange-multiplier solution is equal to the normal stress at the interface and that the motion of the interface is given through rigid body motion. This paper covers the complete stability analysis of the method and an error estimate in the energy norm, under an assumption on the discrete interface velocity. This includes the dynamic error in the domain motion resulting from the discretised ODE and the forces from the discretised PDE. To the best of our knowledge this is the first error analysis of this type of coupled moving domain problem in a fully Eulerian framework. Numerical examples illustrate the theoretical results.

P1$$ {P}_1 $$‐nonconforming quadrilateral finite element space with periodic boundary conditions: Part II. Application to the nonconforming heterogeneous multiscale method

AbstractA homogenization approach is one of effective strategies to solve multiscale elliptic problems approximately. The finite element heterogeneous multiscale method (FEHMM) which is based on the finite element makes possible to simulate such process numerically. In this paper we introduce a FEHMM scheme for multiscale elliptic problems based on nonconforming elements. In particular we use the noconforming element with the periodic boundary condition introduced in the companion paper. Theoretical analysis derives a priori error estimates in the standard Sobolev norms. Several numerical results which confirm our analysis are provided.

Dynamics of Two Interacting Kinks for the $$\phi ^{6}$$ Model

We consider the nonlinear wave equation known as the $$\phi ^{6}$$ model in dimension 1+1. We describe the long-time behavior of this model’s solutions close to a sum of two kinks with energy slightly larger than twice the minimum energy of non-constant stationary solutions. We prove orbital stability of two moving kinks. We show for low energy excess $$\epsilon $$ that these solutions can be described for a long time of order $$-\ln {(\epsilon )}\epsilon ^{-\frac{1}{2}}$$ as the sum of two moving kinks such that each kink’s center is close to an explicit function which is a solution of an ordinary differential system. We give an optimal estimate in the energy norm of the remainder and we prove that this estimate is achieved during a finite instant t of order $$-\ln {(\epsilon )}\epsilon ^{-\frac{1}{2}}.$$

Alternations of neuromagnetic activity across neurocognitive core networks among benign childhood epilepsy with centrotemporal spikes: A multi-frequency MEG study.

We aimed to investigate the alternations of neuromagnetic activity across neurocognitive core networks among early untreated children having benign childhood epilepsy with centrotemporal spikes (BECTS). We recorded the Magnetoencephalography (MEG) resting-state data from 48 untreated children having BECTS and 24 healthy children. The fourth edition of the Wechsler Intelligence Scale for Children (WISC-IV) was utilized to divide the children with BECTS into two groups: the cognitive impairment (CI) group with a full-scale intelligence quotient (FSIQ) of < 90 and the cognitive non-impairment (CNI) group with an FSIQ of > 90. We selected 26 bilateral cognitive-related regions of interest based on the triple network model. The neurocognitive core network spectral power was estimated using a minimum norm estimate (MNE). In the CNI group, the spectral power inside the bilateral anterior cingulate cortex (ACC) and the bilateral caudal middle frontal cortex (CMF) enhanced within the delta band and reduced within the alpha band. Both the CI and the CNI group demonstrated enhanced spectral power inside the bilateral posterior cingulate cortex (PCC), bilateral precuneus (PCu) region, bilateral superior and middle temporal cortex, bilateral inferior parietal lobe (IPL), and bilateral supramarginal cortex (SM) region in the delta band. Moreover, there was decreased spectral power in the alpha band. In addition, there were consistent changes in the high-frequency spectrum (> 90 Hz). The spectral power density within the insula cortex (IC), superior temporal cortex (ST), middle temporal cortex (MT), and parahippocampal cortex (PaH) also decreased. Therefore, studying high-frequency activity could lead to a new understanding of the pathogenesis of BECTS. The alternations of spectral power among neurocognitive core networks could account for CI among early untreated children having BECTS. The dynamic properties of spectral power in different frequency bands could behave as biomarkers for diagnosing new BECTS.

Gradient Flows of Modified Wasserstein Distances and Porous Medium Equations with Nonlocal Pressure

We study families of porous medium equations with nonlocal pressure. We construct their weak solutions via JKO schemes for modified Wasserstein distances. We also establish the regularization effect and decay estimates for the Lp norms.

Upper and lower [formula omitted] estimates for solutions to parabolic equations

In this article we prove results concerning upper and lower decay estimates for homogeneous Sobolev norms of solutions to a rather general family of parabolic equations. Following the ideas of Kreiss, Hagstrom, Lorenz and Zingano, we use eventual regularity of solutions to directly work with smooth solutions in physical space, bootstrapping decay estimates from the L2 norm to higher order derivatives. Besides obtaining upper and lower bounds through this method, we also obtain reverse results: from higher order derivatives decay estimates, we deduce bounds for the L2 norm. We use these general results to prove new decay estimates for some equations and to recover some well known results.

Sharp Strichartz type estimates for the Schrödinger equation associated with harmonic oscillator

In this article we study the Schrödinger equation associated with Harmonic oscillator in the form of Strichartz type inequality. We give simple proofs for Strichartz type inequalities using purely the \(L^2 \rightarrow L^p\) operator norm estimates of the spectral projections associated to the harmonic oscillator proved in [Jeong, Lee and Ryu, 2022 arxiv:2205.03036v1, 2022, and Koch and Tataru 2005 Duke Math. J. 128:369–392, 2005]. Our Strichartz type estimates are sharp in sense of regularity of initial data.

Mathematical problems of dynamical interaction of fluids and multiferroic solids

In the paper, we consider a three-dimensional mathematical problem of fluid-solid dynamical interaction, when an anisotropic elastic body occupying a bounded region is immersed in an inviscid fluid occupying an unbounded domain . In the solid region, we consider the generalized Green–Lindsay's model of the thermo-electro-magneto-elasticity theory. In this case, in the domain we have a six-dimensional thermo-electro-magneto-elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function), while we have a scalar acoustic pressure field in the unbounded domain . The physical kinematic and dynamical relations are described mathematically by the appropriate initial and boundary-transmission conditions. Using the Laplace transform, the dynamical interaction problem is reduced to the corresponding boundary-transmission problem for elliptic pseudo-oscillation equations containing a complex parameter τ. We derive the appropriate norm estimates with respect to the complex parameter τ and construct the solution of the original dynamical problem by the inverse Laplace transform. As a result, we prove the uniqueness, existence, and regularity theorems for the dynamical interaction problem. Actually, the present investigation is a continuation of the paper [Chkadua G, Natroshvili D. Mathematical aspects of fluid-multiferroic solid interaction problems. Math Meth Appl Sci. 2021;44(12):9727–9745], where the fluid-solid interaction problems for elliptic pseudo-oscillation equations associated with the above mentioned generalized thermo-electro-magneto-elasticity theory are studied by the potential method and the theory of pseudodifferential equations.

Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian. The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

One‐Dimensionl Inversion of Magnetotelluric Field Data by SIS and Occam's Inversion Technique: A Case Study from the Indian Himalaya

AbstractElectromagnetic method is one of the most effective geophysical methods in earth crust and mantle structure studies. Magnetotelluric (MT) data are used to obtain resistivity image of earth's surface on the depth scale of few hundred of meters to several tens of km. One‐dimensional (1D) MT inversion is carried out to estimate electrical resistivity variation of earth's surface with depth. In the present research paper one‐dimensional inversion of MT sounding curves is carried out on nine sites in foothill of Himalaya near Roorkee to Hardwar region. A non‐iterative and minimum norm estimator based algorithm Straight forward Inversion Scheme (SIS) is used to obtain continuous smoothest resistivity depth model. Another iterative and minimum norm estimator based algorithm Occam's inversion scheme is also used to obtain smoothest one‐dimensional inversion model. The data is inverted for both TE‐ and TM‐mode of polarization. The inverted data shows one‐dimensional behavior of subsurface resistivity up to depth about 0–15 km with resistivity variation ≈10–100 ohm‐m in both modes of polarization and after that the subsurface structure is two‐dimensional in both modes. 1D inversion of data provides a good starting model to reduce the computational cost of two‐dimensional inversion.

Two-grid virtual element discretization of semilinear elliptic problem

In this paper a two grid algorithm for semilinear elliptic problem based on virtual element method (VEM) discretization is proposed. With this new algorithm the solution of a semilinear elliptic problem on a fine grid is reduced to the solution of a semilinear elliptic problem on a much coarser grid, and the solution of a linear system on the fine grid. A priori error estimates in H1 and L2 norms are derived. Numerical experiments are carried out to illustrate the theoretical findings.

On Norm Estimates for Derivations in Norm-Attainable Classes

In this note, we provide detailed characterization of operators in terms of norm-attainability and norm estimates in Banach algebras. In particular, we establish the necessary and sufficient conditions for norm-attainability of the derivations and also give their norm bounds in the norm-attainable classes.

Noncommutative [formula omitted] functions and Fréchet derivatives of operator functions

Fix a unital C∗-algebra A, and write Asa for the set of self-adjoint elements of A. Also, if f:R→ℂ is a continuous function, then write fA:Asa→A for the operator functiona↦f(a) defined via functional calculus. In this paper, we introduce and study a space NCk(R) of Ck functions f:R→ℂ such that, no matter the choice of A, the operator function fA:Asa→A is k-times continuously Fréchet differentiable. In other words, if f∈NCk(R), then f “lifts” to a Ck map fA:Asa→A, for any (possibly noncommutative) unital C∗-algebra A. For this reason, we call NCk(R) the space of noncommutativeCkfunctions. Our proof that fA∈Ck(Asa;A), which requires only knowledge of the Fréchet derivatives of polynomials and operator norm estimates for “multiple operator integrals” (MOIs), is more elementary than the standard approach; nevertheless, NCk(R) contains all functions for which comparable results are known. Specifically, we prove that NCk(R) contains the homogeneous Besov space Ḃ1k,∞(R) and the Hölder space Clock,ɛ(R). We highlight, however, that the results in this paper are the first of their type to be proven for arbitrary unital C∗-algebras, and that the extension to such a general setting makes use of the author’s recent resolution of certain “separability issues” with the definition of MOIs. Finally, we prove by exhibiting specific examples that Wk(R)loc⊊NCk(R)⊊Ck(R), where Wk(R)loc is the “localized” kth Wiener space.

Design of observers with gain adaptation for systems with disturbances: An orbital symmetry-based approach

We propose a framework for designing global observers for nonlinear systems with disturbances under geometric conditions based on orbital symmetries. Under some additional restrictions these orbital symmetry-based conditions boil down to geometric homogeneity (at infinity) conditions. Our observers are the result of the combination of a first filter, a state norm estimator, with a second filter adaptively tuned by the first and when compared with the existing literature have a completely novel structure, inherited by the orbital symmetry-based conditions. The observers exploit the geometric properties of orbital symmetries which are one-parameter groups of transformations capable of mapping the system state into neighbourhoods with parametrized width.

Sparse Reconstruction Using Hyperbolic Tangent as Smooth l1-Norm Approximation

In the Compressed Sensing (CS) framework, the underdetermined system of linear equation (USLE) can have infinitely many possible solutions. However, we intend to find the sparsest possible solution, which is l0-norm minimization. However, finding an l0 norm solution out of infinitely many possible solutions is NP-hard problem that becomes non-convex optimization problem. It has been a practically proven fact that l0 norm penalty can be adequately estimated by l1 norm, which recasts a non-convex minimization problem to a convex problem. However, l1 norm non-differentiable and gradient-based minimization algorithms are not applicable, due to this very reason there is a need to approximate l1 norm by its smooth approximation. Iterative shrinkage algorithms provide an efficient method to numerically minimize l1-regularized least square optimization problem. These algorithms are required to induce sparsity in their solutions to meet the CS recovery requirement. In this research article, we have developed a novel recovery method that uses hyperbolic tangent function to recover undersampled signal/images in CS framework. In our work, l1 norm and soft thresholding are both approximated with the hyperbolic tangent functions. We have also proposed the criteria to tune optimization parameters to get optimal results. The error bounds for the proposed l1 norm approximation are evaluated. To evaluate performance of our proposed method, we have utilized a dataset comprised of 1-D sparse signal, compressively sampled MR image and cardiac cine MRI. The MRI is an important imaging modality for assessing cardiac vascular function. It provides the ejection fraction and cardiac output of the heart. However, this advantage comes at the cost of a slow acquisition process. Hence, it is essential to speed up the acquisition process to take the full benefits of cardiac cine MRI. Numerical results based on performance metrics, such as Structural Similarity (SSIM), Peak Signal to Noise Ratio (PSNR) and Root Mean Square Error (RMSE) show that the proposed tangent hyperbolic based CS recovery offers a much better performance as compared to the traditional Iterative Soft Thresholding (IST) recovery methods.

Robust Frequency Estimation Under Additive Symmetric α-Stable Gaussian Mixture Noise

Here the estimating problem of a single sinusoidal signal in the additive symmetric α-stable Gaussian (ASαSG) noise is investigated. The ASαSG noise here is expressed as the additive of a Gaussian noise and a symmetric α-stable distributed variable. As the probability density function (PDF) of the ASαSG is complicated, traditional estimators cannot provide optimum estimates. Based on the Metropolis-Hastings (M-H) sampling scheme, a robust frequency estimator is proposed for ASαSG noise. Moreover, to accelerate the convergence rate of the developed algorithm, a new criterion of reconstructing the proposal covariance is derived, whose main idea is updating the proposal variance using several previous samples drawn in each iteration. The approximation PDF of the ASαSG noise, which is referred to the weighted sum of a Voigt function and a Gaussian PDF, is also employed to reduce the computational complexity. The computer simulations show that the performance of our method is better than the maximum likelihood and the -norm estimators.

Quaternion Scalar and Vector Norm Decomposition: Quaternion PCA for Color Face Recognition.

This paper proposes a decomposition called quaternion scalar and vector norm decomposition (QSVND) for approximation problems in color image processing. Different from traditional quaternion norm approximations that are always the single objective models (SOM), QSVND is adopted to transform the SOM into the bi-objective model (BOM). Furthermore, regularization is used to solve the BOM problem as a common scalarization method, which converts the BOM into a more reasonable SOM. This can handle over-fitting or under-fitting problems neglected in this kind of research for quaternion representation (QR) in color image processing. That is how to treat redundancy caused by the extra scalar part when the vector part of a quaternion is used to represent a color pixel. We apply QSVND to quaternion principal component analysis (QPCA) for color face recognition (FR), which can deal with the phenomenon of under-fitting of vector part norm approximation. Comparisons with the competing approaches on AR, FERET, FEI, and KDEF&AKDEF databases consistently show the superiority of the proposed approach for color FR.

On the Estimate of Integral Norm of Solution of the Dirichlet Problem for Quasilinear Elliptic Equation

On the Estimate of Integral Norm of Solution of the Dirichlet Problem for Quasilinear Elliptic Equation

Composition operators from harmonic $ \mathcal{H}^{\infty} $ space into harmonic Zygmund space

&lt;abstract&gt;&lt;p&gt;This research paper sought to characterize the boundedness and compactness of composition operators from the space $ \mathcal{H}^{\infty} $ of bounded harmonic mappings into harmonic Zygmund space $ \mathcal{Z}_H $, on the open unit disk. Furthermore, we obtain an estimate of the essential norms of such an operator. These results extends the similar results that were proven for composition operators on analytic function spaces.&lt;/p&gt;&lt;/abstract&gt;

Unconditional stability and convergence analysis of fully discrete stabilized finite volume method for the time-dependent incompressible MHD flow

In this paper, we consider the stability and convergence of fully discrete stabilized finite volume method for the unsteady incompressible magnetohydrodynamics (MHD) equations on the quasi-uniform and regular triangulations. The spatial discretization is based on the lowest equal-order mixed finite element pair for the velocity, pressure and the magnetic fields, while the time discretization is based on the backward Euler semi-implicit scheme. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection stabilized method on the element is employed. This stabilized method has the advantages of parameter-free, no need to calculate the high-order derivatives or the edge-based data structures and it can be implemented at the element level. The unconditional stability of numerical scheme is established by using the Gronwall lemma, mathematical induction and choosing different test functions. Optimal error estimates of numerical approximations in various norms are also presented by constructing the corresponding dual problem. Finally, some numerical results are proposed to verify the established theoretical findings and show the performances of the considered numerical method.