Neural network method to solve inverse problems for canopy radiative transfer models

Vegetation parameter retrieval is considered as the inverse of modeling canopy radiative transfer. To solve this problem, a new computationally efficient method based on mixture density networks (MDNs) is proposed to estimate the errors of retrieved parameters for each given set of reflectances. The properties of neural networks of traditional architecture and MDNs are considered. The method is tested using a simple model and the PROSPECT leaf radiative transfer model and is validated against real data. The paper is supported by the joint project of the Science and Technology Center in Ukraine (STCU) and National Academy of Sciences of Ukraine (NASU), “Grid Technologies for Multi-Source Data Integration” (No. 4928). Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 159–172, May–June 2009. Original article submitted January 29, 2009.     

An efficient solver for the RANS equations and a one-equation turbulence model

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. Using the algebraic turbulence model of Baldwin and Lomax, this scheme has been used to solve the compressible Reynolds-averaged Navier–Stokes (RANS) equations for transonic and low-speed flows. In this paper we focus on the convergence of the RK/Implicit scheme when the effects of turbulence are represented by the one-equation model of Spalart and Allmaras. With the present scheme the RANS equations and the partial differential equation of the turbulence model are solved in a loosely coupled manner. This approach allows the convergence behavior of each system to be examined. Point symmetric Gauss-Seidel supplemented with local line relaxation is used to approximate the inverse of the implicit operator of the RANS solver. To solve the turbulence equation we consider three alternative methods: diagonally dominant alternating direction implicit (DDADI), symmetric line Gauss-Seidel (SLGS), and a two-stage RK scheme with implicit preconditioning. Computational results are presented for airfoil flows, and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and a transport-type equation for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses RK/Implicit schemes.     

An improved reproducing kernel method for Fredholm integro-differential type two-point boundary value problems

In this work, an improved reproducing kernel method to find the numerical solution of Fredholm integro-differential equation type boundary value problems has been developed. Based on the good properties of reproducing kernel function and the conjugate operator, the solution representation is obtained. Meanwhile, we prove that the approximation converges to the exact solution uniformly. After that the convergence estimates are derived.     

Invariant imbedding and inverse problems with applications to radiative transfer

In this paper, we developed two methods to solve the inverse problem of a nonlinear integro-differential equation. Both methods are based on the principle of invariant imbedding. The first method involves two auxiliary integro-differential equations. The inverse problem is solved by a sequence of approximation solutions of linear equations. The second method involves algebraic equations of scattering matrices under the so-called star-product. An application to a radiative transfer problem such as correction of atmospheric effects on remote sensing is discussed.     

Preconditioned Krylov subspace methods for solving radiative transfer problems with scattering and reflection

Two Krylov subspace methods, the GMRES and the BiCGSTAB, are analyzed for solving the linear systems arising from the mixed finite element discretization of the discrete ordinates radiative transfer equation. To increase their convergence rate and stability, the Jacobi and block Jacobi methods are used as preconditioners for both Krylov subspace methods. Numerical experiments, designed to test the effectiveness of the (preconditioned) GMRES and the BiCGSTAB, are performed on various radiative transfer problems: (i) transparent, (ii) absorption dominant, (iii) scattering dominant, and (iv) with specular reflection. It is observed that the BiCGSTAB is superior to the GMRES, with lower iteration counts, solving times, and memory consumption. In particular, the BiCGSTAB preconditioned by the block Jacobi method performed best amongst the set of other solvers. To better understand the discrete systems for radiative problems (i) to (iv), an eigenvalue spectrum analysis has also been performed. It revealed that the linear system conditioning deteriorates for scattering media problems in comparison to absorbing or transparent media problems. This conditioning further deteriorates when reflection is involved.     

An efficient fractional-order wavelet method for fractional Volterra integro-differential equations

In this paper, an efficient and robust numerical technique is suggested to solve fractional Volterra integro-differential equations (FVIDEs). The proposed method is mainly based on the generalized fractional-order Legendre wavelets (GFLWs), their operational matrices and the Collocation method. The main advantage of the proposed method is that, by using the GFLWs basis, it can provide more efficient and accurate solution for FVIDEs in compare to integer-order wavelet basis. A comparison between the achieved results confirms accuracy and superiority of the proposed GFLWs method for solving FVIDEs. Error analysis and convergence of the GFLWs basis is provided.     

Coupled soil-leaf-canopy and atmosphere radiative transfer modeling to simulate hyperspectral multi-angular surface reflectance and TOA radiance data

Coupling radiative transfer models for the soil background and vegetation canopy layers is facilitated by means of the four-stream flux interaction concept and use of the adding method. Also the coupling to a state-of-the-art atmospheric radiative transfer model like MODTRAN4 can be established in this way, thus enabling the realistic simulation of top-of-atmosphere radiances detected by space-borne remote sensing instruments. Possible applications of coupled modeling vary from mission design to parameter retrieval and data assimilation. This paper introduces a modified Hapke soil BRDF model, a robust version of the PROSPECT leaf model, and a modernized canopy radiative transfer model called 4SAIL2. The latter is a hybrid two-layer version of SAIL accommodating horizontal and vertical heterogeneities, featuring improved modeling of the hot spot effect and output of canopy absorptances. The integrated model is simply called SLC (soil-leaf-canopy) and has been implemented as a speed-optimized Windows DLL which allows efficient use of computer resources even when simulating massive amounts of hyperspectral multi-angular observations. In this paper various examples of possible model output are shown, including simulated satellite image products. First validation results have been obtained from atmospherically corrected hyperspectral multi-angular CHRIS-PROBA data of the Upper Rhine Valley in Germany.     

The integration of directional information and local region information for accurate image segmentation

The original local binary fitting (LBF) model is sensitive to contour initialization and thus easily obtains an inaccurate result due to improper initialization. This paper presents a new method that not only can arrive at sub-pixel accuracy, but also allows for more flexible initialization of the contour. Two important terms play main role in our new method. One is an image gradient alignment term (IGA) which uses the directional information of the image gradient, the other is a local intensity fitting term (LIF) which makes use of local region information. The integration of the above two terms prevents our method from being sensitive to contour initialization. In addition, a global intensity fitting term (GIF) multiplied by a stopping function is included, which can speed up our algorithm while do not influence the accuracy of the segmentation result. Using the simple central difference, the gradient descend flow equation for the level set function can be easily and efficiently implemented. The results on several synthetic and real images demonstrate the effectiveness and accuracy of our method.     

A coupled 1-D atmosphere and 3-D canopy radiative transfer model for canopy reflectance, light environment, and photosynthesis simulation in a heterogeneous landscape

Detailed knowledge of light interactions between the atmosphere and vegetation, and within vegetation are of particular interest for terrestrial carbon cycle studies and optical remote sensing. This study describes a model for 3-D canopy radiative transfer that is directly coupled with an atmospheric radiative transfer model (Forest Light Environmental Simulator, FLiES). The model was developed based on the Monte Carlo ray-tracing method using some existing modeling frameworks. To integrate the canopy radiative transfer model with atmosphere, the same numerical method, sampling technique, and variance reduction technique were employed in both the atmospheric and the canopy modules. Farquhar's leaf photosynthesis model was combined to calculate the canopy level photosynthesis from the light environmental parameters obtained by the radiative transfer calculation. In order to document the quality of the coupled model, we first compared the atmospheric radiative transfer module to well known 1-D atmospheric radiative transfer models, and then evaluated the 3-D canopy radiative transfer module against a series of test cases provided by the RAMI On-line Model Checker (ROMC). We used the model to show the impact of atmospheric properties and 3-D canopy structure on the directionality of downward photosynthetically active radiation (PAR) at the top of canopy, the 3-D distribution of absorbed PAR (APAR), and overall canopy photosynthesis. The results indicate the importance to consider angular geometry of incident light at TOC and 3-D canopy structure.     

A fast and accurate computational technique for efficient numerical solution of nonlinear singular boundary value problems

This paper is concerned with design and implementation of a computational technique for the efficient solution of a class of singular boundary value problems. The method is based on a modified homotopy analysis method. The method is illustrated by six examples, two of which arise in chemical engineering: the first problem arises in the study of thermal explosions, while the second problem arises in the study of heat and mass transfer within the porous catalyst particles. Numerical results reveal that our method provides better results as compared to some existing methods. Furthermore, it is a powerful tool for dealing with different types of problems with strong nonlinearity.     

An efficient NFEM for optimal control problems governed by a bilinear state equation

In this paper, a nonconforming finite element method (NFEM) is proposed for the constrained optimal control problems (OCPs) governed by a bilinear state equation. The state and adjoint state are approximated by the nonconforming $E{Q}_1^{rot}$ element, and the control is approximated by the orthogonal projection through the state and adjoint state. Some superclose and superconvergence properties are obtained by full use of the distinguish characters of this $E{Q}_1^{rot}$ element, such as the interpolation operator equals the Ritz projection, and the consistency error is one order higher than its interpolation error in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.     

An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices

An efficient method for computing a given number of leading eigenvalues (i.e., having largest real parts) and the corresponding eigenvectors of a large asymmetric matrixM is presented. The method consists of three main steps. The first is a filtering process in which the equationx = Mx is solved for an arbitrary initial conditionx(0) yielding:x(t)=e ^Mt x(0). The second step is the construction of (n+1) linearly independent vectorsv _m =M ^m x, 0mn orv _m =e ^mMt x ( being a short time interval). By construction, the vectorsv _m are combinations of only a small number of leading eigenvectors ofM. The third step consists of an analysis of the vectors {v _m } that yields the eigenvalues and eigenvectors. The proposed method has been successfully tested on several systems. Here we present results pertaining to the Orr-Sommerfeld equation. The method should be useful for many computations in which present methods are too slow or necessitate excessive memory. In particular, we believe it is well suited for hydrodynamic and mechanical stability investigations.     

An implicit matrix-free Discontinuous Galerkin solver for viscous and turbulent aerodynamic simulations

This paper presents some recent advancements of the computational efficiency of a Discontinuous Galerkin (DG) solver for the Navier–Stokes (NS) and Reynolds Averaged Navier Stokes (RANS) equations. The implementation and the performance of a Newton–Krylov matrix-free (MF) method is presented and compared with the matrix based (MB) counterpart. Moreover two solution strategies, developed in order to increase the solver efficiency, are discussed and experimented. Numerical results of some test cases proposed within the EU ADIGMA (Adaptive Higher-Order Variational Methods for Aerodynamic Applications in Industry) project demonstrate the capabilities of the method.     

An efficient and accurate quantum lattice-gas model for the many-body Schrödinger wave equation

Presented is quantum lattice-gas model for simulating the time-dependent evolution of a many-body quantum mechanical system of particles governed by the non-relativistic Schrödinger wave equation with an external scalar potential. A variety of computational demonstrations are given where the numerical predictions are compared with exact analytical solutions. In all cases, the model results accurately agree with the analytical predictions and we show that the model's error is second order in the temporal discretization and fourth order in the spatial discretization. The difficult problem of simulating a system of fermionic particles is also treated and a general computational formulation of this problem is given. For pedagogical purposes, the two-particle case is presented and the numerical dispersion of the simulated wave packets is compared with the analytical solutions.     

Inversion of a radiative transfer model for estimating vegetation LAI and chlorophyll in a heterogeneous grassland

Radiative transfer models have seldom been applied for studying heterogeneous grassland canopies. Here, the potential of radiative transfer modeling to predict LAI and leaf and canopy chlorophyll contents in a heterogeneous Mediterranean grassland is investigated. The widely used PROSAIL model was inverted with canopy spectral reflectance measurements by means of a look-up table (LUT). Canopy spectral measurements were acquired in the field using a GER 3700 spectroradiometer, along with simultaneous in situ measurements of LAI and leaf chlorophyll content. We tested the impact of using multiple solutions, stratification (according to species richness), and spectral subsetting on parameter retrieval. To assess the performance of the model inversion, the normalized RMSE and R² between independent in situ measurements and estimated parameters were used. Of the three investigated plant characteristics, canopy chlorophyll content was estimated with the highest accuracy (R² = 0.70, NRMSE = 0.18). Leaf chlorophyll content, on the other hand, could not be estimated with acceptable accuracy, while LAI was estimated with intermediate accuracy (R² = 0.59, NRMSE = 0.18). When only sample plots with up to two species were considered (n = 107), the estimation accuracy for all investigated variables (LAI, canopy chlorophyll content and leaf chlorophyll content) increased (NRMSE = 0.14, 0.16, 0.19, respectively). This shows the limits of the PROSAIL radiative transfer model in the case of very heterogeneous conditions. We also found that a carefully selected spectral subset contains sufficient information for a successful model inversion. Our results confirm the potential of model inversion for estimating vegetation biophysical parameters at the canopy scale in (moderately) heterogeneous grasslands using hyperspectral measurements.     

An interactive method for the selection of design criteria and the formulation of optimization problems in computer aided optimal design

This paper presents an interactive method for the selection of design criteria and the formulation of optimization problems within a computer aided optimization process of engineering systems. The key component of the proposed method is the formulation of an inverse optimization problem for the purpose of determining the design preferences of the engineer. These preferences are identified based on an interactive modification of a preliminary optimization result that is the solution of an initial problem statement. A formulation of the inverse optimization problem is presented, which is based on a weighted-sum multi-objective approach and leads to an explicit optimization problem that is computationally inexpensive to solve. Numerical studies on structural shape optimization problems show that the proposed method is able to identify the optimization criteria and the formulation of the optimization problem which drive the interactive user modifications.     

An efficient branch and bound algorithm for assembly line balancing problems with parallel multi-manned workstations

In the event that big-sized complex products (containing a large number of assembly tasks most of which have long task times) are produced in simple or two-sided assembly lines, hundreds of stations are essentially required. Long product flow time, a large area for establishment of the line, a high budget for the investment of equipment, and tools in stations and several work-in-process are also required for these kinds of products. In order to avoid these disadvantages, assembly lines with parallel multi-manned workstations can be utilized. In this paper, these lines and one of their balancing problems are addressed, and a branch and bound algorithm is proposed. The algorithm is composed of a branching scheme, some efficient dominance and feasibility criteria based on a problem-specific knowledge. A heuristic-based guidance for enumeration process is included as an efficient component of the algorithm as well. VWSolver algorithm proposed for a special version of the problem in the literature has been modified and compared with the proposed algorithm. Results show that proposed algorithm outperforms VWSolver in terms of both CPU times and quality of feasible solutions found.     

An efficient approach for solving the straightness and the flatness problems at large number of data points

The construction of the convex hull as a critical step of solving the straightness and the flatness errors needs a great amount of computation, especially, when the number of data point is large, that limits the computational efficiency. To establish an efficient algorithm to solve these problems, three theorems are developed in this paper to show that the straightness and the flatness errors can be obtained without the construction of the whole convex hull. Theorem 1 shows how to identify the redundant data points that gives an efficient way to reduce the unnecessary computations. The optimum criterion defined in Theorem 2 shows that the optimum solution can be obtained by a small number of data points if these points meet the criterion. Theorem 3 offers an easy way to identify the potential candidates of the solution holders, which are data points on the vertices of the convex hull. The efficiency of the proposed algorithm is validated at the end of this paper. The validation results show that the computational efficiency of solving the straightness and the flatness problems is improved significantly through the proposed algorithm, especially, when the number of data point is large.