2. 6 Preliminary Numerical Essay

The model parameters are estimated from the EP and therefore the AR can be calculated within the TP (Strong, 1992). Explicitly, the AR which

3) What do the rate of change values you just calculated represent? Why are some positive and some negative?

\KwIn{nodal value of solution $\mathbf{u} = \left(p, \mathbf{v} \right)$, volume geometric factors $\partial (rst)/ \partial (xyz)$, 1D derivative operator $D_{ij} = \partial \hat{l}_j /\partial x_i$, model parameters $\rho, c$}

A hierarchical Bayesian model is developed in the inverse problem setup. The Bayesian approach contains a natural mechanism for regularization in the form of a prior distribution, and a LASSO type prior distribution is used to strongly induce sparseness. We propose a variational type algorithm by minimizing the Kullback–Leibler divergence between the true posterior distribution and a separable approximation. The proposed method is illustrated on several two-dimensional linear and nonlinear inverse problems, e.g., Cauchy problem and permeability estimation problem. The proposed method performs comparably with full Markov chain Monte Carlo (MCMC) in terms of accuracy and is computationally

A SWAT model for simulation under surface flow using from the reserve kinetic model. This model simulation subsurface flow in tow-dimensional section and flow down the slope that calculation from Equation (5).

Data which exhibit none constant variance is considered. Smoothing procedures are applied to estimate these none constant variances. In these smoothing methods the problem is to establish how much to smooth. The choice of the smoother and the choice of the bandwidth are explored. Kernel and Spline smoothers are compared using simulated data as well as real data. Although the two seem to work very closely, Kernel smoother comes out to be slightly better.

In many practical applications the case of solving an inverse problem is encountered when com- ponent controlling the pde system (1) is unknown or partially known. If κ(x,t) = κ(x) is an

The presence of a convective term means that the data of the problem across the entire domain has an influence on the outflow boundary layer, in contrast to reaction diffusion problems where it is only the data local to a boundary that has an influence on the boundary layer. Hegarty and O'Riordan \cite{115} constructed a parameter-uniform numerical methods for the singularly perturbed problem posed on a circular domain. Asymptotic expansions for the solutions to such a problem have been established in \cite{120,121,122,123}. Analytical expressions for the exact solution, in the case of constant data, are given in \cite{123} as a Fourier series with coefficients written in terms of modified Bessel functions. In \cite{120,121,122} sufficient compatibility conditions are identified so that the accuracy of the asymptotic expansion can be estimated in the $L^2-$norm or in a suitably weighted energy norm. These expansions are derived without recourse to a maximum principle. The smooth case, where significant compatibility is assumed, is examined in \cite{120}; the non-compatible case with a polynomial source term is studied in \cite{125} and for a more general source term in \cite{122}. Based on these asymptotic expansions, a numerical method is constructed in \cite{124} for the problem, which uses a quasi-uniform mesh (which is, hence, not layer-adapted). By enriching the finite element subspace, with certain exponential boundary-layer basis

Some of the smoothing techniques include Kernel, Spline, Locally weighted regression, Recursive Regressogram, Convolution, Median, Split linear fit and K-Nearest Neighbor among others. One of the most active research areas in Statistics in the last 20 years has been the search for a method to find the "optimal" bandwidth for a smoother. There are now a great number of methods to do this. Unfortunately none of them is fully satisfactory. Here a comparative study of the two mostly used and easy to implement smoothers is presented. The Kernel and the cubic spline smoothers .The comparison is preformed on a simulated data set. Looking at

In this paper, we introduce two new functions on the semi-infinite interval namely Rational Gegenbauer and Exponential Gegenbauer functions and we apply them as basis functions in Tau method to solve the boundary layer flow of a magneto-micropolar fluid on a continuous moving plate with suction and injection. These functions are a general case of rational Chebyshev and Legendre functions and this is the first time that they are used in Tau method. The operational matrices of derivative and product of rational and exponential Gegenbauer functions are also presented. These new matrices together with the Tau method are then utilized to reduce the solution of the governing equations to the solution of a system of nonlinear algebraic

The Richardson extrapolation technique has been used to approximate the solution of singularly perturbed problems, improving the numerical approximation of a basic scheme in the case of 1D steady problems of convection diffusion type \cite{253}, parabolic problems of reaction–diffusion and convection–diffusion type \cite{254,255}, and elliptic problems of convection–diffusion and reaction diffusion type \cite{138,256}. In all that papers, the main difficulties in the analysis of the uniform convergence of the extrapolated approximation were related with the

For different values of κ the multiscale basis changes as the pde solution changes. To address this issue offline-online computation is used where in offline stage for representative values over a grid of κ’s the offline basis and the corresponding linear space is constructed. The online basis then constructed solving a local problem for each candidate κ∗, using the principal component direction of the spectral problem. The mathematical details of the online-offline technique can be found in Guha and Tan (2016), Efendiev et al( 2011).

In almost any quantitative field of research (as well as in applied science), the researcher (or, e.g., engineer or economist) frequently needs to fit a parametrized function to observed data. In some cases to make interpolations or extrapolations; the engineer may be interested in values between expensive measurement points, and the economist may be interested in giving a prognosis for the future. In other cases, the parameters (-- removed HTML --) themselves (-- removed HTML --) can be the primary interest. In nuclear physics, it can be of interest to know the fraction of nuclear reactions yielding a particular reaction product; this is an example we will return to repeatedly throughout this paper, starting in Sec. (-- removed HTML

• For level l = 1, sample κ from proposal q1 = q(κ|κli), where κli is the current value at level l = 1.

The solution u, the parameter κ can have oscillatory nature (both in temporal and spatial scale) with multiple scales/periods. A numerical solution that captures the local property of this solution requires capturing the local structure which involves solving a homogenous version of (1) locally and use these solutions as basis to capture the global solution, which is known as multiscale solution (Fish et al., 2012; Franca et al., 2005). A highly oscillatory κ(x, t) = κ(x) is given for a two dimensional domain in Figure 2 .