Adaptive Wavelet Collocation Method Research Articles

Wavelet Transforms and Machine Learning Methods for the Study of Turbulence

This article investigates the applications of wavelet transforms and machine learning methods in studying turbulent flows. The wavelet-based hierarchical eddy-capturing framework is built upon first principle physical models. Specifically, the coherent vortex simulation method is based on the Taylor hypothesis, which suggests that the energy cascade occurs through vortex stretching. In contrast, the adaptive wavelet collocation method relies on the Richardson hypothesis, where the self-amplification of the strain field and a hierarchical breakdown of large eddies drive the energy cascade. Wavelet transforms are computational learning architectures that propagate the input data across a sequence of linear operators to learn the underlying nonlinearity and coherent structure. Machine learning offers a wealth of data-driven algorithms that can heavily use statistical concepts to extract valuable insights into turbulent flows. Supervised machine learning needs “perfect” turbulent flow data to train data-driven turbulence models. The current advancement of artificial intelligence in turbulence modeling primarily focuses on accelerating turbulent flow simulations by learning the underlying coherence over a low-dimensional manifold. Physics-informed neural networks offer a fertile ground for augmenting first principle physics to automate specific learning tasks, e.g., via wavelet transforms. Besides machine learning, there is room for developing a common computational framework to provide a rich cross-fertilization between learning the data coherence and the first principles of multiscale physics.

A particle resolved simulation approach for studying shock interactions with moving, colliding solid particles

This work applies a new combination of techniques for the fully resolved simulation of compressible, gas–particle multiphase flows. The adaptive wavelet collocation method is used to dynamically, and efficiently, adapt the computational grid to localized flow features and the particles. A characteristic-based volume penalization method that imposes arbitrary Dirichlet, Neumann, or Robin-type immersed boundary conditions, is used to enforce the no-slip condition at particle surfaces. A hard-sphere collision model is applied to capture the particle–particle collisions. Proof of concept test cases are presented, showcasing the dynamic grid adaptation and fully resolved two-way coupling between the phases that is possible with this approach. Results for a shock-driven single cylinder under viscous and inviscid conditions are presented along with a demonstration of a shock interacting with a cloud of randomly distributed cylinders and spheres.

An adaptive wavelet collocation method for the optimal heat source problem

PurposeThis paper aims to propose a new adaptive numerical method to find more accurate numerical solution for the heat source optimal control problem (OCP).Design/methodology/approachThe main aim of this paper is to present an adaptive collocation approach based on the interpolating wavelets to solve an OCP for finding optimal heat source, in a two-dimensional domain. This problem arises when the domain is heated by microwaves or by electromagnetic induction.FindingsThis paper shows that combination of interpolating wavelet basis and finite difference method makes an accurate structure to design adaptive algorithm for such problems which usually have non-smooth solution.Originality/valueThe proposed numerical technique is flexible for different OCP governed by a partial differential equation with box constraint over the control or the state function.

Computational Fluid Dynamics Using the Adaptive Wavelet-Collocation Method

Advancements to the adaptive wavelet-collocation method over the last decade have opened up a number of new possible areas for active research. Volume penalization techniques allow complex immersed boundary conditions to be used with high efficiency for both internal and external flows. Anisotropic methods make it possible to use body-fitted meshes while still taking advantage of the dynamic adaptability properties wavelet-based methods provide. The parallelization of the approach has made it possible to perform large high-resolution simulations of detonation initiation and fluid instabilities to uncover new physical insights that would otherwise be difficult to discover. Other developments include space-time adaptive methods and nonreflecting boundary conditions. This article summarizes the work performed using the adaptive wavelet-collocation method developed by Vasilyev and coworkers over the past decade.

Hierarchical Adaptive Eddy-Capturing Approach for Modeling and Simulation of Turbulent Flows

A short review of wavelet-based adaptive methods for modeling and simulation of incompressible turbulent flows is presented. Wavelet-based computational modeling approaches of different fidelities are recast into an integrated hierarchical adaptive eddy-capturing turbulence modeling framework. The wavelet threshold filtering procedure and the associated wavelet-filtered Navier–Stokes equations are briefly discussed, along with the adaptive wavelet collocation method that is used for numerical computations. Depending on the level of wavelet thresholding, the simulation is possibly supplemented with a localized closure model. The latest advancements in spatiotemporally varying wavelet thresholding procedures along with the adaptive-anisotropic wavelet-collocation method make the development of a fully adaptive approach feasible with potential applications for complex turbulent flows.

Wavelet-based adaptive large-eddy simulation of supersonic channel flow

Abstract

An Adaptive Wavelet Collocation Method for Solution of the Convection-Dominated Problem on a Sphere

In this paper, we present a numerical simulation of a convection-dominated problem using adaptive wavelet collocation method. This is based on second-generation spherical wavelets on a recursively refined spherical geodesic grid. The power lies in the fact that they only require small number of coefficients to represent this problem more accurately, which allows compression and efficient computation. In contrast with other approximate schemes, this approach reduced the number of collocation points used by factors of many orders of magnitude when compared to uniform grids of equivalent resolution. Numerical results demonstrate the efficiency of our approach.

Stabilized conservative level set method

This paper addresses one of the main challenges of the conservative level set method, namely the ill-conditioned behavior of the normal vector away from the interface. An alternative formulation for reconstruction of the interface is proposed. Unlike commonly used methods, which rely on a unit normal vector, the Stabilized Conservative Level Set (SCLS) makes use of a modified normal vector with diminishing magnitude away from the interface. With the new formulation, in the vicinity of the interface the reinitialization procedure utilizes compressive flux and diffusive terms only in normal direction with respect to the interface, thus, preserving the conservative level set properties, while away from the interface the directional diffusion mechanism automatically switches to homogeneous diffusion. The proposed formulation is robust and general. It is especially well suited for use with the adaptive mesh refinement (AMR) approaches, since for computational accuracy high resolution is only required in the vicinity of the interface, while away from the interface low resolution simulations might be sufficient. All of the results reported in this paper are obtained using the Adaptive Wavelet Collocation Method, a general arbitrary order AMR-type method, which utilizes wavelet decomposition to adapt on steep gradients in the solution while retaining a predetermined order of accuracy. Numerical solution for a number of benchmark problems has been carried out to demonstrate the performance of the SCLS method.

Wavelet multi-resolution approximation for multiobjective optimal control.

A new sequential method based on multi-resolution approximation is proposed for solving computationally expensive multi-objective optimization problems. A traditional strategy is to decompose a multi-objective optimization problem into a number of single-objective optimization problems, whereby the PF can be regarded as a function of weights. Therefore, it is very natural to use wavelet multi-resolution approximation techniques for setting weight vectors. In our framework, the sequential approach starts with sampling aggressive functions on the initial coarsest grid with a few collocation points; once a rough PF is obtained, new points are automatically added on the basis of an adaptive wavelet collocation method. Therefore, the PF can be approximated with a relatively small number of weights. The efficiency of our method is demonstrated on two examples: a typical multi-objective optimization problem and an expensive multi-objective control optimal problem.

An adaptive wavelet collocation method for solving optimal control of elliptic variational inequalities of the obstacle type

This paper presents a fast computational technique based on the wavelet collocation method for the numerical solution of an optimal control problem governed by elliptic variational inequalities of obstacle type. In this problem, the solution divides the domain into contact and noncontact sets. The boundary between the contact and noncontact sets is a free boundary, which is not known a priori and the solution is not smooth on it. Accordingly, a very fine grid is needed in order to obtain a solution with a reasonable accuracy. In this paper, our aim is to propose an adaptive scheme in order to generate an appropriate and economic irregular dyadic mesh for finding the optimal control and state functions. The irregular mesh will be generated such that its density around the free boundary is higher than in other places and high-resolution computations are focused on these zones. To this aim, we use an adaptive wavelet collocation method and take advantage of the fast wavelet transform of compact-supported interpolating wavelets to develop a multi-level algorithm, which generates an adaptive computational grid. Using this adaptive grid takes less CPU time than using a full regular mesh. At each step of the algorithm, the active set method is used for solving the optimality system of the obstacle problem on the adapted mesh. Finally, the numerical examples are presented to show the validity and efficiency of the technique.

Adaptive-Anisotropic Wavelet Collocation Method on general curvilinear coordinate systems

A new general framework for an Adaptive-Anisotropic Wavelet Collocation Method (A-AWCM) for the solution of partial differential equations is developed. This proposed framework addresses two major shortcomings of existing wavelet-based adaptive numerical methodologies, namely the reliance on a rectangular domain and the “curse of anisotropy”, i.e. drastic over-resolution of sheet- and filament-like features arising from the inability of the wavelet refinement mechanism to distinguish highly correlated directional information in the solution. The A-AWCM addresses both of these challenges by incorporating coordinate transforms into the Adaptive Wavelet Collocation Method for the solution of PDEs. The resulting integrated framework leverages the advantages of both the curvilinear anisotropic meshes and wavelet-based adaptive refinement in a complimentary fashion, resulting in greatly reduced cost of resolution for anisotropic features. The proposed Adaptive-Anisotropic Wavelet Collocation Method retains the a priori error control of the solution and fully automated mesh refinement, while offering new abilities through the flexible mesh geometry, including body-fitting. The new A-AWCM is demonstrated for a variety of cases, including parabolic diffusion, acoustic scattering, and unsteady external flow.

A hybrid wavelet-based adaptive immersed boundary finite-difference lattice Boltzmann method for two-dimensional fluid–structure interaction

A second generation wavelet-based adaptive finite-difference Lattice Boltzmann method (FD-LBM) is developed in this paper. In this approach, the adaptive wavelet collocation method (AWCM) is firstly, to the best of our knowledge, incorporated into the FD-LBM. According to the grid refinement criterion based on the wavelet amplitudes of density distribution functions, an adaptive sparse grid is generated by the omission and addition of collocation points. On the sparse grid, the finite differences are used to approximate the derivatives. To eliminate the special treatments in using the FD-based derivative approximation near boundaries, the immersed boundary method (IBM) is also introduced into FD-LBM. By using the adaptive technique, the adaptive code requires much less grid points as compared to the uniform-mesh code. As a consequence, the computational efficiency can be improved. To justify the proposed method, a series of test cases, including fixed boundary cases and moving boundary cases, are invested. A good agreement between the present results and the data in previous literatures is obtained, which demonstrates the accuracy and effectiveness of the present AWCM-IB-LBM.

Adaptive wavelet collocation method for simulation of a 2D micro-ring resonator

In this paper, we use adaptive wavelet collocation method (AWCM) to solve the 2D time domain Maxwell's equations. Our motivation of choosing AWCM is because it effectively adapts numerical grid points along moving signals according to the requirements of resolution levels of them. In the region where moving signals are intensive, there will be assigned more grid points; and in the region where signals are sparse, there will be arranged less grid points. Since signals are moving during time stepping, the grid changes dynamically at each time step. We verified that AWCM is an efficient method especially for problems in which signals are highly concentrated inside and guided through optical waveguides such as micro-ring resonators.

Approximation of the differential operators on an adaptive spherical geodesic grid using spherical wavelets

In this work, a new adaptive multi-level approximation of surface divergence and scalar-valued surface curl operator on a recursively refined spherical geodesic grid is presented. A hierarchical finite volume scheme based on the wavelet multi-level decomposition is used to approximate the surface divergence and scalar-valued surface curl operator. The multi-level structure provides a simple way to adapt the computation to the local structure of the surface divergence and scalar-value surface curl operator so that the high resolution computations are performed only in regions where singularities or sharp transitions occur. This multi-level approximation of the surface divergence operator is then used in an adaptive wavelet collocation method (AWCM) to solve two standard advection tests, solid-body rotation and divergent flow on the sphere. In contrast with other approximate schemes, this approach can be extended easily to other curved manifolds by considering appropriate coarse approximation to the desired manifold (here we used the icosahedral approximation to the sphere at the coarsest level) and using recursive surface subdivision.

Numerical solution of parabolic partial differential equations using adaptive gird Haar wavelet collocation method

In this paper, we applied the adaptive grid Haar wavelet collocation method (AGHWCM) for the numerical solution of parabolic partial differential equations (PDEs). The approach of AGHWCM for the numerical solution of parabolic PDEs is mentioned, the obtained numerical results, error analysis are presented in figures and tables. This shows that, the AGHWCM gives better accuracy than the HWCM and FDM. Some of the test problems are taken for demonstrating the validity and applicability of the AGHWCM.

Comprehensive numerical methodology for direct numerical simulations of compressible Rayleigh–Taylor instability

An investigation of compressible Rayleigh-Taylor instability (RTI) using Direct Numerical Simulations (DNS) requires efficient numerical methods, advanced boundary conditions, and consistent initialization in order to capture the wide range of scales and vortex dynamics present in the system, while reducing the computational impact associated with acoustic wave generation and the subsequent interaction with the flow. An advanced computational framework is presented that handles the challenges introduced by considering the compressive nature of RTI systems, which include sharp interfacial density gradients on strongly stratified background states, acoustic wave generation and removal at computational boundaries, and stratification dependent vorticity production. The foundation of the numerical methodology described here is the wavelet-based grid adaptivity of the Parallel Adaptive Wavelet Collocation Method (PAWCM) that maintains symmetry in single-mode RTI systems to extreme late-times. PAWCM is combined with a consistent initialization, which reduces the generation of acoustic disturbances, and effective boundary treatments, which prevent acoustic reflections. A dynamic time integration scheme that can handle highly nonlinear and potentially stiff systems, such as compressible RTI, completes the computational framework. The numerical methodology is used to simulate two-dimensional single-mode RTI to extreme late-times for a wide range of flow compressibility and variable density effects. The results show that flow compressibility acts to reduce the growth of RTI for low Atwood numbers, as predicted from linear stability analysis.

Parallel adaptive wavelet collocation method for PDEs

A parallel adaptive wavelet collocation method for solving a large class of Partial Differential Equations is presented. The parallelization is achieved by developing an asynchronous parallel wavelet transform, which allows one to perform parallel wavelet transform and derivative calculations with only one data synchronization at the highest level of resolution. The data are stored using tree-like structure with tree roots starting at a priori defined level of resolution. Both static and dynamic domain partitioning approaches are developed. For the dynamic domain partitioning, trees are considered to be the minimum quanta of data to be migrated between the processes. This allows fully automated and efficient handling of non-simply connected partitioning of a computational domain. Dynamic load balancing is achieved via domain repartitioning during the grid adaptation step and reassigning trees to the appropriate processes to ensure approximately the same number of grid points on each process. The parallel efficiency of the approach is discussed based on parallel adaptive wavelet-based Coherent Vortex Simulations of homogeneous turbulence with linear forcing at effective non-adaptive resolutions up to 20483 using as many as 2048 CPU cores.

An adaptive wavelet collocation method for solving optimal control problem

A sequential solution approach based on an adaptive wavelet collocation method is proposed for solving optimal control problems. By expanding the state and control variables with wavelet multi-resolution decomposition, an original optimal control problem can be transcribed into a nonlinear programming problem that can be solved by general methods. In the proposed framework, an iterative algorithm starts from an initial coarse grid with few collocations. Subsequent solutions can thus be obtained on a dynamically refined grid by an adaptive wavelet collocation method, while the previous coarser level solution can be taken as the initial guess for the next iteration. Consequently, the computational grid for direct numerical optimization method is able to automatically adapt to any irregularities or discontinuities in the solution. The efficiency and accuracy of the proposed algorithm are verified by two typical examples.

Adaptive wavelet collocation method on the shallow water model

This paper presents an integrated approach for modeling several ocean test problems on adaptive grids using novel boundary techniques. The adaptive wavelet collocation method solves the governing equations on temporally and spatially varying meshes, which allows higher effective resolution to be obtained with less computational cost. It is a general method for the solving a large class of partial differential equations, but is applied to the shallow water equations here. In addition to developing wavelet-based computational models, this work also uses an extension of the Brinkman penalization method to represent irregular and non-uniform continental boundaries. This technique is used to enforce no slip boundary conditions through the addition of a term to the field equations. When coupled with the adaptive wavelet collocation method, the flow near the boundary can be well resolved. It is especially useful for simulations of boundary currents and tsunamis, where flow and the boundary is important, thus, those are the test cases presented here.

Integration of barotropic vorticity equation over spherical geodesic grid using multilevel adaptive wavelet collocation method

In this paper, we present the multilevel adaptive wavelet collocation method for solving non-divergent barotropic vorticity equation over spherical geodesic grid. This method is based on multi-dimensional second generation wavelet over a spherical geodesic grid. The method is more useful in capturing, identifying, and analyzing local structure [1] than any other traditional methods (i.e. finite difference, spectral method), because those methods are either full or partial miss important phenomena such as trends, breakdown points, discontinuities in higher derivatives of the solution. Wavelet decomposition is used for interpolation and adaptive grid refinement on different levels.