Optimal Estimates Research Articles

Asymptotically efficient Quasi-Newton type identification with quantized observations under bounded persistent excitations

This paper is concerned with the optimal identification problem of dynamical systems in which only quantized output observations are available under the assumption of fixed thresholds and bounded persistent excitations. Based on a time-varying projection, a weighted Quasi-Newton type projection (WQNP) algorithm is proposed. With some mild conditions on the weight coefficients, the algorithm is proved to be mean square and almost surely convergent, and the convergence rate can be the reciprocal of the number of observations, which is the same order as the optimal estimate under accurate measurements. Furthermore, inspired by the structure of the Cramér–Rao lower bound, an information-based identification (IBID) algorithm is constructed with an adaptive design about weight coefficients of the WQNP algorithm, where the weight coefficients are related to the parameter estimates which leads to the essential difficulty of algorithm analysis. Beyond the convergence properties, this paper demonstrates that the IBID algorithm tends asymptotically to the Cramér–Rao lower bound, and hence is asymptotically efficient. A numerical example is simulated to show the effectiveness of the proposed algorithms.

The Effect of Foreign Direct Investment and Economic Growth on Health: An Empirical Investigation for Türkiye

This study aims to investigate the effect of foreign direct investment (FDI) on health for the period of 1975-2018 in Türkiye by using a health production function approach. Life expectancy at birth which is the dependent variable in the study is used as a proxy for the average health status of Turkish citizens. FDI is defined as the ratio of foreign direct investment to GDP. In order to provide optimal estimates of cointegrating relationship, the Fully Modified OLS (FMOLS) developed by Phillips & Hansen (1990) is preferred as the main estimation method, as it allows for endogeneity of explanatory variables. The findings of the study point out that foreign direct investment inflows reduce life expectancy in Türkiye.

Discrete anisotropic curve shortening flow in higher codimension

Abstract We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${{\mathbb{R}}}^{d}$, $d\geq 2$. The reformulation hinges on a suitable manipulation of the parameterization’s tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method. For a fully discrete finite element approximation based on piecewise linear elements we prove optimal error estimates. Numerical simulations confirm the theoretical results and demonstrate the practicality of the method.

A stabilized SAV difference scheme and its accelerated solver for spatial fractional Cahn–Hilliard equations

A novel energy-stable scheme is proposed to solve the spatial fractional Cahn–Hilliard equations, using the idea of scalar auxiliary variable (SAV) approach and stabilization technique. Thanks to the stabilization technique, it is shown that larger temporal stepsizes can be applied in numerical simulations. Moreover, the proposed SAV finite difference scheme is non-coupled and linearly implicit, which can be efficiently solved by the preconditioned conjugate gradient (PCG) method with a sine transform based preconditioner. Optimal error estimates of the fully-discrete scheme are obtained rigorously. Numerical examples are given to confirm the theoretical results and show the higher efficiency of the proposed scheme than the previous SAV schemes.

Single trial Bayesian inference by population vector readout in the barn owl's sound localization system.

Bayesian models have proven effective in characterizing perception, behavior, and neural encoding across diverse species and systems. The neural implementation of Bayesian inference in the barn owl's sound localization system and behavior has been previously explained by a non-uniform population code model. This model specifies the neural population activity pattern required for a population vector readout to match the optimal Bayesian estimate. While prior analyses focused on trial-averaged comparisons of model predictions with behavior and single-neuron responses, it remains unknown whether this model can accurately approximate Bayesian inference on single trials under varying sensory reliability, a fundamental condition for natural perception and behavior. In this study, we utilized mathematical analysis and simulations to demonstrate that decoding a non-uniform population code via a population vector readout approximates the Bayesian estimate on single trials for varying sensory reliabilities. Our findings provide additional support for the non-uniform population code model as a viable explanation for the barn owl's sound localization pathway and behavior.

An Augmented Mixed DG Scheme for the Electric Field

In this paper, a new augmented mixed DG formulation for the numerical approximation of the electrostatic field was introduced and studied. Its error analysis was carried out and an optimal error estimates as a function of the mesh size was obtained. Some numerical tests confirming the theoretical convergence were given.

Optimal Approximation of Unique Continuation

AbstractWe consider numerical approximations of ill-posed elliptic problems with conditional stability. The notion of optimal error estimates is defined including both convergence with respect to discretisation and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the approximation space. A proof is given that no approximation can converge at a better rate than that given by the definition without increasing the sensitivity to perturbations, thus justifying the concept. A recently introduced class of primal-dual finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be optimal in the sense of this definition.

Fully decoupled, linearized and stabilized finite volume method for the time-dependent incompressible MHD equations

In this paper, we consider the stability and convergence of the fully decoupled and linearized numerical scheme for the time-dependent incompressible magnetohydrodynamic equations based on the finite volume method. The lowest equal-order mixed finite element pair (P1-P1-P1) is used to approximate the velocity, pressure and magnetic fields, and the pressure projection stabilization is introduced to bypass the restriction of the discrete inf-sup condition. The semi-implicit treatment is used to linearize the nonlinear terms and the first order projection scheme is adopted to split the velocity and pressure, then a series of fully decoupled and linearized subproblems are formed. From the view of theoretical analysis, some novel stability results of the numerical solutions in both spatial semi-discrete scheme and time–space fully discrete scheme are provided, the optimal error estimates are also presented by using the energy method and choosing different test functions. From the view of computational results, the fully decoupled and linearized numerical scheme not only keeps good accuracy, but also saves a lot of computational cost.

An implicit–explicit relaxation extrapolated Runge–Kutta and energy-preserving finite element method for Klein–Gordon–Schrödinger equations

An implicit–explicit (IMEX) relaxation extrapolated Runge–Kutta (RERK) and energy-preserving finite element method is designed for Klein–Gordon–Schrödinger (KGS) equations with periodic boundary conditions. First, the RERK method is employed to the discretization in time, which can keep the unconditionally stable and energy conservation after selecting the appropriate relaxation parameters γn. An IMEX time-marching discretization is constructed so that only a system of linear equations needs to be solved at any time step. Next, the finite element method is utilized to the discretization in space. By the temporal–spatial splitting technique, optimal error estimates in the L2-norm and H1-norm are obtained with the convergent order O(τs+hk+1) and O(τs+hk), without any time stepsize restrictions. At last, some numerical examples are presented to demonstrate the theoretical results.

Optimal error estimates of a decoupled finite element scheme for the unsteady inductionless MHD equations

AbstractThis article focuses on a new and optimal error analysis of a decoupled finite element scheme for the inductionless magnetohydrodynamic (MHD) equations. The method uses the classical inf‐sup stable Mini/Taylor‐Hood (Mini/TH) finite element pairs to appropriate the velocity and pressure, and Raviart–Thomas (RT) face element to discretize the current density spatially, and the semi‐implicit Euler scheme with an additional stabilized term and some delicate implicit–explicit handling for the coupling terms temporally. The method enjoys some impressive features that it is linear, decoupled, unconditional energy stable and charge‐conservative. Due to the errors from the explicit handing of the coupling terms and the existence of the artificial stabilized term, and the contamination of the lower‐order RT face discretization in the error analysis, the existing theoretical results are not unconditional and optimal. By utilizing the anti‐symmetric structure of the coupling terms and the existence of the extra dissipative term, and the negative‐norm estimate for the mixed Poisson projection, we establish the unconditional and optimal error estimates for all the variables. Numerical tests are presented to illustrate our theoretical findings.

The posteriori analysis of the spectral element discretization of the wave equation

This work focuses on discretizing a second-order linear wave equation using the implicit Euler scheme for time discretization and the spectral element method for spatial discretization. We prove that optimal adaptivity can be achieved by combining adaptive time steps with adaptive spectral mesh. We introduce two sets of error indicators for time and space, respectively, and derive optimal estimates.

Optimal convergence analysis of two RPC-SAV schemes for the unsteady incompressible magnetohydrodynamics equations

Abstract In this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressure-correction (RPC) approach is adopted to decouple the system, and the recently developed scalar auxiliary variable (SAV) method is used to treat the nonlinear terms explicitly and keep energy stability. One is the first-order RPC-SAV-Euler and the other one is generalized Crank–Nicolson-type scheme: GRPC-SAV-CN. For the RPC-SAV-Euler scheme, both unconditionally energy stability and optimal convergence are derived. The new GRPC-SAV-CN is constructed and can be regarded as a parameterized scheme, which includes PC-SAV-CN when the parameter $\beta =0$ and RPC-SAV-CN when $\beta \in (0,\frac {1}{2}]$; see Algorithm 3.2. However, Jiang and Yang (Jiang, N. & Yang, H. (2021) SIAM J. Sci. Comput., 43, A2869–A2896) point out that the SAV method has low accuracy by several commonly tested benchmark flow problem when solving Navier–Stokes equations. To improve the accuracy, we added two stabilization $-\alpha _{1}\varDelta t\nu \varDelta (\widetilde {\textbf {u}}^{n+1}-{\textbf {u}}^{n})$ and $\alpha _{2}\varDelta t\sigma ^{-1}\mbox {curl}\mbox {curl} (\textbf {H}^{n+1}-\textbf {H}^{n})$ in the GRPC-SAV-CN scheme, which play decisive roles in giving optimal error estimates. The unconditionally energy stability of the proposed scheme is given. We prove that the PC-SAV-CN scheme has second-order convergence speed, and the RPC-SAV-CN one has 1.5-order convergence rate. Finally, some numerical examples are presented to verify the validity and convergence of the numerical schemes.

Optimal error estimates of a second-order fully decoupled finite element method for the nonstationary generalized Boussinesq model

In this paper, we develop and analyze a fully decoupled finite element method for the non-stationary generalized Boussinesq equations, where the viscosity and thermal conductivity depend on the temperature. Based on some subtle implicit-explicit treatments for the nonlinear coupling terms, we develop a second-order in time, fully decoupled, linear and unconditionally energy stable scheme for solving this system. The unconditional stability of the fully discrete scheme with finite element approximation is proved. The optimal L2-error estimates are analyzed for temperature-dependent thermal conductivity system. Numerical experiments are presented to illustrate the convergence, accuracy and applicability of the proposed numerical scheme.

Improving Tree Cover Estimation for Sparse Trees Mixed with Herbaceous Vegetation in Drylands Using Texture Features of High-Resolution Imagery

Tree cover is a crucial vegetation structural parameter for simulating ecological, hydrological, and soil erosion processes on the Chinese Loess Plateau, especially after the implementation of the Grain for Green project in 1999. However, current tree cover products performed poorly across most of the Loess Plateau, which is characterized by grasslands with sparse trees. In this study, we first acquired high-accuracy samples of 0.5 m tree canopy and 30 m tree cover using a combination of unmanned aerial vehicle imagery and WorldView-2 (WV-2) imagery. The spectral and textural features derived from Landsat 8 and WV-2 were then used to estimate tree cover with a random forest model. Finally, the tree cover estimated using WV-2, Landsat 8, and their combination were compared, and the optimal tree cover estimates were also compared with current products and tree cover derived from canopy classification. The results show that (1) the normalized difference moisture index using Landsat 8 shortwave infrared and the standard deviation of correlation metric calculated by means of gray-level co-occurrence matrix using the WV-2 near-infrared band are the optimal spectral feature and textural feature for estimating tree cover, respectively. (2) The accuracy of tree cover estimated using only WV-2 is highest (RMSE = 7.44%), indicating that high-resolution textural features are more sensitive to tree cover than the Landsat spectral features (RMSE = 11.53%) on grasslands with sparse trees. (3) Textural features with a resolution higher than 8 m perform better than the combination of Landsat 8 and textural features, and the optimal resolution is 2 m (RMSE = 7.21%) for estimating tree cover, whereas the opposite is observed when the resolution of textural features is lower than 8 m. (4) The current global product seriously underestimates tree cover on the Loess Plateau, and the tree cover calculation using the canopy classification of high-resolution imagery performs worse than the method of directly using remote sensing features.

Local Path Planner for Mobile Robot Considering Future Positions of Obstacles

Local path planning is a necessary ability for mobile robot navigation, but existing planners are not sufficiently effective at dynamic obstacle avoidance. In this article, an improved timed elastic band (TEB) planner based on the requirements of mobile robot navigation in dynamic environments is proposed. The dynamic obstacle velocities and TEB poses are fully integrated through two-dimensional (2D) lidar and multi-obstacle tracking. First, background point filtering and clustering are performed on the lidar points to obtain obstacle clusters. Then, we calculate the data association matrix of the obstacle clusters of the current and previous frame so that the clusters can be matched. Thirdly, a Kalman filter is adopted to track clusters and obtain the optimal estimates of their velocities. Finally, the TEB poses and obstacle velocities are associated: we predict the obstacle position corresponding to the TEB pose through the detected obstacle velocity and add this constraint to the corresponding TEB pose vertex. Then, a pose sequence considering the future positions of obstacles is obtained through a graph optimization algorithm. Compared with the original TEB, our method reduces the total running time by 22.87%, reduces the running distance by 19.23%, and increases the success rate by 21.05%. Simulations and experiments indicate that the improved TEB enables robots to efficiently avoid dynamic obstacles and reach the goal as quickly as possible.

Convergence analysis of a second order numerical scheme for the Flory–Huggins–Cahn–Hilliard–Navier–Stokes system

We present an optimal rate convergence analysis for a second order accurate in time, fully discrete finite difference scheme for the Cahn–Hilliard–Navier–Stokes (CHNS) system, combined with logarithmic Flory–Huggins energy potential. The numerical scheme has been recently proposed, and the positivity-preserving property of the logarithmic arguments, as well as the total energy stability, have been theoretically justified. In this paper, we rigorously prove second order convergence of the proposed numerical scheme, in both time and space. Since the CHNS is a coupled system, the standard ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh2) error estimate could not be easily derived, due to the lack of regularity to control the numerical error associated with the coupled terms. Instead, the ℓ∞(0,T;Hh1)∩ℓ2(0,T;Hh3) error analysis for the phase variable and the ℓ∞(0,T;ℓ2) analysis for the velocity vector, which shares the same regularity as the energy estimate, is more suitable to pass through the nonlinear analysis for the error terms associated with the coupled physical process. Furthermore, the highly nonlinear and singular nature of the logarithmic error terms makes the convergence analysis even more challenging, since a uniform distance between the numerical solution and the singular limit values of is needed for the associated error estimate. Many highly non-standard estimates, such as a higher order asymptotic expansion of the numerical solution (up to the third order accuracy in time and fourth order in space), combined with a rough error estimate (to establish the maximum norm bound for the phase variable), as well as a refined error estimate, have to be carried out to conclude the desired convergence result. To our knowledge, it will be the first work to establish an optimal rate convergence estimate for the Cahn–Hilliard–Navier–Stokes system with a singular energy potential.

Variational data assimilation with finite-element discretization for second-order parabolic interface equation

Abstract In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.

Optimal [formula omitted] error estimates of stabilizer-free weak Galerkin finite element method for the drift-diffusion problem

We continue our effort in Li et al. (2024) to develop a stabilizer-free weak Galerkin finite element method (SFWG-FEM) for the two-dimensional unsteady-state drift-diffusion (DD) model. The transient state equation includes both the first derivative convection and the second derivative diffusion terms, and it couples with a Poisson potential equation. The piecewise Pk(k≥1) polynomials elements in the interior and on the boundaries are utilized for both electron concentration and electric potential functions. The vector function of [Pj(K)]2(j≥j0) for the discrete weak gradient space is used on each element K. The main difficulty is the treatment of the non-linearity and coupling of the system. Based on an introduced Ritz projection and two L2 projection operators, optimal error estimates of O(Δt+hk+1) in the L2 norm and O(Δt+hk) in the energy-like norm are derived. Numerical experiments are presented to illustrate the theoretical analysis.

Convergence analysis of a decoupled pressure-correction SAV-FEM for the Cahn–Hilliard–Navier–Stokes model

In this paper, a linear, decoupled and unconditional energy-stable numerical scheme of Cahn–Hilliard–Navier–Stokes (CHNS) model is constructed and analyzed. We reformulate the CHNS model into an equivalent system based on the scalar auxiliary variable approach. The Euler implicit/explicit scheme is used for time discretization, that is, linear terms are implicitly processed, nonlinear terms are explicitly processed, and pressure-correction method of the Navier–Stokes equations is also adopted. In this way, we achieve the decoupling of phase field, velocity and pressure, and only need to solve a series of linear equations with constant coefficients at each time step, thereby improving computational efficiency. Then the finite element method is used for space discretization, and the finite element spaces of phase field, velocity and pressure are taken as Pl−Pl−Pl−1 respectively. We also verify that the proposed scheme satisfies the discrete energy dissipation law. The optimal error estimates of the fully discrete scheme is proved, which the phase field, velocity and pressure satisfy the first-order accuracy in time and the (ℓ+1,ℓ+1,ℓ)th order accuracy in space, respectively. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients

Abstract The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ \mathcal{D} \in{\mathbb{R}}^{d}$, $ d \leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d \le 3$, under more relaxed assumptions on the stochastic convolution. This improves relevant results in the literature and covers both the space-time white noise ($d=1$) and the trace-class noises ($\text{Tr} (Q) < \infty $) in multiple dimensions $d=2,3$. Such an improvement is achieved based on a key perturbation estimate for a perturbed PDE, with the aid of which we prove the convergence and uniform regularity of a spectral approximation of the SPDEs and thus get the improved regularity results. The second contribution of the paper is to propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linearly implicit nonlinearity-tamed Euler method for the temporal discretization. The proposed time-stepping scheme is linearly implicit and does not suffer from solving nonlinear algebra equations as the backward Euler scheme does. Based on the improved regularity results, we recover the expected strong convergence rates of the fully discrete scheme and reveal how the convergence rates rely on the regularity of the noise process. In particular, a classical convergence rate of order $O(h^{2} +\tau )$ can be obtained even in high dimension $d=3$, as the driven noise is of trace class and satisfies certain regularity assumptions. The optimal error estimates turn out to be challenging and face some essential difficulties when the tamed time-stepping scheme meets the finite element spatial discretization, particularly in the context of low regularity and multiple dimensions $d \le 3$. Some highly nontrivial arguments are introduced to overcome the difficulties. Finally, numerical examples corroborate the claimed strong orders of convergence.