Multiresolution Basis Research Articles

Nonlinear Estimators for Hypersonic Heat Flux Reconstruction

A novel class of nonlinear estimators that minimizes the projection of the error on a multiresolution basis is developed to improve the accuracy and efficiency of the multidimensional reconstruction of the heat flux from discrete temperature sensors in nonplanar hypersonic boundary layers. This approach uses the local temperature time derivative as the pre-estimator of the heat flux estimation risk in the inverse heat conduction problem. This new approach leads to improved results both in terms of computational efficiency and signal-to-noise ratio when compared to both regularization of the linear response operator and filtering of the temperature measurements. The approach is verified against an analytical verification test case, which shows significant accuracy improvement compared to optimal linear estimators and L2 regularization. It is then applied to experimental data to resolve the heat flux from noisy temperature data collected using fast-response sensors in the Virginia Tech hypersonic wind tunnel at Mach 6. The heat flux agrees very well with the Fay–Riddell approximation. Moreover, the heat transfer coefficient variation over the surface of the test is consistent with computational fluid dynamics simulations of the Reynolds-averaged Navier–Stokes equations.

A New Multi-resolution Analysis Method for Electrooculography Signals.

Electrooculography (EOG) signals indicate the degree and direction of eye movements. Hence, EOG signals have been useful in eye movement controlled rehabilitation systems. Denoising and accurate identification of the type of eye movement in EOG signals are the major challenges in their analysis. The state-of-the-art techniques for EOG signal analysis concerning denoising and eye movement extraction are based on multi-resolution analysis using wavelet bases, such as Haar or Daubechies. However, these wavelets are designed for general purpose signal processing applications and hence are not optimized for the EOG signal structures. In this paper, we propose a new multi-resolution basis specific to the analysis of EOG signals. The scaling and wavelet functions for the basis are derived from the signatures of blinks and saccades respectively, and hence we name them as blinklets and saclets accordingly, thereby forming a new multi-resolution basis. These descriptors are found to be more effective than standard wavelets for EOG signals, signal denoising, and for identifying the different eye movement signatures such as saccades, blinks, smooth pursuits, and fixations, as tested on the Physiosig and Centre for Biomedical Cybernetics Eye Movement (CBC-EM) EOG Databases.

Additive dynamic models for correcting numerical model outputs

Numerical air quality models are pivotal for the prediction and assessment of air pollution, but numerical model outputs may be systematically biased. An additive dynamic model is proposed to correct large-scale raw model outputs using data from other sources, including readings collected at ground monitoring networks and weather outputs from other numerical models. An additive partially linear model specification is employed for the nonlinear relationships between air pollutants and covariates. In addition, a multi-resolution basis function approximate is proposed to capture the different small-scale variations of biases, and a discretized stochastic integro-differential equation is constructed to characterize the dynamic evolution of the random coefficients at each spatial resolution. An expectation-maximization algorithm is developed for parameter estimation and a multi-resolution ensemble-based scheme is embedded to accelerate the computation. For statistical inference, a conditional simulation technique is applied to quantify the uncertainty of parameter estimates and bias correction results. The proposed approach is used to correct the biased raw outputs of PM2.5 from the Community Multiscale Air Quality (CMAQ) system for China's Beijing-Tianjin-Hebei region. Our method improves the root mean squared error and continuous rank probability score by 43.70% and 34.76%, respectively. Compared to other statistical methods under different metrics, our model has advantages in both correction accuracy and computational efficiency.

Spatially varying coefficient models using reduced-rank thin-plate splines

Spatially varying coefficient (SVC) regression models are concerned about regression for spatial data, where regression coefficients may vary in space. This paper proposes a new approach for SVC modeling by representing regression coefficients using a class of multiresolution spline basis functions in a generalized-linear model framework. The proposed method provides flexible and parsimonious representations for regression coefficients. It enables commonly used (generalized) linear-regression packages for estimation, testing, and constructing confidence levels. We develop a fast estimation algorithm that simultaneously performs variable selection, detects spatial heterogeneity for each variable, and determines its complexity. We provide numerical examples and an application to a real estate dataset to demonstrate the proposed method’s effectiveness.

Forecasting of high-resolution electricity consumption with stochastic climatic covariates via a functional time series approach

This paper proposes a functional autoregressive model with stochastic functional covariates, denoted by FARSX, to depict high-resolution data dynamics. An easy-to-implement procedure is proposed to estimate the model parameters under the frameworks of an expansion of multiresolution B-spline basis functions and an adaptive lasso criterion with a two-layer sparsity assumption. We derive the consistency of the proposed estimators under mild conditions. The effectiveness of the estimation procedure allows us to further construct a FARSX model with time-varying parameters under a rolling window framework to capture stochastic effects of functional covariates timely and enhance the prediction accuracy. In the empirical study, the FARSX method with time-varying parameters is applied to the high-resolution electricity consumption and intraday temperatures in Belgium and the U.S. separately. The investigation results reveal that the FARSX model with time-varying parameters provides more reliable day-ahead predictions than several existing models.

Multi-resolution wavelet basis for solving steady forced Korteweg\u2013de Vries model

A steady forced Korteweg–de Vries (fKdV) model which includes gravity, capillary, and pressure distributions is solved numerically using the wavelet Galerkin method. The anti-derivatives of Daubechies wavelets are developed as the basis of the solution subspaces for the mixed boundary condition type. Accuracy of numerical solutions can be improved by increasing the number of wavelet levels in the multi-resolution analysis. The theoretical result of convergence rate is also shown. The problem can be viewed as gravity-capillary wave flows over an applied pressure distribution. The flow regime can be characterized by subcritical, supercritical, and critical flows depending on the value of the Froude number. Trapped depression and elevation waves are found over the pressure distribution. For a near-critical flow regime, a generalized solitary wave with ripples is presented. This shows a capillary effect in balance to gravity and the pressure force on the free surface.

Reduction of Hartree-Fock Wavefunctions to Kohn-Sham Effective Potentials Using Multiresolution Analysis.

We present a highly accurate numerical implementation for computing the Kohn-Sham effective potentials for molecules based on a Hartree-Fock wavefunction and density, following the RKS approach of Staroverov and co-workers [ J. Chem. Phys. 2014, 140, 18A535]. Potentials and orbitals are represented in a multiresolution wavelet basis, avoiding basis set incompleteness-related issues. Together with the RKS method, the often occurring problems of oscillating potentials are removed. The MRA implementation of the RKS method allows the generation of molecular Kohn-Sham potentials of benchmark quality. Numerical data for atoms up to Kr and a number of molecules are given, with a special emphasis on the role of nodal planes in the calculations, as showcased in HCN and benzene.

Testing Independence Between Two Spatial Random Fields

In this article, we consider testing independence between two spatial Gaussian random fields evaluated, respectively, at p and q locations with sample size n, where both p and q are allowed to be larger than n. We impose no spatial stationarity and no parametric structure for the two random fields. Our approach is based on canonical correlation analysis (CCA). But instead of applying CCA directly to the two random fields, which is not feasible for high-dimensional testing considered, we adopt a dimension-reduction approach using a special class of multiresolution spline basis functions. These functions are ordered in terms of their degrees of smoothness. By projecting the data to the function space spanned by a few leading basis functions, the spatial variation of the data can be effectively preserved. The test statistic is constructed from the first sample canonical correlation coefficient in the projected space and is shown to have an asymptotic Tracy–Widom distribution under the null hypothesis. Our proposed method automatically detects the signal between the two random fields and is designed to handle irregularly spaced data directly. In addition, we show that our test is consistent under mild conditions and provide three simulation experiments to demonstrate its powers. Moreover, we apply our method to investigate whether the precipitation in continental East Africa is related to the sea surface temperature (SST) in the Indian Ocean and whether the precipitation in west Australia is related to the SST in the North Atlantic Ocean.

Orhonormal Wavelet Bases on The 3D Ball Via Volume Preserving Map from the Regular Octahedron

We construct a new volume preserving map from the unit ball B 3 to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded cells having the same volume. On this 3D uniform grid, we construct a multiresolution analysis and orthonormal wavelet bases of L 2 ( B 3 ) , consisting in piecewise constant functions with small local support.

Multiresolution Grid Connectivity-Based Transform for Efficient History Matching of Unconventional Reservoirs

Summary Proper characterization of heterogeneous rock properties and hydraulic fracture parameters is essential for optimization of well spacing and reliable estimation of estimated ultimate recovery (EUR) in unconventional reservoirs. High resolution characterization of matrix properties and complex fracture parameters require efficient history matching of well production and pressure response. We propose a novel reservoir model parameterization method to reduce the number of unknowns, regularize the ill-posed problem, and enhance the efficiency of history matching of unconventional reservoirs. The proposed method makes a low-rank approximation of the spatial distribution of reservoir properties taking into account the varying model resolution of the matrix and hydraulic fractures. Typically, hydraulic fractures are represented with much higher resolution through local grid refinements compared to the matrix properties. In our approach, the spatial property distribution of both matrix and fractures is represented using a few parameters via a linear transformation with multiresolution basis functions. The parameters in transform domain are then updated during model calibrations, substantially reducing the number of unknowns. The multiresolution basis functions are constructed by using Eigen-decomposition of an adaptively coarsened grid Laplacian corresponding to the data resolution. Higher property resolution at the area of interest through the adaptive resolution control while keeping the original grid structure improves quality of history matching, reduces simulation runtime, and improves the efficiency of history matching. We demonstrate the power and efficacy of our method using synthetic and field examples. First, we illustrate the effectiveness of the proposed multiresolution parameterization by comparing it with traditional methods. For the field application, an unconventional tight oil reservoir model with a multistage hydraulic fractured well is calibrated using bottomhole pressure and water cut history data. The hydraulic fractures as well as the stimulated reservoir volume (SRV) near the well are represented with higher grid resolution. In addition to matrix and fracture properties, the extent of the SRV and hydraulic fractures are also adjusted through history matching using a multiobjective genetic algorithm. The calibrated ensemble of models are used to obtain bounds of production forecast. Our proposed method is designed to calibrate reservoir and fracture properties with higher resolution in regions that have improved data resolution and higher sensitivity to the well performance data, for example the SRV region and the hydraulic fractures. This leads to a fast and efficient history matching workflow and enables us to make optimal development/completion plans in a reasonable time frame.

Direct determination of optimal pair-natural orbitals in a real-space representation: The second-order Moller-Plesset energy.

An efficient representation of molecular correlated wave functions is proposed, which features regularization of the Coulomb electron-electron singularities via the F12-style explicit correlation and a pair-natural orbital factorization of the correlation components of the wave function expressed in the real space. The pair-natural orbitals are expressed in an adaptive multiresolution basis and computed directly by iterative variational optimization. The approach is demonstrated by computing the second-order Moller-Plesset energies of small- and medium-sized molecules. The resulting MRA-PNO-MP2-F12 method allows for the first time to compute correlated wave functions in a real-space representation for systems with dozens of atoms (as demonstrated here by computations on alkanes as large as C10H22), with precision exceeding what is achievable with the conventional explicitly correlated MP2 approaches based on the atomic orbital representations.

Wavelet-enriched adaptive hierarchical FE model for coupled crystal plasticity-phase field modeling of crack propagation in polycrystalline microstructures

This paper develops a novel, wavelet-enriched adaptive finite element model for solving coupled crystal plasticity-phase field models to simulate crack propagation in polycrystalline microstructures. No a-priori assumption of the crack path is needed. Crack propagation under conditions of finite deformation is driven by stored elastic energy that accounts for material anisotropy and tension–compression asymmetry, and defect energy resulting from slip system dislocation glide and hardening. The resulting finite element model is capable of simulating both brittle and ductile crack propagation in material microstructures. A major contribution of this work is the creation of the adaptive, multi-resolution wavelet-based hierarchical enrichment of the FE model. The adapted enrichment follows the path of crack growth and is able to successfully overcome the challenges of high resolution required for the regularized crack in the coupled model. The multi-resolution wavelet basis functions adaptively construct optimal enrichment basis for the high gradients in the phase field order parameter near the crack path. The wavelet-enriched adaptive finite element model is found to be robust with excellent convergence characteristics in multiple validation tests conducted with the polycrystalline Ti–6V–4Al alloy.

A Bayesian Spatial Categorical Model for Prediction to Overlapping Geographical Areas in Sample Surveys

Summary Motivated by the Australian National University poll, we consider a situation where survey data have been collected from respondents for several categorical variables and a primary geographic classification, e.g. postcode. Here, a common and important problem is to obtain estimates for a second target geography that overlaps with the primary geography but has not been collected from the respondents. We examine this problem when areal level census information is available for both geographic classifications. Such a situation is challenging from a small area estimation perspective for several reasons: there is a misalignment between the census and survey information as well as the geographical classifications; the geographic areas are potentially small and so prediction can be difficult because of the sparse or spatially missing data issue; and there is the possibility of non-stationary spatial dependence. To address these problems we develop a Bayesian model using latent processes, underpinned by a non-stationary spatial basis that combines Moran's I and multiresolution basis functions with a small but representative set of knots. The study results based on simulated data demonstrate that the model can be highly effective and gives more accurate estimates for areas defined by the target geography than several existing models. The model also performs well for the Australian National University poll data to predict on a second geographic classification: statistical area level 2.

Physically constrained spatiotemporal modeling: generating clear-sky constructions of land surface temperature from sparse, remotely sensed satellite data

Satellite remote-sensing is used to collect important atmospheric and geophysical data at various spatial resolutions, providing insight into spatiotemporal surface and climate variability globally. These observations are often plagued with missing spatial and temporal information of Earth's surface due to (1) cloud cover at the time of a satellite passing and (2) infrequent passing of polar-orbiting satellites. While many methods are available to model missing data in space and time, in the case of land surface temperature (LST) from thermal infrared remote sensing, these approaches generally ignore the temporal pattern called the ‘diurnal cycle’ which physically constrains temperatures to peak in the early afternoon and reach a minimum at sunrise. In order to infill an LST dataset, we parameterize the diurnal cycle into a functional form with unknown spatiotemporal parameters. Using multiresolution spatial basis functions, we estimate these parameters from sparse satellite observations to reconstruct an LST field with continuous spatial and temporal distributions. These estimations may then be used to better inform scientists of spatiotemporal thermal patterns over relatively complex domains. The methodology is demonstrated using data collected by MODIS on NASA's Aqua and Terra satellites over both Houston, TX and Phoenix, AZ USA.

Adaptive algorithm for electronic structure calculations using reduction of Gaussian mixtures

We present a new adaptive method for electronic structure calculations based on novel fast algorithms for reduction of multivariate mixtures. In our calculations, spatial orbitals are maintained as Gaussian mixtures whose terms are selected in the process of solving equations. Using a fixed basis leads to the so-called basis error since orbitals may not lie entirely within the linear span of the basis. To avoid such an error, multiresolution bases are used in adaptive algorithms so that basis functions are selected from a fixed collection of functions, large enough to approximate solutions within any user-selected accuracy. Our new method achieves adaptivity without using a multiresolution basis. Instead, as part of an iteration to solve nonlinear equations, our algorithm selects the ‘best’ subset of linearly independent terms of a Gaussian mixture from a collection that is much larger than any possible basis since the locations and shapes of the Gaussian terms are not fixed in advance. Approximating an orbital within a given accuracy, our algorithm yields significantly fewer terms than methods using multiresolution bases. We demonstrate our approach by solving the Hartree–Fock equations for two diatomic molecules, HeH + and LiH, matching the accuracy previously obtained using multiwavelet bases.

MIST: A multi-resolution parcellation of functional brain networks

Functional brain connectomics investigates functional connectivity between distinct brain parcels. There is an increasing interest to investigate connectivity across several levels of spatial resolution, from networks down to localized areas. Here we present the Multiresolution Intrinsic Segmentation Template (MIST), a multi-resolution parcellation of the cortical, subcortical and cerebellar gray matter. We provide annotated functional parcellations at nine resolutions from 7 to 444 functional parcels. The MIST parcellations compare well with prior work in terms of homogeneity and generalizability. We found that parcels at higher resolutions largely fell within the boundaries of larger parcels at lower resolutions. This allowed us to provide an overlap based pseudo-hierarchical decomposition tree that relates parcels across resolutions in a meaningful way. We provide an interactive web interface to explore the MIST parcellations and also made it accessible in the neuroimaging library nilearn. We believe that the MIST parcellation will facilitate future investigations of the multiresolution basis of brain function.

Bounds and Conditions for Compressive Digital Holography Using Wavelet Sparsifying Bases

Numerous experiments have been conducted with success in the field of compressive digital holography, but the theory to determine optimal measurement conditions is lagging behind. In contrast to a prior study that expects object wavefields to be sparse in the spatial domain, we investigate how the configuration of the interferometer influences the reconstruction of wavefields that are sparse in a multiresolution orthogonal wavelet basis. In particular, we derive expressions for the coherence between the free-space wave propagation operator and the basis functions of a Shannon multiresolution representation as a function of the wavelength, the propagation distance, the image sensor's pixel pitch, and the scale of the basis functions. These expressions reveal that the coherence as a function of the Fresnel number is subject to specific scaling and translating rules as the scale of the basis functions changes. For a multiresolution orthogonal wavelet representation and digital holograms that are recorded in the near field, we deduce subsequently the optimal configuration of the interferometer and we show by means of hypothesis testing that the associated phase transition bound coincides with the weak threshold for block-sparse compressive sensing with a block length of 2, which is an optimal bound for the class of complex-valued compressive sensing problems. By means of experiments with a USAF 1951 resolution target and an angle grid, we validate our findings and demonstrate that the reconstructed object wavefields are resilient to sparsity defects and additive noise.

Coupled-Cluster in Real Space. 1. CC2 Ground State Energies Using Multiresolution Analysis.

A framework to calculate CC2 approximated coupled-cluster ground state correlation energies in a multiresolution basis is derived and implemented into the MADNESS library. The CC2 working equations are formulated in first quantization which makes them suitable for real-space methods. The first quantized equations can be interpreted diagrammatically using the usual diagrams from second quantization with adjusted interpretation rules. Singularities arising from the nuclear and electronic potentials are regularized by explicitly taking the nuclear and electronic cusps into account. The regularized three- and six-dimensional cluster functions are represented directly on an adaptive grid. The resulting equations are free of singularities and virtual orbitals, which results in a low intrinsic scaling. Correlation energies close to the basis set limit are computed for small molecules. This work is the first step toward CC2 excitation energies in a multiresolution basis.

Wavelet-enriched adaptive crystal plasticity finite element model for polycrystalline microstructures

Micromechanical analysis of polycrystalline microstructures of metals and alloys, using crystal plasticity finite element (CPFE) models is extensively used for predicting deformation and failure under various conditions of strain-rates, creep and fatigue loading. Many CPFE models involve a large number of degrees of freedom for accurate representation of realistic polycrystalline microstructures. This can lead to prohibitively high computational costs to conduct meaningful analyses of phenomena of interest. To overcome this limitation, the authors have recently developed a wavelet enrichment adapted finite element model in Azdoud and Ghosh (2017) for elastic materials. The method adaptively creates an optimal discretization space conforming to the solution profile by projecting the solution field onto a set of scaling and multi-resolution wavelet basis functions. This paper extends this wavelet adapted FE model to finite deformation, crystal plasticity analysis of polycrystalline microstructures. After presenting the formulations, various validation tests are conducted to examine the convergence rates and computational efficiency of this method.

Improved Multiresolution Preconditioner With Loop-Flower Basis Functions

In this letter, an improved multiresolution preconditioner, based on the loop-flower basis functions, has been proposed for the low-frequency problems. By using loop-flower basis function instead of loop-tree basis function as the coarse-level multiresolution basis function, the condition number of the system matrix is largely reduced, and, thus, the convergence speed of the iterative solver is greatly improved. Some numerical examples have been given to show the validity and the performance of the proposed method.