Multiresolution Approximation Research Articles

Nonstationary Spatial Modeling of Massive Global Satellite Data

Earth-observing satellite instruments obtain a massive number of observations every day. Despite their size, such data sets are incomplete and noisy, necessitating spatial statistical inference to obtain complete, high-resolution fields with quantified uncertainties. Such inference is challenging due to the high computational cost and the nonstationary behavior of environmental processes on a global scale. Classical methods for constructing complete maps of environmental variables can be overly simplistic and result in opaque uncertainties, thereby limiting the scientific application of these data. In this work, we address the need to construct complete, high resolution maps of geophysical processes (Level 3 data) from incomplete, noisy, georeferenced observations (Level 2 data). We develop a multi-resolution approximation (M-RA) of a Gaussian process (GP) whose nonstationary, global covariance function is obtained using local fits. We consider sea surface temperature (SST) as an application to illustrate the workflow and our computational implementation. The M-RA requires domain partitioning, which can be set up application-specifically. In the SST case, we partition the domain purposefully to account for and weaken dependence across land barriers. Our M-RA implementation is tailored to distributed-memory computation in high-performance-computing environments. We analyze a Moderate Resolution Imaging Spectroradiometer (MODIS) SST data set consisting of more than 43 million observations, to our knowledge the largest dataset ever analyzed using a probabilistic GP model. We show that our nonstationary model based on local fits provides substantially improved predictive performance relative to a stationary approach, which already improves upon current methods.

Multiresolution approximation for shallow water equations using summation-by-parts finite differences

Abstract We present spatial approximation for shallow water equations on a mesh of multiple rectangular blocks with different resolution in Cartesian geometry. The approximation is based on finite-difference operators that fulfill Summation By Parts (SBP) property – a discrete analogue of integration by parts. The solution continuity conditions between mesh blocks are imposed in a weak form using Simultaneous Approximation Terms (SAT) method.We show that the resulting discrete divergence and gradient operators are anti-conjugate. The important consequences are the discrete analogues for mass and energy conservation laws along with the proof of stability for linearized equations. The numerical shallow water equations model based on the presented spatial approximation is tested using problems with meteorological context. Test results prove high-order accuracy of SBP-SAT discretization. The interfaces between mesh blocks of different resolution produce no significant noise. The local mesh refinement is shown to have positive effect on the solution both locally inside the refined region and globally in the dynamically coupled areas.

A multiresolution layerwise method with intrinsic damage detection capabilities for the simulation of guided waves in composite strips

A high-order layerwise multiresolution method that utilizes Daubechies wavelet and scaling functions for the approximation of state variables is presented for the enhanced simulation of guided waves in composite strips. The multiresolution approximation yields a hierarchical set of equations of motion involving the coarse component of generalized displacements, and finer components can be added on the coarse one, forming improved predictions until the desired precision is accomplished. The multiresolution approach is combined with a high-order layerwise laminate theory, enabling the accurate prediction of both symmetric and antisymmetric wave modes, the modeling of surface traction, and localized intra-ply and delamination damage types. Numerical results for the simulation of guided waves in laminated composite strips are presented, exhibiting significant reduction in computing times and remarkable convergence rates compared to single-resolution approaches and traditional finite element methods. Moreover, it is shown that each resolution can model specific bandwidths of wavenumbers, thus providing unique inherent capabilities to localize and isolate coexisting wave modes and detect converted and reflected waves, induced by degraded material regions and delaminations.

Multiresolution finite wavelet domain method for efficient modeling of guided waves in composite beams

A Multiresolution finite wavelet domain meshless approach is presented for the simulation of guided waves in composite beams. The Daubechies wavelet and scaling functions are both employed as basis functions and their remarkable properties for the hierarchical spatial approximation of state variables are explored. The first level of the multiresolution approximation provides the coarse solution, while finer resolution components are sequentially calculated and superimposed on the coarse solution, until the desired level of accuracy is achieved. The development of the multiresolution wavelet-based beam element that encompasses first-order shear deformation theory kinematic assumptions is described, and the multiresolution equations of motion including stiffness and mass matrices are formulated. Numerical results for the simulation of guided wave propagation in aluminum and laminated composite beams are presented, demonstrating substantial reduction in computational effort compared to single resolution approaches and traditional finite elements. Convergence studies using various Daubechies wavelet interpolation functions, are evincing extraordinary convergence rates with very low requirements of grid points per wavelength. Furthermore, additional benefits of the proposed method are encompassed in the analysis of multiple guided wave modes, attributed to the low-pass filter behavior of the scaling functions and the band-pass filter behavior of the wavelet functions that can result in isolation of different wave modes by the individual resolution components of the total solution.

Regularization of ill-posed problems involving constant-coefficient pseudo-differential operators

This paper deals with the wavelet regularization for ill-posed problems involving linear constant-coefficient pseudo-differential operators. We concentrate on solving ill-posed equations involving these operators, which are behaving badly in theory and practice. Since a wide range of ill-posed and inverse problems in mathematical physics can be described and rewritten by the language of these operators, it has gathered significant attention in the literature. Based on a general framework, we classify ill-posed problems in terms of their degree of ill-posedness into mildly, moderately, and severely ill-posed problems in a certain Sobolev scale. Using wavelet multi-resolution approximations, it is shown that wavelet regularizers can achieve order-optimal rates of convergence for pseudo-differential operators in special Sobolev space both for the a priori and the a posteriori choice rules. Our strategy, however, turns out that both schemes yield comparable convergence rates. In this setting, ultimately, we provided some prototype examples for which our theoretical results correctly predict improved rates of convergence.

Multiresolution approximation and consistent estimation of a multivariate density function

In this paper we extend multiresolution analysis structures on to approximate multivariate probability density functions. We propose a consistent estimator for a multivariate multiresolution approximation (MMR) of a multivariate pdf. And we also develop an algorithm to estimate the MMR pdf that behaves well when handling big data. This algorithm performs better, in terms of running time, than traditional optimization algorithms. For large samples, the estimations are as good as those obtained by maximum likelihood. Numerical results are provided to illustrate the method.

Identifying regions of inhomogeneities in spatial processes via an M-RA and mixture priors.

Soils have been heralded as a hidden resource that can be leveraged to mitigate and address some of the major global environmental challenges. Specifically, the organic carbon stored in soils, called soil organic carbon (SOC), can, through proper soil management, help offset fuel emissions, increase food productivity, and improve water quality. As collecting data on SOC are costly and time-consuming, not much data on SOC are available, although understanding the spatial variability in SOC is of fundamental importance for effective soil management. In this manuscript, we propose a modeling framework that can be used to gain a better understanding of the dependence structure of a spatial process by identifying regions within a spatial domain where the process displays the same spatial correlation range. To achieve this goal, we propose a generalization of the multiresolution approximation (M-RA) modeling framework of Katzfuss originally introduced as a strategy to reduce the computational burden encountered when analyzing massive spatial datasets. To allow for the possibility that the correlation of a spatial process might be characterized by a different range in different subregions of a spatial domain, we provide the M-RA basis functions weights with a two-component mixture prior with one of the mixture components a shrinking prior. We call our approach the mixture M-RA. Application of the mixture M-RA model to both stationary and nonstationary data show that the mixture M-RA model can handle both types of data, can correctly establish the type of spatial dependence structure in the data (e.g., stationary versus not), and can identify regions of localstationarity.

A multi-resolution approximation via linear projection for large spatial datasets

Recent technical advances in collecting spatial data have been increasing the demand for methods to analyze large spatial datasets. The statistical analysis for these types of datasets can provide useful knowledge in various fields. However, conventional spatial statistical methods, such as maximum-likelihood estimation and kriging, are impractically time-consuming for large spatial datasets due to the necessary matrix inversions. To cope with this problem, we propose a multi-resolution approximation via linear projection (M-RA-lp). The M-RA-lp conducts a linear projection approach on each subregion whenever a spatial domain is subdivided, which leads to an approximated covariance function capturing both the large- and small-scale spatial variations. Moreover, we elicit the algorithms for fast computation of the log-likelihood function and predictive distribution with the approximated covariance function obtained by the M-RA-lp. Simulation studies and a real data analysis for air dose rates demonstrate that our proposed M-RA-lp works well relative to the related existing methods.

Multi-scale process modelling and distributed computation for spatial data

Recent years have seen a huge development in spatial modelling and prediction methodology, driven by the increased availability of remote-sensing data and the reduced cost of distributed-processing technology. It is well known that modelling and prediction using infinite-dimensional process models is not possible with large data sets, and that both approximate models and, often, approximate-inference methods, are needed. The problem of fitting simple global spatial models to large data sets has been solved through the likes of multi-resolution approximations and nearest-neighbour techniques. Here we tackle the next challenge, that of fitting complex, nonstationary, multi-scale models to large data sets. We propose doing this through the use of superpositions of spatial processes with increasing spatial scale and increasing degrees of nonstationarity. Computation is facilitated through the use of Gaussian Markov random fields and parallel Markov chain Monte Carlo based on graph colouring. The resulting model allows for both distributed computing and distributed data. Importantly, it provides opportunities for genuine model and data scalability and yet is still able to borrow strength across large spatial scales. We illustrate a two-scale version on a data set of sea-surface temperature containing on the order of one million observations, and compare our approach to state-of-the-art spatial modelling and prediction methods.

Spatiotemporal multi-resolution approximations for analyzing global environmental data

Technological developments and open data policies have made large, global environmental datasets accessible to everyone. For analyzing such datasets, including spatiotemporal correlations using traditional models based on Gaussian processes does not scale with data volume and requires strong assumptions about stationarity, separability, and distance measures of covariance functions that are often unrealistic for global data. Only very few modeling approaches suitably model spatiotemporal correlations while addressing both computational scalability as well as flexible covariance models. In this paper, we provide an extension to the multi-resolution approximation (MRA) approach for spatiotemporal modeling of global datasets. MRA has been shown to be computationally scalable in distributed computing environments and allows for integrating arbitrary user-defined covariance functions. Our extension adds a spatiotemporal partitioning, and fitting of complex covariance models including nonstationarity with kernel convolutions and spherical distances. We evaluate the effect of the MRA parameters on estimation and spatiotemporal prediction using simulated data, where computation times reduced around two orders of magnitude with an increase of the root-mean-square prediction error of around five percent. This allows for trading off computation times against prediction errors, and we derive a practical strategy for selecting the MRA parameters. We demonstrate how the approach can be practically used for analyzing daily sea surface temperature data on global scale and compare models with different complexities in the covariance function.

A wavelet multi-resolution enabled interpolation Galerkin method for two-dimensional solids

A novel wavelet multi-resolution formulation in which arbitrary specified nodal coefficients can be exactly the function values at the corresponding nodes is proposed to approximate field functions for solid mechanics problems. This wavelet approximation is explicitly constructed based only on properly scattered nodes without any matrix inversion or ad-hoc parameters. It allows easy and automatic node generation and further refinement at any targeted zones, with a loose and explicit criterion. Using this wavelet multi-resolution approximation to create trial and weighted functions, a wavelet multi-resolution enabled interpolation Galerkin method (WMEIGM) is developed. In the proposed WMEIGM, the stiffness matrix can be efficiently evaluated through semi-analytical method. The essential boundary conditions can be imposed with ease as in the finite element method (FEM). The accuracy and convergence of the proposed WMEIGM are examined theoretically and numerically. Numerical results show that the present WMEIGM has an excellent stability against irregular nodal distribution even with an extremely large ratio of the maximum grid size to the minimum, and a strong capacity for handling irregular problem domains with complicated shape. Moreover, a robust multi-resolution enrichment technique is developed for the present WMEIGM for improving local accuracy and for capturing localized steep gradients.

Electromagnetic device modeling using a new adaptive wavelet finite element method

Wavelets provide sparse representations of integral and differential operators through a multiresolution approximation. This property is well-suited for adaptive implementation of numerical techniques, which is a common topic among researchers nowadays. The current state of the wavelet theory also allows flexible construction of transforms customized to certain applications, which is an important feature considering that the generalization of wavelet techniques to a large class of real-world numerical simulations is still limited. In this paper, it is proposed an adaptive wavelet finite element method for achieving operator-orthogonality between approximation spaces, a property which eliminates coupling among levels of resolution – a common shortcoming of adaptive methods. The resulting method is easily extended to any given finite element application and requires less computation power when compared to other adaptive techniques. These features are showcased by comparing the proposed technique to the popular h-refinement method using electromagnetic device simulations involving Dirichlet and Neumann boundary conditions, inhomogeneous and non-linear media, and irregular meshes.

Кратномасштабная аппроксимация решения интегрального уравнения типа свертки на основе новых частотно-модифицированных вейвлетов Кравченко

Кратномасштабная аппроксимация решения интегрального уравнения типа свертки на основе новых частотно-модифицированных вейвлетов Кравченко

A class of multi-resolution approximations for large spatial datasets

Gaussian processes are popular and flexible models for spatial, temporal, and functional data, but they are computationally infeasible for large datasets. We discuss Gaussian-process approximations that use basis functions at multiple resolutions to achieve fast inference and that can (approximately) represent any spatial covariance structure. We consider two special cases of this multi-resolution-approximation framework, a taper version and a domain-partitioning (block) version. We describe theoretical properties and inference procedures, and study the computational complexity of the methods. Numerical comparisons and an application to satellite data are also provided.

A Multi-resolution Approximation for Time Series

Time series is a common and well-known way for describing temporal data. However, most of the state-of-the-art techniques for analysing time series have focused on generating a representation for a single level of resolution. For analysing of a time series at several levels of resolutions, one would require to compute different representations, one for each resolution level. We introduce a multi-resolution representation for time series based on local trends and mean values. We require the level of resolution as parameter, but it can be automatically computed if we consider the maximum resolution of the time series. Our technique represents a time series using trend-value pairs on each segment belonging to a resolution level. To provide a useful representation for data mining tasks, we also propose dissimilarity measures and a symbolic representation based on the SAX technique for efficient similarity search using a multi-resolution indexing scheme. We evaluate our method for classification and discord discovery tasks over a diversity of data domains, achieving a better performance in terms of efficiency and effectiveness compared with some of the best-known classic techniques. Indeed, for some of the experiments, the time series mining algorithms using our multi-resolution representation were an order of magnitude faster, in terms of distance computations, than the state of the art.

Multiresolution analysis and adaptive estimation on a sphere using stereographic wavelets

We construct an adaptive estimator of a density function on d dimensional unit sphere Sd (d≥2), using a new type of spherical frames. The frames, or as we call them, stereografic wavelets are obtained by transforming a wavelet system, namely Daubechies, using some stereographic operators. We prove that our estimator achieves an optimal rate of convergence on some Besov type class of functions by adapting to unknown smoothness. Our new construction of stereografic wavelet system gives us a multiresolution approximation of L2(Sd) which can be used in many approximation and estimation problems. In this paper we also demonstrate how to implement the density estimator in S2 and we present a finite sample behavior of that estimator in a numerical experiment.

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.

Wavelet multi-resolution approximation of time-varying frame structure

Engineering structures usually exhibit time-varying behavior when subjected to strong excitation or due to material deterioration. This behavior is one of the key properties affecting the structural performance. Hence, reasonable description and timely tracking of time-varying characteristics of engineering structures are necessary for their safety assessment and life-cycle management. Due to its powerful ability of approximating functions in the time–frequency domain, wavelet multi-resolution approximation has been widely applied in the field of parameter estimation. Considering that the damage levels of beams and columns are usually different, identification of time-varying structural parameters of frame structure under seismic excitation using wavelet multi-resolution approximation is studied in this article. A time-varying dynamical model including both the translational and rotational degrees of freedom is established so as to estimate the stiffness coefficients of beams and columns separately. By decomposing each time-varying structural parameter using one wavelet multi-resolution approximation, the time-varying parametric identification problem is transformed into a time-invariant non-parametric one. In solving the high number of regressors in the non-parametric regression program, the modified orthogonal forward regression algorithm is proposed for significant term selection and parameter estimation. This work is demonstrated through numerical examples which consider both gradual variation and abrupt changes in the structural parameters.

Improving Particle Filter Performance by Smoothing Observations

AbstractThis article shows that increasing the observation variance at small scales can reduce the ensemble size required to avoid collapse in particle filtering of spatially extended dynamics and improve the resulting uncertainty quantification at large scales. Particle filter weights depend on how well ensemble members agree with observations, and collapse occurs when a few ensemble members receive most of the weight. Collapse causes catastrophic variance underestimation. Increasing small-scale variance in the observation error model reduces the incidence of collapse by de-emphasizing small-scale differences between the ensemble members and the observations. Doing so smooths the posterior mean, though it does not smooth the individual ensemble members. Two options for implementing the proposed observation error model are described. Taking a discretized elliptic differential operator as an observation error covariance matrix provides the desired property of a spectrum that grows in the approach to small scales. This choice also introduces structure exploitable by scalable computation techniques, including multigrid solvers and multiresolution approximations to the corresponding integral operator. Alternatively the observations can be smoothed and then assimilated under the assumption of independent errors, which is equivalent to assuming large errors at small scales. The method is demonstrated on a linear stochastic partial differential equation, where it significantly reduces the occurrence of particle filter collapse while maintaining accuracy. It also improves continuous ranked probability scores by as much as 25%, indicating that the weighted ensemble more accurately represents the true distribution. The method is compatible with other techniques for improving the performance of particle filters.

Discrete linear canonical wavelet transform and its applications

The continuous generalized wavelet transform (GWT) which is regarded as a kind of time-linear canonical domain (LCD)-frequency representation has recently been proposed. Its constant-Q property can rectify the limitations of the wavelet transform (WT) and the linear canonical transform (LCT). However, the GWT is highly redundant in signal reconstruction. The discrete linear canonical wavelet transform (DLCWT) is proposed in this paper to solve this problem. First, the continuous linear canonical wavelet transform (LCWT) is obtained with a modification of the GWT. Then, in order to eliminate the redundancy, two aspects of the DLCWT are considered: the multi-resolution approximation (MRA) associated with the LCT and the construction of orthogonal linear canonical wavelets. The necessary and sufficient conditions pertaining to LCD are derived, under which the integer shifts of a chirp-modulated function form a Riesz basis or an orthonormal basis for a multi-resolution subspace. A fast algorithm that computes the discrete orthogonal LCWT (DOLCWT) is proposed by exploiting two-channel conjugate orthogonal mirror filter banks associated with the LCT. Finally, three potential applications are discussed, including shift sampling in multi-resolution subspaces, denoising of non-stationary signals, and multi-focus image fusion. Simulations verify the validity of the proposed algorithms.