Nonlinear Approximation Scheme Research Articles

Error estimate of the cell-centered nonlinear positivity-preserving two-point flux approximation schemes

We present the error estimate of nonlinear positivity-preserving finite volume schemes for diffusion problems. These schemes are also named by nonlinear two-point flux approximation (NTPFA) methods. Due to their conservation and monotonicity, such schemes have attracted wide attention. Under suitable regularity assumptions on the exact solution, we prove a discrete H1 error estimate through the analysis of consistent errors. Some numerical examples are presented to demonstrate the theoretical results.

SUHMO: an adaptive mesh refinement SUbglacial Hydrology MOdel v1.0

Abstract. Water flowing under ice sheets and glaciers can have a strong influence on ice dynamics, particularly through pressure changes, suggesting that a comprehensive ice sheet model should include the effect of basal hydrology. Modeling subglacial hydrology remains a challenge, however, mainly due to the range of spatial and temporal scales involved – from subglacial channels to vast subglacial lakes. Additionally, networks of subglacial drainage channels dynamically evolve over time. To address some of these challenges, we have developed an adaptive mesh refinement (AMR) model based on the Chombo software framework. We extend the model proposed by Sommers et al. (2018) with a small but significant change to accommodate the transition from unresolved to resolved flow features. We handle the strong nonlinearities present in the equations by resorting to an efficient nonlinear full approximation scheme multigrid (FAS-MG) algorithm. We outline the details of the algorithm and present convergence analysis results demonstrating its good performance. Additionally, we present results validating our approach, using test cases from the Subglacial Hydrology Model Intercomparison Project (SHMIP) (de Fleurian et al., 2018). We finish by presenting a more complex, 100 km-by-100 km synthetic test case with peaks and valleys that we use to investigate the effective pressure distribution as the number of AMR levels increases. These preliminary results suggest that a minimum spatial resolution is needed to properly capture channel features, but additional work is required to precisely quantify this and its impact on accurately modeling the coupled ice sheet–hydrology system. The efficiency of our approach, relying on localized refinement, is also demonstrated. Future work will include coupling the SUbglacial Hydrology MOdel (SUHMO) with the BISICLES AMR ice sheet model (Cornford et al., 2013), both built on the same numerical framework.

Construction of Nonlinear Approximation Schemes for Piecewise Smooth Data

Construction of Nonlinear Approximation Schemes for Piecewise Smooth Data

Regularity in Sobolev and Besov Spaces for Parabolic Problems on Domains of Polyhedral Type

This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in Dahlke and Schneider (Anal Appl 17(2):235–291, 2019, Thms. 4.5, 4.9, 4.12, 4.14) to domains of polyhedral type. In particular, we study the smoothness in the specific scale B^r_{tau ,tau }, frac{1}{tau }=frac{r}{d}+frac{1}{p} of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms.

Stabilizing Q Learning Via Soft Mellowmax Operator

Learning complicated value functions in high dimensional state space by function approximation is a challenging task, partially due to that the max-operator used in temporal difference updates can theoretically cause instability for most linear or non-linear approximation schemes. Mellowmax is a recently proposed differentiable and non-expansion softmax operator that allows a convergent behavior in learning and planning. Unfortunately, the performance bound for the fixed point it converges to remains unclear, and in practice, its parameter is sensitive to various domains and has to be tuned case by case. Finally, the Mellowmax operator may suffer from oversmoothing as it ignores the probability being taken for each action when aggregating them. In this paper we address all the above issues with an enhanced Mellowmax operator, named SM2 (Soft Mellowmax). Particularly, the proposed operator is reliable, easy to implement, and has provable performance guarantee, while preserving all the advantages of Mellowmax. Furthermore, we show that our SM2 operator can be applied to the challenging multi-agent reinforcement learning scenarios, leading to stable value function approximation and state of the art performance.

Projection-based monotone embedded discrete fracture method for flow and transport in porous media

The extension of the monotone embedded discrete fracture method (monotone EDFM) (Nikitin and Yanbarisov, 2020) is considered in this work. The new method is based on the two coupled discretization schemes: projection-based EDFM (pEDFM) (Tene et al., 2017) and nonlinear compact multi-point flux approximation scheme (Lipnikov et al., 2012). The main novelty of the extended method is the satisfaction of the discrete maximum principle and applicability to the anisotropic porous media with conductive and blocking fractures. The resulting projection-based monotone EDFM combines effectiveness and simplicity of pEDFM approach with accuracy and physical relevance of the nonlinear FV schemes on non-orthogonal grids and anisotropic media.

A Computational Workflow for Flow and Transport in Fractured Porous Media Based on a Hierarchical Nonlinear Discrete Fracture Modeling Approach

Modeling flow and transport in fractured porous media has been a topic of intensive research for a number of energy- and environment-related industries. The presence of multiscale fractures makes it an extremely challenging task to resolve accurately and efficiently the flow dynamics at both the local and global scales. To tackle this challenge, we developed a computational workflow that adopts a two-level hierarchical strategy based on fracture length partitioning. This was achieved by specifying a partition length to split the discrete fracture network (DFN) into small-scale fractures and large-scale fractures. Flow-based numerical upscaling was then employed to homogenize the small-scale fractures and the porous matrix into an equivalent/effective single medium, whereas the large-scale fractures were modeled explicitly. As the effective medium properties can be fully tensorial, the developed hierarchical framework constructed the discrete systems for the explicit fracture–matrix sub-domains using the nonlinear two-point flux approximation (NTPFA) scheme. This led to a significant reduction of grid orientation effects, thus developing a robust, applicable, and field-relevant framework. To assess the efficacy of the proposed hierarchical workflow, several numerical simulations were carried out to systematically analyze the effects of the homogenized explicit cutoff length scale, as well as the fracture length and orientation distributions. The effect of different boundary conditions, namely, the constant pressure drop boundary condition and the linear pressure boundary condition, for the numerical upscaling on the accuracy of the workflow was investigated. The results show that when the partition length is much larger than the characteristic length of the grid block, and when the DFN has a predominant orientation that is often the case in practical simulations, the workflow employing linear pressure boundary conditions for numerical upscaling give closer results to the full-model reference solutions. Our findings shed new light on the development of meaningful computational frameworks for highly fractured, heterogeneous geological media where fractures are present at multiple scales.

Bifurcation analysis and circuit implementation for a tabu learning neuron model

This paper investigates two-dimensional non-autonomous tabu learning neuron model. The equilibrium state is modulated by an externally applied input and the stability transitions of the stable and unstable equilibrium states via fold and Hopf bifurcations are disclosed. Complex neuron firing activities such as chaotic/periodic spiking/bursting firing patterns and coexisting bi-stable firing patterns are revealed by several numerical measures. Moreover, a nonlinear approximation scheme is presented to implement the neuron model using analog discrete components, which is achieved by utilizing an approximation activation function to replace the original activation function. Based on the circuit implementation, Multisim circuit simulations are performed to validate the feasibility of nonlinear approximation scheme and correctness of the numerical simulations.

On Besov regularity of solutions to nonlinear elliptic partial differential equations

In this paper, we study the regularity of the solutions of some nonlinear elliptic equations in Kondratiev spaces on certain domains of polyhedral type. General embedding theorems between Kondratiev spaces and Besov spaces will allow to avoid drawbacks to the standard Sobolev regularity theory for those nonsmooth domains. This will give us the opportunity to derive optimal rates for certain nonlinear approximation schemes.

Besov regularity of parabolic and hyperbolic PDEs

This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations on nonsmooth domains. In particular, we study the smoothness in the specific scale [Formula: see text] of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms.

Besov regularity for the stationary Navier–Stokes equation on bounded Lipschitz domains

We use the scale , , , to study the regularity of the stationary Stokes equation on bounded Lipschitz domains , , with connected boundary. The regularity in these Besov spaces determines the order of convergence of nonlinear approximation schemes. Our proofs rely on a combination of weighted Sobolev estimates and wavelet characterizations of Besov spaces. Using Banach’s fixed point theorem, we extend this analysis to the stationary Navier–Stokes equation with suitable Reynolds number and data, respectively.

An efficient, unconditionally energy stable local discontinuous Galerkin scheme for the Cahn–Hilliard–Brinkman system

In this paper, we present an efficient and unconditionally energy stable fully-discrete local discontinuous Galerkin (LDG) method for approximating the Cahn–Hilliard–Brinkman (CHB) system, which is comprised of a Cahn–Hilliard type equation and a generalized Brinkman equation modeling fluid flow. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction (Δt=O(Δx4)) of explicit time discretization methods for stability, we introduce a semi-implicit scheme which consists of the implicit Euler method combined with a convex splitting of the discrete Cahn–Hilliard energy strategy for the temporal discretization. The unconditional energy stability of this fully-discrete convex splitting scheme is also proved. Obviously, the fully-discrete equations at the implicit time level are nonlinear, and to enhance the efficiency of the proposed approach, the nonlinear Full Approximation Scheme (FAS) multigrid method has been employed to solve this system of algebraic equations. We also show the nearly optimal complexity numerically. Numerical experiments based on the overall solution method of combining the proposed LDG method, convex splitting scheme and the nonlinear multigrid solver are given to validate the theoretical results and to show the effectiveness of the proposed approach for the CHB system.

PHLST with adaptive tiling and its application to antarctic remote sensing image approximation

We propose an efficient nonlinear approximation scheme using thePolyharmonic Local Sine Transform (PHLST) of Saito and Remy combinedwith an algorithm to tile a given image automatically and adaptivelyaccording to its local smoothness and singularities. To measure suchlocal smoothness, we introduce the so-called local Besov indices ofan image, which is based on the pointwise modulus of smoothness ofthe image. Such an adaptive tiling of an image is important forimage approximation using PHLST because PHLST stores the corner andboundary information of each tile and consequently it is wasteful todivide a smooth region of a given image into a set of smaller tiles.We demonstrate the superiority of the proposed algorithm usingAntarctic remote sensing images over the PHLST using the uniformtiling. Analysis of such images including their efficientapproximation and compression has gained its importance due to theglobal climate change.

An efficient fully-discrete local discontinuous Galerkin method for the Cahn–Hilliard–Hele–Shaw system

In this paper, we develop an efficient and energy stable fully-discrete local discontinuous Galerkin (LDG) method for the Cahn–Hilliard–Hele–Shaw (CHHS) system. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction (Δt=O(Δx4)) of explicit time discretization methods for stability, we introduce a semi-implicit time integration scheme which is based on a convex splitting of the discrete Cahn–Hilliard energy. The unconditional energy stability has been proved for this fully-discrete LDG scheme. The fully-discrete equations at the implicit time level are nonlinear. Thus, the nonlinear Full Approximation Scheme (FAS) multigrid method has been applied to solve this system of algebraic equations, which has been shown the nearly optimal complexity numerically. Numerical results are also given to illustrate the accuracy and capability of the LDG method coupled with the multigrid solver.

Efficient standing-wave model for DFB lasers by nonlinear carrier approximation

A nonlinear approximation scheme is proposed to describe the time-dependent longitudinal carrier distribution with a view toward finding the efficient solution scheme of the previously proposed standing-wave model for simulation of DFB lasers. It shows its advantage over the existing linear scheme in terms of the modal accuracy in cases where the optical power is large enough to trigger the saturation effect. The validity of this improved scheme is demonstrated through comparisons made on the static and dynamic performances of \(\lambda /4\)-phase shifted DFB lasers.

On the $L_q(L_p)$-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $O \subset R^d$ with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobolev spaces $\mathfrak{H}^{\gamma,q}_{p,\theta}(O;T)$. The summability parameters p and q in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the H\"older regularity in time is analysed. Moreover, we prove a general embedding of weighted Lp(O)-Sobolev spaces into the scale of Besov spaces $B^\alpha_{\tau,\tau}(O), 1/\tau=\alpha/d+1/p, \alpha > 0$. This leads to a H\"older-Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.

Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains

We study the spatial regularity of semilinear parabolic stochastic partial differential equations on bounded Lipschitz domains 𝒪⊆ ℝ d in the scale , 1/τ=α/d+1/p, p≥2 fixed. The Besov smoothness in this scale determines the order of convergence that can be achieved by adaptive numerical algorithms and other nonlinear approximation schemes. The proofs are performed by establishing weighted Sobolev estimates and combining them with wavelet characterizations of Besov spaces.

Rate Distortion Behavior of Sparse Sources

The rate distortion behavior of sparse memoryless sources is studied. These serve as models of sparse signal representations and facilitate the performance analysis of “sparsifying” transforms like the wavelet transform and nonlinear approximation schemes. For strictly sparse binary sources with Hamming distortion, R(D) is shown to be almost linear. For nonstrictly sparse continuous-valued sources, termed compressible, two measures of compressibility are introduced: incomplete moments and geometric mean. The former lead to low- and high-rate upper bounds on mean squared error D(R), while the latter yields lower and upper bounds on source entropy, thereby characterizing asymptotic R(D) behavior. Thus, the notion of compressibility is quantitatively connected with actual lossy compression. These bounding techniques are applied to two source models: Gaussian mixtures and power laws matching the approximately scale-invariant decay of wavelet coefficients. The former are versatile models for sparse data, which in particular allow to bound high-rate compression performance of a scalar mixture compared to a corresponding unmixed transform coding system. Such a comparison is interesting for transforms with known coefficient decay, but unknown coefficient ordering, e.g., when positions of highest-variance coefficients are unknown. The use of these models and results in distributed coding and compressed sensing scenarios are also discussed.

Dynamic Programming and Value-Function Approximation in Sequential Decision Problems: Error Analysis and Numerical Results

Value-function approximation is investigated for the solution via Dynamic Programming (DP) of continuous-state sequential N-stage decision problems, in which the reward to be maximized has an additive structure over a finite number of stages. Conditions that guarantee smoothness properties of the value function at each stage are derived. These properties are exploited to approximate such functions by means of certain nonlinear approximation schemes, which include splines of suitable order and Gaussian radial-basis networks with variable centers and widths. The accuracies of suboptimal solutions obtained by combining DP with these approximation tools are estimated. The results provide insights into the successful performances appeared in the literature about the use of value-function approximators in DP. The theoretical analysis is applied to a problem of optimal consumption, with simulation results illustrating the use of the proposed solution methodology. Numerical comparisons with classical linear approximators are presented.

On Besov regularity of solutions to nonlinear elliptic partial differential equations

In this paper, we study the regularity of the solutions to nonlinear elliptic equations. In particular, we are interested in smoothness estimates in the specific scale \(B^{\alpha}_{\tau}(L_{\tau})\), τ=(α/d+1/2)−1, of Besov spaces which determines the approximation order of adaptive and other nonlinear numerical approximation schemes with respect to the L 2-norm. We show that the Besov regularity is high enough to justify the use of adaptive schemes.