L1-norm Research Articles

Existence of solutions to a nonlinear fractional diffusion equation with exponential growth

In this paper, we study a Cauchy problem for a space–time fractional diffusion equation with exponential nonlinearity. Based on the standard Lp-Lq estimates of strongly continuous semigroup generated by fractional Laplace operator, we investigate the existence of global solutions for initial data with small norm in Orlicz space exp Lp(Rd) and a time weighted Lr(Rd) space. In the framework of the Hölder interpolation inequality, we also discuss the existence of local solutions without the Orlicz space.

The lowest-order weak Galerkin finite element method for linear elasticity problems on convex polygonal grids

This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of CW0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H1 and L2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property.

On Coarse Isometries and Linear Isometries between Banach Spaces

Let X,Y be two Banach spaces and f:X→Y be a standard coarse isometry. In this paper, we first show a sufficient and necessary condition for the coarse left-inverse operator of general Banach spaces to admit a linearly isometric right inverse. Furthermore, by using the well-known simultaneous extension operator, we obtain an asymptotical stability result when Y is a space of continuous functions. In addition, we also prove that every coarse left-inverse operator does admit a linear isometric right inverse without other assumptions when Y is a Lp(1<p<∞) space, or both X and Y are finite dimensional spaces of the same dimension. Making use of the results mentioned above, we generalize several results of isometric embeddings and give a stability result of coarse isometries between Banach spaces.

Method for Sparse Representation of Complex Data Based on Overcomplete Basis, l1 Norm, and Neural MFNN-like Network

The article presents the results of research into a method for representing complex data based on an overcomplete basis and l0/l1 norms. The proposed method is an extended modification of the neural-like MFNN (minimum fuel neural network) for the case of complex data. The influence of the choice of activation function on the performance of the method is analyzed. The results of the numerical simulation demonstrate the effectiveness of the proposed method for the case of sparse representation of complex data and can be used to determine the direction of arrival (DOA) for a uniform linear array (ULA).

A color image fusion model by saturation-value total variation

In this paper, we propose and develop a novel color image fusion model by using saturation-value total variation. In the proposed model, we develop a variational approach containing an energy functional to determine the weighting mask functions and the fused image together. The objective fused image is modeled by using the saturation-value total variation regularization. The data-fitting term is formulated based on the weighting L2 norm of the objective fused image and the input observed images. The objective weighting mask functions are modeled by using total variation for the piecewise-smoothness purpose. Meanwhile, we add another fidelity term to capture more texture and color information from the input images by forcing the weighting local mean images to be close to the clear input image. The existence of a solution of the proposed minimizing model is shown. Numerically, we apply alternating direction method of multipliers (ADMM) to solve the proposed minimization problem, meanwhile, we give the convergence analysis of the proposed algorithm. Numerical examples are presented to demonstrate that the performance of the proposed color image fusion model is better than that of other testing methods in terms of visual quality and some criterias such as structure similarity (SSIM), average local contrast (ALC), discrete entropy (DE), natural image quality evaluator (NIQE) and perception-based image quality evaluator (PIQE).

Data and measurement mechanism integrated imaging method for electrical capacitance tomography

This study presents a new imaging paradigm for overcoming the challenges limiting the improvement of the imaging quality in the electrical capacitance technique. The new imaging model enables the integration of measurement physics, sparsity-induced prior and physics-informed multi-fidelity learning prior (PIMFLP), as well as the synergy between data-driven and measurement physics modeling paradigms. The transformed L1 norm is used to model the PIMFLP, and the maximum correntropy criterion is used as a data misfit term to restrict the adverse impact of noises or outliers. The PIMFLP is learned from data and characterizes the structural details of imaging targets. The half-quadratic optimization method is developed to overcome computational challenges in solving the novel imaging model. A new physics-informed multi-fidelity learning method is developed to predict PIMFLP by synergizing deep convolutional neural network with measurement physics, and a new bilevel optimization model solved by a new nested algorithm that merges the genetic algorithm and the split Bregman method is proposed for training. Comparisons with other popular reconstruction algorithms confirm that the new imaging method leads to the improved reconstruction accuracy and noise immunity, and opens up new possibilities for unlocking the potential of the measurement technology.

A regularized orthogonal activated inverse-learning neural network for regression and classification with outliers

A novel regularized orthogonal activated inverse-learning (ROAIL) neural network is proposed and investigated for reducing the impact of outliers in regression and classification fields. The proposed ROAIL network does not require extensive iterative computations. Instead, it can achieve the desired results with a single step of computation, allowing for the efficient acquisition of network weights. By extending the Gegenbauer polynomials to a multi-variate version, and integrating the ℓ2 regularization and Welsch loss function into the orthogonal activated inverse-learning framework, two forms of ROAIL are obtained, i.e., ℓ2 norm ROAIL (ℓ2-ROAIL) and Welsch-ROAIL (W-ROAIL). ℓ2-ROAIL neural network is proposed to minimize the empirical and structural risk simultaneously since taking the structural risk as a part of loss function can effectively reduce the complexity of the model and thus improve the generalization ability. W-ROAIL neural network further improves the robustness of the ℓ2-ROAIL neural network by replacing the original two-norm in loss function with Welsch function. The Welsch function can determine the weights of each sample according to its output error, and influence of outliers could be weakened since the weights of outliers would be reduced. Both regression and classification experiments show that W-ROAIL neural network has strong ability to suppress the influence of outliers.

A posteriori error analysis and mesh adaptivity for a virtual element method solving the Stokes equations

We investigate an adaptive mesh strategy for the conforming virtual element method (VEM) of the Stokes equations proposed in Manzini and Mazzia (2022). The VEM generalizes the finite element approach to polygonal and polyehedral meshes in the framework of Galerkin approximation. The scheme of Manzini and Mazzia (2022) is inf-sup stable, converges optimally in the L2 and energy norm for all polynomial orders k≥1, and the Stokes velocity is weakly divergence-free at the machine precision level. Our adaptive mesh strategy is based on a suitable residual-based a posteriori indicator. A posteriori analysis shows that such an indicator is theoretically efficient and reliable. Our numerical experiments show that it can be an efficient tool for solving scientific and engineering problems by applying it to a set of representative situations, including the case of a weakly singular solution as that of the “L-shape” domain.

Global well-posedness and decay estimates for the one-dimensional models of blood flow with a general parabolic velocity profile

In this paper, we study the one-dimensional models of blood flow arising from the hemodynamics of aorta, which are derived from the averaging of the Navier–Stokes equations. We establish the global well-posedness and long-time behavior of the viscid 1D models of blood flow in the Sobolev space framework, where a general parabolic velocity profile is considered. Precisely speaking, we prove the global existence of smooth solution when the initial data is sufficiently small. Moreover, by combining the time-weighted energy estimates with the Green function method, we obtain the optimal time decay rate in Lp (2≤p≤∞) norm. In addition, one can see the area-averaged axial velocity decays faster than the cross-sectional area of vessel from the Green function of the linearized system. This observation is essential to study the decay rates of our nonlinear system.

A NEW MULTI-STEP BDF ENERGY STABLE TECHNIQUE FOR THE EXTENDED FISHER-KOLMOGOROV EQUATION

The multi-step backward difference formulas of order k (BDF-k) for 3 ≤ k ≤ 5 are proposed for solving the extended Fisher–Kolmogorov equation. Based upon the careful discrete gradient structures of the BDF-k formulas, the suggested numerical schemes are proved to preserve the energy dissipation laws at the discrete levels. The maximum norm priori estimate of the numerical solution is established by means of the energy stable property. With the help of discrete orthogonal convolution kernels techniques, the L2 norm error estimates of the implicit BDF-k schemes are established. Several numerical experiments are included to illustrate our theoretical results.

A stochastic process representation for time warping functions

Time warping function provides a mathematical representation to measure phase variability in functional data. Recent studies have developed various approaches to estimate optimal warping between functions. However, a principled, linear, generative representation on time warping functions is still under-explored. This is highly challenging because the warping functions are non-linear in the conventional L2 space. To address this problem, a new linear warping space is defined and a stochastic process representation is proposed to characterize time warping functions. The key is to define an inner-product structure on the time warping space, followed by a transformation which maps the warping functions into a sub-space of the L2 space. With certain constraints on the warping functions, this transformation is an isometric isomorphism. In the transformed space, the L2 basis in the Hilbert space is adopted for representation, which can be easily utilized to generate time warping functions by using different types of stochastic process. The effectiveness of this representation is demonstrated through its use as a new penalty in the penalized function registration, accompanied by an efficient gradient method to minimize the cost function. The new penalized method is illustrated through simulations that properly characterize nonuniform and correlated constraints in the time domain. Furthermore, this representation is utilized to develop a boxplot for warping functions, which can estimate templates and identify warping outliers. Finally, this representation is applied to a Covid-19 dataset to construct boxplots and identify states with outlying growth patterns.

Rotation number and eigenvalues of two-component modified Camassa–Holm equations

In this paper, we study the spectral problem y1y2x=−1212λm−12λn12y1y2for the two-component modified Camassa–Holm equation, where m,n are two potentials. By introducing the rotation number ρ(λ) and studying its properties, we prove that for any integer k, the periodic or anti-periodic eigenvalues are the endpoints of the interval λ∈R:ρ(λ)=−k2. Moreover, we prove that as nonlinear functionals of potentials, such eigenvalues are continuous in potentials with respect to the weak topologies in the Lebesgue space L1[0,T]. Finally, we apply the trace formula to give some estimates of the periodic eigenvalues when the L1 norms of potentials are given.

Reiterated homogenization of parabolic systems with several spatial and temporal scales

We consider quantitative estimates in the homogenization of second-order parabolic systems with periodic coefficients that oscillate on multiple spatial and temporal scales,∂t−div(A(x,t,x/ε1,…,x/εn,t/ε1′,…,t/εm′)∇), where εℓ=εαℓ,εk′=εβk,ℓ=1,...,n,k=1,...,m, with 0<α1<...<αn<∞ and 0<β1<...<βm<∞. The convergence rate in the homogenization is derived in the L2 space, and the large-scale interior and boundary Lipschitz estimates are also established. In the case n=m=1, such issues have been addressed by Geng and Shen (2020) [12] based on an interesting scale reduction technique developed therein. Our investigation relies on a quantitative reiterated homogenization theory.

H[formula omitted] finite-time prescribed performance trajectory tracking control of MSVs with self-adjusting neural networks

The paper proposes a robust adaptive finite-time prescribed performance trajectory tracking control scheme for marine surface vessels (MSVs) under unknown external disturbance and model uncertainty, based on H∞ control and self-adjusting neural networks (SANN). Firstly, to tackle the challenge of unknown velocity, a high-gain observer is introduced. Secondly, SANN is designed to tackle model uncertainty, achieving a balance between the optimal number of neurons and the best expected performance, thus saving network resources. Simultaneously, finite-time (FT) auxiliary functions, error transformation functions, and H functions are introduced, ensuring a bounded expected decay level with L2 norm within FT. Finally, by combining backstepping, SANN, prescribed performance control and FTH∞ control methods, the system ensures the achievement of practically finite-time stable using fewer network resources under prescribed performance conditions. Simulation results validate the effectiveness of the proposed controller.

Solution of planar elastic stress problems using stress basis functions

The use of global displacement basis functions to solve boundary-value problems in linear elasticity is well established. No prior work uses a global stress tensor basis for such solutions. We present two such methods for solving stress problems in linear elasticity. In both methods, we split the sought stress σ into two parts, where neither part is required to satisfy strain compatibility. The first part, σ p , is any stress in equilibrium with the loading. The second part, σ h is a self-equilibrated stress field on the unloaded body. In both methods, σ h is expanded using tensor-valued global stress basis functions developed elsewhere. In the first method, the coefficients in the expansion are found by minimizing the strain energy based on the well-known complementary energy principle. For the second method, which is restricted to planar homogeneous isotropic bodies, we show that we merely need to minimize the squared L2 norm of the trace of stress. For demonstration, we solve nine stress problems involving sharp corners, multiple-connectedness, non-zero net force and/or moment on an internal hole, body force, discontinuous surface traction, material inhomogeneity, and anisotropy. The first method presents a new application of a known principle. The second method presents a hitherto unreported principle, to the best of our knowledge.

An adaptive finite element method for distributed elliptic optimal control problems with variable energy regularization

We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual L2(Ω) norm regularization term with a constant regularization parameter ϱ is replaced by a suitable representation of the energy norm in H−1(Ω) involving a variable, mesh-dependent regularization parameter ϱ(x). It turns out that the error between the computed finite element state u˜ϱh and the desired state u‾ (target) is optimal in the L2(Ω) norm provided that ϱ(x) behaves like the local mesh size squared. This is especially important when adaptive meshes are used in order to approximate discontinuous target functions. The adaptive scheme can be driven by the computable and localizable error norm ‖u˜ϱh−u‾‖L2(Ω) between the finite element state u˜ϱh and the target u‾. The numerical results not only illustrate our theoretical findings, but also show that the iterative solvers for the discretized reduced optimality system are very efficient and robust.

Annual Solar Geoengineering: Mitigating Yearly Global Warming Increases

Solar geoengineering (SG) solutions have many advantages compared to the difficulty of carbon dioxide removal (CDR): SG produces fast results, is shown here to have much higher efficiency than CDR, is not related to fossil fuel legislation, reduces the GHG effect including water vapor, and is something we all can participate in by brightening the Earth with cool roofs and roads. SG requirements detailed previously to mitigate global warming (GW) have been concerning primarily because of overwhelming goals and climate circulation issues. In this paper, annual solar geoengineering (ASG) equations and estimated requirements for yearly solar radiation modification (SRM) of areas are provided along with the advantages of annual solar geoengineering (ASG) to mitigate yearly global warming temperature increases. The ASG albedo area modification requirements found here are generally 50 to potentially more than 150 times less compared to the challenge of full SG GW albedo mitigation, reducing circulation concerns and increasing feasibility. These reductions are applied to L1 space sunshading, Earth brightening, and stratosphere aerosol injection (SAI) SRM annual area requirements. However, SAI coverage compared to other methods will have higher yearly increasing maintenance costs in the annual approach. Results also show that because ASG Earth albedo brightening area requirements are much smaller than those needed for full mitigation, there are concerns that worldwide negative SG would interfere with making positive advances for several reasons. That is, negative SG currently dominates yearly practices with the application of dark asphalt roads, roofs, and building sides. This issue is discussed.

Deep Spectral Meshes: Multi-Frequency Facial Mesh Processing with Graph Neural Networks

With the rising popularity of virtual worlds, the importance of data-driven parametric models of 3D meshes has grown rapidly. Numerous applications, such as computer vision, procedural generation, and mesh editing, vastly rely on these models. However, current approaches do not allow for independent editing of deformations at different frequency levels. They also do not benefit from representing deformations at different frequencies with dedicated representations, which would better expose their properties and improve the generated meshes’ geometric and perceptual quality. In this work, spectral meshes are introduced as a method to decompose mesh deformations into low-frequency and high-frequency deformations. These features of low- and high-frequency deformations are used for representation learning with graph convolutional networks. A parametric model for 3D facial mesh synthesis is built upon the proposed framework, exposing user parameters that control disentangled high- and low-frequency deformations. Independent control of deformations at different frequencies and generation of plausible synthetic examples are mutually exclusive objectives. A Conditioning Factor is introduced to leverage these objectives. Our model takes further advantage of spectral partitioning by representing different frequency levels with disparate, more suitable representations. Low frequencies are represented with standardised Euclidean coordinates, and high frequencies with a normalised deformation representation (DR). This paper investigates applications of our proposed approach in mesh reconstruction, mesh interpolation, and multi-frequency editing. It is demonstrated that our method improves the overall quality of generated meshes on most datasets when considering both the L1 norm and perceptual Dihedral Angle Mesh Error (DAME) metrics.

Modeling fluid flow in fractured porous media: a comparative analysis between Darcy–Darcy model and Stokes–Brinkman model

AbstractThere are limited comparative studies on modeling fluid transport in fractured porous media. Hence, this paper systematically compares the steady-state creeping flow Stokes–Brinkman and Darcy–Darcy models for computational efficiency and accuracy. Sensitivity analyses were also conducted on the effect of fracture orientations, fracture sizes, mesh resolution, and fractures with Local Grid Refinement (LGR) under the FEniCS computational framework. Both models were validated numerically, and the accuracy of their solution is compared using the R-squared metric and L2 norm estimates. Key results showed that both models have similar pressure and velocity field solutions for a given fracture orientation. The computational time required for solving the Stokes–Brinkman models for a single fracture case was unusually lower than that of the Darcy–Darcy model when the pressure and velocity terms in the Darcy–Darcy model were solved simultaneously using two equations, contrary to where only one equation solves for the pressure and the velocity is obtained by projecting the gradient of pressure onto a vector space. The Stokes–Brinkman model is more sensitive to mesh resolution, and as a result, the Darcy–Darcy model tends to be more accurate than the Stokes–Brinkman model at low resolutions. Local Grid Refinement (LGR) can improve the Stokes–Brinkman model's accuracy at low mesh resolution. Furthermore, both models showed similar results when compared for complex fracture systems such as multiple fracture cases: interconnecting and isolated fractured porous media systems under low-velocity and steady-state creeping flow conditions. The FEniCS code in this paper is shared for future researchers to reproduce results or extend the research work.

Synthesis of ship systems optimal regulators based on matrix inequalities

The output and state regulators synthesis for ship control systems based on linear matrix inequalities using the CVX software package for solving optimization problems by the method of semi-definite programming is considered. Estimates parameters for regulators providing asymptotically stable systems regimes on Lyapunov inequalities solutions sets are obtained. An algorithm for controlling the ship course using a robust regulator in state feedback, followed by the use of CVX technologies to implement the minimum energy consumption control algorithm, is presented. The control vector estimation is reduced to the minimization of the L2 norm and it is performed in two stages, namely, the best mode is first determined, and then the mode under constraints is defined. The use of Lyapunov inequalities with the calculations implementation in CVX format on proprietary solvers has allowed us to obtain compact program texts and expand the set of restrictions introduced. In the algorithm for solving the boundary value problem of controlling a discrete model of an object, in contrast to existing optimization methods, a mode of transfer from the initial state to the final one with the right boundary, not necessarily equal to zero, is implemented. For a class of observable and controllable time–invariant systems, using the Krylov’s function contained in the MATLAB gallery of functions, L2‑norms to be minimized are formed. Examples of regulator calculations with a given degree of stability and optimal systems, confirming proposed solutions correctness, are given.