text:
\section{Introduction}

The reactive transport in porous media is important component of many industrial and environmental problems like water purification, soil pollution and remediation, catalytic filters, $CO_2$ storage, oil recovery, etc., to name just a few. Historically, most of the theoretical and experimental research on transport in porous media in general, and on reactive transport in particular, has been carried out at macroscopic Darcy scale \cite{bear2013dynamics,helmig1997multiphase}. In many cases, the bottleneck in performing computational modeling of reactive transport is the absence of data for the pore scale adsorption and desorption rate (or in more general, the parameters of the heterogeneous reactions). Despite the progress in developing devices to perform experimental measurements at the pore–scale, experimental characterization of these rates is still a very challenging task.

In the case of heterogeneous (surface) reactions at pore scale, the species transport is coupled to surface reaction via boundary conditions. When the reaction rates are not known, their identification falls into the class of boundary value inverse problems, \cite{lavrentev1986ill,alifanov2011inverse,isakovinverse,samarskii2007numerical}. The additional information which is needed to identify the parameters is often provided in form of dynamic change of the  concentration at the outlet (e.g., so called breakthrough curves).  In the literature, inverse problems for porous media flow are discussed mainly in connection with parameter identification for macroscopic, Darcy scale problems. An overview on inverse problems in groundwater Darcy scale modeling can be found in  \cite{sun2013inverse}. Identification of parameters for pore scale models is discussed in this paper, and the algorithms from \cite{sun2013inverse} and other papers discussing parameter identification at macroscale can not be applied here without modification. Let us shortly mention some general approaches for solving inverse problems.

Different algorithms can be applied for solving parameter identification problems, see, e.g.
\cite{tarantola2005inverse,Aster2013}. Many of the algorithms exploit deterministic methods based on Tikhonov regularization technique  \cite{tikhonov1977solutions,engl2014inverse} and target at minimizing a functional of the difference between measured and computed quantities. An important part of such algorithms is the definition of feasible set of parameters on which the functional is minimized. Local or global optimization procedures are used in the optimization \cite{horst2013handbook,nocedal2006numerical}.
In this sense it could be pointed out that there is certain similarity between the mathematical formulation of an optimization problem and of a parameter identification one.

Stochastic-deterministic methods are also popular approach for solving parameter identification problems. A variant of the method based on deterministic sampling of points looks appropriate for the topic considered here. A stochastic approach for global optimization in its simplest form consists only of a random search and it is called Pure Random Search \cite{Zhigljavsky2008}. In this case the residual functional is evaluated at randomly chosen points from the feasible set.
Sobol sequences \cite{sobol1976uniformly,sobol1979systematic} can be used for sampling. The sensitivity analysis tool SALib \cite{Herman2017} has shown to be appropriate tool for this. Such an approach is successfully used, for example, in multicriteria parameter identification \cite{sobol1981choosing}.

The solution of the inverse problems we are interested in, is composed of two ingredients: (multiple) solution of the direct (called also forward) problem, and the parameter identification algorithm.

\begin{comment}
The direct pore scale simulation of reactive flow in porous media requires four essential components, namely,
\\ \textbf{(i)} model of the porous geometry; \textbf{(ii)} mathematical model of the processes to be investigated;  \textbf{(iii)} algorithms and \textbf{(iv)} software for solving these models.
Let us give some very short comments on the above components.
\\ \textbf{(i)} The pore scale geometry in the simplest case can be represented by periodic arrangement of obstacles. Another class of pore scale geometries are the stochastic ones. Variety of approaches exist for generation of stochastic porous media \cite{}. Finally, nowadays 2D and 3D images of samples of real porous media (rocks, nonwoven, textiles, ceramic membranes etc.) can be used \cite{}. Let us also mention that there are a large number of papers and software tools (for example Aviso \cite{}, GeoDict \cite{GeoDict},  Ingrain \cite{}, etc.) dedicated to simulating a number of problems describing flow and (passive) transport in 3D computerized tomography images \cite{ComputedTomography}. For the needs of this paper we restrict our considerations to periodic porous media, the case of computer generated stochastic geometry or porous geometry coming from 3D imaging will be subject of forthcoming papers.
\\ \textbf{(ii)} The single phase or multiphase flow at pore scale in most of the cases is described by laminar Navier-Stokes equations \cite{Bear,Kaviany}. Single phase flow is considered in this paper. Several models exist for describing passive transport of substance(s) \cite{concentration, random walk, Bear, ...}. Variety of papers consider different models of volumetric and/or surface reactions (called also homogeneous and heterogeneous reactions, respectively) \cite{}. A relatively simple surface reaction model is considered in this paper. The reaction rates are constants, the pore scale geometry does not change for the considered reactions and time scales.;
\\ \textbf{(iii)} and \textbf{(iv)} Finite volume, finite element, meshfree methods have been developed for solving the above class of problems. Finite element method is used here, mainly aiming to benefit from availability of well developed software tools for FEM. The software platform FEniCS is exploited \cite{LoggMardalEtAl2012a}.
}
\end{comment}

Note that the considered here parameter identification approach is different from approaches like determining the single fibre (or collector) efficiency. Originally \cite{Baron} this additive approach used analytical solution for flow and transport around a single sphere or cylinder to evaluate the contribution of the convection and diffusion to the particles deposition and to propose a formula for this deposition depending on Pe number and fiber diameter. Later on several modification were proposed, for example in \cite{Mesina} a convection diffusion equation with zero Dirichlet boundary condition on the surface of a single fiber, in order to calculate the specie flux to the surface and to fit an algebraic formula for the single fiber efficiency. In \cite{Wood} it was stated that zero Dirichlet boundary condition on the surface of the pore is simplification and introduced a jump described by Henry isotherm. The adsorption and desorption parameters in this isotherm were derived from the solution of Smoluchovski equation at nanoscale. This was further embedded in volume averaging upscaling procedure from nanoscale to Darcy scale. As mentioned above, our aim is to have an identification procedure applicable to any periodic and random porous media, for various process regimes, various kinetics of the heterogeneous reactions, and thus to go beyond the limited applicability of formula like single collector efficiency. On the other hand, we assume that it is known which isotherm has to be used for the processes we model, and we do not plan to go to nanoscale to derive isotherms.

The goal of this paper is to contribute to the understanding of the formulation and the solution of a class of parameter identification problems for pore scale reactive transport in the case when the measured concentration of the specie at the outlet of the domain is provided as extra information in order to carry out the identification procedure. Deterministic and stochastic parameter identification approaches are considered. The influence of the noise in the measurements on the accuracy of the identified parameters is discussed. Multistage identification procedure is suggested for the considered class of problems. The proposed identification approach is applicable for different geometries (random and periodic) and for a range of process parameters. In this paper the potential of the approach is demonstrated in identifying parameters of Langmuir isotherm for low Peclet and low Damkoler numbers reactive flow in a 2D periodic porous media with circular inclusions. It is supposed that this paper is the first one in series of papers dedicated to this topic. Simulation results for random porous media and other regime parameters are subject of follow up papers.

\begin{comment}
Periodic porous media in 2D and simple surface reaction are considered here to reduce the presentation and to keep track on the main focus of the paper. Suitable basic deterministic and stochastic algorithms for identification of unknown adsorption and desorption rates are discussed. Significant attention is paid to issues important for the solution of practical problems. Such issues are the influence of the noise in the measurement on the accuracy of the identification, the identification of admissible set of parameters (unlike the typical attempt to identify fixed values for parameters), the usage of prediction-refinement approach to parameter identification. The sensitivity with respect to important parameters is also studied.
\end{comment}

The reminder of the paper is organized as follows.
The direct problem is considered in Section 2. At pore scale, single phase laminar flow described by incompressible Stokes equations, and solute transport described by convection-diffusion equation, are considered. The surface reaction is accounted for in the boundary conditions. Henry and Langmuir adsorption isotherms are considered here \cite{kralchevsky2008chemical}, identification is carried out for  adsorption and desorption parameters in the Langmuir isotherm. Section 3 is dedicated to description of the used computational algorithm. Finite element method is exploited after triangulation of the computational domain. The numerical investigation of the grid convergence and sensitivity with respect to parameters is also presented in this Section. The set up of the parameter identification problem for reactive flow in porous media is described in Section 4. Computational algorithm for statistical evaluation of adsorption parameters, based on multiple solution of direct problems, is presented there.
Finally, Section 5 summarizes the results presented in this paper.

\section{Mathematical model}

Two dimensional pore scale transport of a dissolved substance in the presence of adsorption and desorption is considered. The coordinate along the porous domain (the vertical one) is denoted as $x_2$, while the coordinate across the porous domain (the horizontal one, coinciding with mean velocity direction) is denoted as $x_1$). Sketch of the flow is shown on Fig.\ref{f-1}.
The porous media geometry is modeled as periodic arrangement of cylinders.
A part of the domain, namely $\Omega_f$ is occupied by a fluid, while the other part is occupied be the obstacles $\Omega_s$. The obstacle surfaces (where the reaction occurs) are denoted by $\Gamma_s$, while symmetry lines are denoted by $\Gamma_{sim}$. It is supposed that dissolved substance is introduced via the inlet boundary
$\Gamma_{in}$, and the part of the substance which did not react flows out via $\Gamma_{out}$. The computational domain consists from $\Omega_f$.

\begin{figure}[ht]
  \begin{center}
    \begin{tikzpicture}
       \shade[top color=blue!5, bottom color=blue!5] (7,0) rectangle +(3,5);
       \shade[left color=blue!50, right color=blue!5] (3,0) rectangle +(4,5);
       \shade[top color=blue!50, bottom color=blue!50] (0,0) rectangle +(3,5);
       \draw [dashed] (0, 0) rectangle +(10,5);
       \draw [color=red] (0,4.25) rectangle +(10,0.5);
       \foreach \y in {3,...,6}
	\foreach \x in {0,...,4} {
  	\filldraw [fill=green,draw=black] (\y,\x+0.25) circle (0.24);
  	\filldraw [fill=green,draw=black] (\y+0.5,\x+0.75) circle (0.24);
       }
       \draw [<-, line width=2, color=blue] (1,1.5) -- (-1,1.5);
       \draw [<-, line width=2, color=blue] (1,2.5) -- (-1,2.5);
       \draw [<-, line width=2, color=blue] (1,3.5) -- (-1,3.5);
       \draw [<-, line width=2, color=blue] (11,1.5) -- (9,1.5);
       \draw [<-, line width=2, color=blue] (11,2.5) -- (9,2.5);
       \draw [<-, line width=2, color=blue] (11,3.5) -- (9,3.5);
       \draw [-] (2,4.5) -- (2,6);
       \draw  (2,6.4) node {$\Omega_f$};
       \draw [-] (4,4.375) -- (4,6);
       \draw  (4,6.4) node {$\Omega_s$};
       \draw [-] (6,4.5) -- (6,6);
       \draw  (6,6.4) node {$\Gamma_s$};
       \draw [-] (0,4.5) -- (-0.5,6);
       \draw  (-0.5,6.4) node {$\Gamma_{in}$};
       \draw [-] (7.5,4.25) -- (8,6);
       \draw [-] (8.5,4.75) -- (8,6);
       \draw  (8,6.4) node {$\Gamma_{sim}$};
       \draw [-] (10,4.5) -- (10.5,6);
       \draw  (10.5,6.4) node {$\Gamma_{out}$};
       \draw [->] (-0.5,-0.5) -- (1.5,-0.5);
       \draw  (1.9,-0.5) node {$x_1$};
       \draw [->] (-0.5,-0.5) -- (-0.5,0.5);
       \draw  (-0.5,0.9) node {$x_2$};
    \end{tikzpicture}
    \caption{Sketch of the pore scale domain}
   \label{f-1}
  \end{center}
\end{figure}

\subsection{Flow problem}

The flow in the pores is often slow, it is described here by the steady state incompressible Stokes equations:
\begin{equation}\label{1}
\nabla p - \mu \nabla^2 \bm{u} = 0,
\end{equation}
\begin{equation}\label{2}
 \nabla \cdot \bm{u} = 0,
 \quad  \bm{x} \in \Omega_f,
 \quad  t > 0,
\end{equation}
where $\bm{u}(\bm{x})$ and $p(\bm{x})$ are the fluid velocity and pressure, respectively,
while $\mu > 0$ and $\rho > 0$ are the viscosity and the density, which we assume to be constants \cite{bear2013dynamics,acheson2005elementary}.

Denote by $\bm n$ the outer normal vector to the boundary.
Suitable boundary conditions on $\partial \Omega_f$ are specified.
The velocity of the fluid $\bar{u}$ is prescribed at the inlet
\begin{equation}\label{3}
 \bm{u} \cdot \bm n = \bar{u},
 \quad \bm{u} \times  \bm n = 0,
 \quad \bm x \in \Gamma_{in} .
\end{equation}
At the outlet, pressure and absence of tangential force are prescribed
\begin{equation}\label{4}
 p - \bm \sigma \bm n \cdot \bm n = \bar{p},
 \quad \bm \sigma \bm n \times \bm n = 0,
 \quad \bm x \in \Gamma_{out} .
\end{equation}
Standard no-slip and no-penetration conditions are prescribed on the solid walls:
\begin{equation}\label{5}
 \bm{u} \cdot \bm n = 0,
 \quad \bm{u} \times  \bm n = 0,
 \quad \bm x \in \Gamma_{s} .
\end{equation}
Symmetry conditions are prescribed on the symmetry boundary of the computational domain:
\begin{equation}\label{6}
 \bm{u} \cdot \bm n = 0,
 \quad \bm \sigma \bm n \times \bm n = 0,
 \quad \bm x \in \Gamma_{sim} .
\end{equation}

\subsection{Species Transport}

The concentration of the solute in the fluid is denoted by $c(\bm x, t)$.
The unsteady solute transport in absence of homogeneous reactions is governed by convection diffusion equation
\begin{equation}\label{7}
 \frac{\partial c }{\partial t} + \nabla (\bm{u} c)
 - D \nabla^2 c = 0 ,
 \quad  \bm{x} \in \Omega_f,
 \quad  t > 0,
\end{equation}
where $D > 0$ is the solute diffusion coefficient which is assumed to be scalar and constant.

The concentration of the solute at the inlet is assumed to be known:
\begin{equation}\label{8}
 c(\bm x, t) = \bar{c},
 \quad \bm x \in \Gamma_{in} ,
\end{equation}
where $\bar{c} > 0$ is assumed to be constant.
Zero diffusive flux of the solute at the outlet and on
the external boundaries of the domain is prescribed as follows:
\begin{equation}\label{9}
 D  \nabla c \cdot \bm{n} = 0,
 \quad \bm x \in \Gamma_{sim} \cup  \Gamma_{out} .
\end{equation}
Note that convective flux via the outlet is implicitly allowed by the above equations.
The surface reactions that occur at the obstacles' surface $\Gamma_s$ satisfy the mass conservation law, in this particular case meaning that the change in adsorbed surface concentration is equal to the flux from the fluid to the surface. This is described as
\begin{equation}\label{10}
 \frac{\partial m}{\partial t} = - D  \nabla c \cdot \bm{n} ,
 \quad \bm x \in \Gamma_{s},
\end{equation}
where $m$ is the surface concentration of adsorbed solute \cite{kralchevsky2008chemical}.
A mixed kinetic–diffusion adsorption description is used:
\begin{equation}\label{11}
 \frac{\partial m}{\partial t} = f(c, m) .
\end{equation}
For reactive boundaries, the choice of $f$  and its dependence on $c$ and $m$ is critical for a correct description of the reaction dynamics
at the solid–fluid interface. A number of different isotherms (i.e., different functions $f(c,m)$) exist for describing these dynamics,
dependent on the solute attributes, the order of the reaction, and the interface type.

The simplest of these is the Henry isotherm, which assumes a linear relationship between the near surface
concentration and the surface concentration of the adsorbed particles, and takes the form
\begin{equation}\label{12}
 f(c, m) = k_a c - k_d m,
\end{equation}
Here $k_a \geq 0$ is the rate of adsorption, measured in unit length per unit time,
and $k_d \geq 0$ is the rate of desorption, measured per unit time.
The Langmuir adsorption isotherm is a more complicated, three parameter model:
\begin{equation}\label{13}
 f(c, m) = k_a c \left (1 - \frac{m}{m_\infty} \right )  - k_d m .
\end{equation}
Here $m_\infty > 0$  is the maximal possible adsorbed surface concentration.
In comparison to the Henry isotherm (\ref{12}), the
Langmuir isotherm (\ref{13}) predicts a decrease in the rate of adsorption as the adsorbed concentration increases
due to the reduction in available adsorption surface.

The formulation of the initial -- boundary value problem in addition to the governing equations
(\ref{1}), (\ref{2}), (\ref{7}), boundary conditions (\ref{3})--(\ref{6}), (\ref{8})--(\ref{11}), and specified isotherm (\ref{12}) or (\ref{13}), requires specification of initial conditions:
\[
 \bm{u}(\bm x, 0) = \bm{\bar{u}}(\bm x),
 \quad \bm x \in \Omega_f ,
\]
\begin{equation}\label{14}
 c(\bm x, 0) = c_0(\bm x),
 \quad \bm x \in \Omega_f .
\end{equation}
\begin{equation}\label{15}
 m(\bm x, 0) = m_0(\bm x),
 \quad \bm x \in \Gamma_s.
\end{equation}

\subsection{Dimensionless form of the equations}

When a problem like the above one need to be solved for a range of parameters (what is our goal here),
working with dimensionless form of the equations give definitive advantages.
For the dimensionless variables (velocity, pressure, concentration) below, the same notations are used as for the dimensional ones. The height of the computational domain $\Omega_f$, namely $l$, is used for scaling spatial sizes, the scaling of the velocity is done by the inlet velocity $\bar{u}$, and the scaling of the concentration is done by the inlet concentration $\bar{c}$.

The Stokes Eq.(\ref{3}) and its boundary conditions in dimensionless remain unchanged, keeping mind that in this case they are written with respect to dimensionless velocity and pressure, and considering the dimensionless viscosity to be equal to one.

In dimensionless form Eq. (\ref{7}) reads
\begin{equation}\label{20}
 \frac{\partial c }{\partial t} + \nabla (\bm{u} c)
 - \frac{1}{\mathrm{Pe}}  \nabla^2 c = 0 ,
 \quad  \bm{x} \in \Omega_f,
 \quad  t > 0,
\end{equation}
where
\[
 \mathrm{Pe} =  \frac{l \bar{u} }{D}
\]
is the Peclet number.

Further on, Eq. (\ref{8}) is transformed into
\begin{equation}\label{21}
 c = 1,
 \quad \bm x \in \Gamma_{in} ,
\end{equation}
while the boundary condition (\ref{9}) take the form
\begin{equation}\label{22}
 \nabla c \cdot \bm{n} = 0,
 \quad \bm x \in \Gamma_{sim} \cup  \Gamma_{out} .
\end{equation}
The dimensionless form of Eq.(\ref{10}) is given by
\begin{equation}\label{23}
 \frac{\partial m}{\partial t} = - \nabla c \cdot \bm{n} ,
 \quad \bm x \in \Gamma_{s},
\end{equation}
where $m$ is scaled as follows:
\[
 \bar{m} =  l \bar{c} .
\]
In dimensionless form the adsorption relations, in the case of Henry isotherm, are written as follows
\begin{equation}\label{24}
 \frac{\partial m}{\partial t} = \mathrm{Da}_a c - \mathrm{Da}_d m,
 \quad \bm x \in \Gamma_{s},
\end{equation}
where the adsorption and desorption Damkoler numbers are given by
\[
 \mathrm{Da}_a = \frac{k_a}{\bar{u}} ,
 \quad  \mathrm{Da}_d = \frac{k_d l}{\bar{u}} .
\]
In the case when we consider Langmuir isotherm, (\ref{11}), (\ref{13}), the following dimensionless relation is used
\begin{equation}\label{25}
 \frac{\partial m}{\partial t} = \mathrm{Da}_a c \left (1- \frac{m}{\mathrm{M}} \right )   - \mathrm{Da}_d m,
 \quad \bm x \in \Gamma_{s},
\end{equation}
where the dimensionless parameter $M$ is given by
\[
 \mathrm{M} = \frac{m_\infty }{l \bar{c}} .
\]

\section{Numerical solution of the direct problem}

Finite Element Method, FEM, is used for space discretization of the above problem, together with implicit discretization in time.
The algorithm used here for solving the direct problem is practically identical of the algorithm used to study oxidation in \cite{oxidation}. Therefore some details, e.g., grid convergence studies, will be omitted here.

\subsection{Geometry and grid}

The computational domain is a rectangle with a dimensionless height of $x_2=1$ and dimensionless length of $x_1=17.5$, in which ten half cylinders are embedded.
The distance between the centers of the cylinders in $x_1$ direction is 1.5 dimensionless units, the radius of cylinders is 0.4 dimensionless units.

The computational domain $\Omega_f$ is triangulated using the grid generator Gmsh (website gmsh.info) \cite{Gmsh}.
The script for preparing the geometry is written in Python. In order to control the accuracy of results, computations on consecutively refined grids are performed.

\subsection{Computation of steady state single phase fluid flow}

One way coupling is considered here. The fluid flow influences the species transport, but there is no back influence of the species concentration on the fluid flow. Based on this, the flow is computed in advance. The FEM approximation of the steady state flow problem \cite{gresho200incompressible} is based on variational formulation of the considered boundary value problem (\ref{1}), (\ref{2}), (\ref{3}), (\ref{4}), (\ref{5}), (\ref{6}).
The following functional space $\bm V$ is defined for the velocity $\bm u$  ($\bm u \in \bm V$):
\[
\begin{split}
 \bm V = \{ \bm v \in \bm H^1(\Omega) : \  & \bm{u} \cdot \bm n = 1, \
 \bm{u} \times  \bm n = 0 \ \mathrm{on} \ \Gamma_{in} ,
 \\ & \bm{u} = 0 \ \mathrm{on} \ \Gamma_{s}, \ \bm{u} \cdot \bm n = 0 \
 \ \mathrm{on} \ \Gamma_{sim} \} .
\end{split}
\]
Test function $\bm v \in \hat{\bm V}$, where
\[
 \hat{\bm V} = \{ \bm v \in \bm H^1(\Omega_f) : \ \bm{u} = 0 \ \mathrm{on} \ \Gamma_{in} , \
 \bm{u} = 0 \ \mathrm{on} \ \Gamma_{s}, \ \bm{u} \cdot \bm n = 0 \
 \ \mathrm{on} \ \Gamma_{sim} \} .
\]
For the pressure $p$ and the related test functions $q$, it is required that $p, q \in Q$, where
\[
 Q = \{ g \in L_2(\Omega_f) : \ q = 0 \ \mathrm{on} \  \Gamma_{out} \} .
\]

Let us multiply Eq.(\ref{1}) by $\bm v$, Eq.(\ref{2}) by $q$, and integrate over the computational domain. Taking into account the boundary conditions (\ref{3}), (\ref{4}), (\ref{5}), (\ref{6}), the following system of equations is obtained
with respect to $\bm v \in \bm V$, $q \in Q$
\begin{equation}\label{26}
 a(\bm u, \bm v) - b(\bm v, p) = 0 \ \forall \bm v \in \hat{\bm V} ,
\end{equation}
\begin{equation}\label{27}
 b(\bm u, q) = 0 \ \forall q \in Q .
\end{equation}
Here
\[
  a(\bm u, \bm v) := \int_{\Omega_f} \nabla \bm u \cdot \nabla \bm v \, d \bm x ,
\]
\[
 b(\bm v, p) := \int_{\Omega_f}(\nabla \cdot \bm v) q \, d \bm x .
\]

For the FEM approximation of the velocity, the pressure, and the respective test functions, the following finite dimensional subspaces are selected
$\bm V_h \subset \bm V$, $\hat{\bm V}_h \subset \hat{\bm V}$
and $Q_h \subset Q$.
Taylor-Hood $P_2-P_1$ elements \cite{taylor1973numerical} are used here.
These are continuous $P_2$ Lagrange elements for the velocity components
and continuous $P_1$ Lagrange elements for the pressure field.
The computations are carried out using the computing platform for partial differential equations FEniCS (website fenicsproject.org) \cite{LoggMardalEtAl2012a,AlnaesBlechta2015a}.

As mentioned above, mainly slow flows are of interest for the current study, therefore the basic considered variant is characterized by $\mathrm{Re} =1$. Computed velocity components and pressure are shown on Fig.\ref{f-2}. Convergence of the solution with respect to refinement of the grid is illustrated on Fig.\ref{f-3}.
We have used three computational grids: basic grid with 18743 nodes and 35958 triangles, coarse grid with 4760  nodes and 8754 triangles and fine grid with 72745  nodes and 142460 triangles.
Fig.\ref{f-3} shows velocity components and pressure on the middle line of the computational domain, computed on the fine grid. The differences ($\delta u_1, \delta u_2, \delta p$) between solutions computed on the fine grid and solutions computed on the basic and on the coarse grid are presented to illustrate the convergence. For convenience, in the visualization these differences are multiplied by 100. From the results, it can be concluded that the basic grid provides good accuracy for the numerical solution.

\begin{sidewaysfigure}
  \begin{center}
    \includegraphics[width=0.5\linewidth] {2-1l.png} \vspace{-8mm} \\
    \includegraphics[width=1\linewidth] {2-1.png} \\
	a \\
    \includegraphics[width=0.5\linewidth] {2-2l.png}  \vspace{-8mm} \\
    \includegraphics[width=1\linewidth] {2-2.png} \\
	b \\
    \includegraphics[width=0.5\linewidth] {2-3l.png}  \vspace{-8mm} \\
    \includegraphics[width=1\linewidth] {2-3.png} \\
	c
  \caption{Velocity and pressure for $\mathrm{Re} = 1$: a --- horizontal component of velocity vector $u_1$,
           b --- vertical component of velocity vector $u_2$, c ---pressure}
  \label{f-2}
  \end{center}
\end{sidewaysfigure}

\begin{sidewaysfigure}
  \begin{center}
    \includegraphics[width=1.1\linewidth] {3-1.png} \\
	a \\
    \includegraphics[width=1.1\linewidth] {3-2.png} \\
	b \\
    \includegraphics[width=1.1\linewidth] {3-3.png} \\
	c \\
  \caption{Velocity components and pressure along the middle line of of the computational domain, computed on fine grid,
     as well as differences with the solution on coarser grids:
     a --- velocity component $u_1$, b --- velocity component $u_2$, c --- pressure.   }
  \label{f-3}
  \end{center}
\end{sidewaysfigure}

\subsection{Simulation of reactive transport}

The unsteady species transport problem (\ref{20}),  (\ref{15}), (\ref{21})-- (\ref{23}) is solved numerically using standard Lagrangian $P_1$ finite elements. Let us define
\[
 S = \{ s \in H^1(\Omega_f) : \ q = 1 \ \mathrm{on} \  \Gamma_{in} \} ,
\]
\[
 \hat{S} = \{ s \in H^1(\Omega_f) : \ q = 0 \ \mathrm{on} \  \Gamma_{in} \} .
\]
The approximate solution $c \in S$ is sought from
\begin{equation}\label{28}
\left (\frac{\partial c}{\partial t}, s \right) + d(c,s) =
 \left (\frac{\partial m}{\partial t} ,s \right )_s
 \quad \forall s \in \hat{S} ,
\end{equation}
where the following notations are used
\[
  d(c,s) := - \int_{\Omega_f} c \bm u \cdot \nabla  s \, d \bm x
  + \frac{1}{\mathrm{Pe}} \int_{\Omega_f} \nabla c \cdot \nabla  s \, d \bm x
  + \int_{\Gamma_{out}} (\bm u \cdot \bm n) c s \, d \bm x ,
\]
\[
  (\varphi,s)_s := - \int_{\Gamma_{s}} \varphi   s \, d \bm x .
\]
For determining $m \in G = L_2(\Gamma_{s})$ (see (\ref{11})) we use
\begin{equation}\label{29}
 \left (\frac{\partial m}{\partial t} , g \right )_s -
 (f(c,m), g)_s = 0,
 \quad g \in G .
\end{equation}
The discretization in time is based on symmetric discretization (Crank–Nicolson method), which is second order accurate (see, e.g., \cite{Samarskii,Ascher2008}.
Let $\tau$  be a step-size of a uniform grid in time such that
$c^n = c(t^n), t^n = n\tau, \ n = 0, 1, ...$.

Eq.(\ref{28}) is approximated in time as follows
\[
\left (\frac{c^{n+1} - c^n}{\tau }, s \right) + d\left (\frac{c^{n+1} + c^n}{2} ,s\right ) =
 \left (\frac{m^{n+1} - m^n}{\tau } ,s \right )_s .
\]
Similarly, for (\ref{29}) we get
\[
\left (\frac{m^{n+1} - m^n}{\tau }  , g \right )_s -
 \left (f\left (\frac{c^{n+1} + c^n}{2} ,\frac{m^{n+1} + m^n}{2} \right ), g\right )_s = 0 ,
 \quad n = 0, 1, ... ,
\]

In the considered here case, zero initial conditions are posed
\[
 c^0 = 0, \quad \bm x \in \Omega_f,
\]
\[
 m^0 = 0, \quad \bm x \in \Gamma_s .
\]

Henry adsorption isotherm (\ref{24}) is used in the simulations reported below. The basic set of parameters reads as follows:
\[
 \mathrm{Pe} = 10,
 \quad \mathrm{Da}_a = 0.005,
 \quad \mathrm{Da}_d = 0.05,
\]
The unsteady problem is solved on dimensionless time interval $(0,T)$, $T = 40$, using time step $\tau =0.1$.
Breakthrough curve (i.e., average outlet concentration) is used to characterize the reactive transport:
\begin{equation}\label{30}
 c_{out}(t) = \frac{\int_{\Gamma_{out}} c(\bm x, t) d \bm x}{\int_{\Gamma_{out}} d \bm x} .
\end{equation}

\begin{sidewaysfigure}
  \begin{center}
    \includegraphics[width=0.5\linewidth] {4-0.png} \vspace{-8mm} \\
    \includegraphics[width=1.1\linewidth] {4-1.png} \\
	 a \\
    \includegraphics[width=1.1\linewidth] {4-2.png} \\
	 b \\
    \includegraphics[width=1.1\linewidth] {4-3.png} \\
	 c
  \caption{Concentration $c$ at different time moments, a --- $t = 5$,	b --- $t = 10$, c --- $t = 15$}
  \label{f-4}
  \end{center}
\end{sidewaysfigure}

\begin{sidewaysfigure}
  \begin{center}
    \includegraphics[width=1.1\linewidth] {5-1.png} \\
	a \\
    \includegraphics[width=1.1\linewidth] {5-2.png} \\
	b \\
    \includegraphics[width=1.1\linewidth] {5-3.png} \\
	c \\
  \caption{Concentration $c$ computed with $\tau = 0.05$ and its difference $\delta c$
           from the concentrations computed with $\tau = 0.1$   and $\tau = 0.2$ at different time moments:
           a --- $t = 5$, b --- $t = 10$, c --- $t = 15$}
  \label{f-5}
  \end{center}
\end{sidewaysfigure}

Species concentration at different time moments are shown on Fig.\ref{f-4}.
The accuracy of the approximate solution with respect to the time step is illustrated on Fig.\ref{f-5}.
The solution along the middle line of the computational domain is shown there for the case $\tau = 0.05$,
as well as the magnified by factor 100 differences between this solution and solutions obtained
with $\tau = 0.1$ and $\tau = 0.2$.
The results show that a good accuracy can be achieved using $\tau = 0.1$,
and the further simulations are performed using this time step.

Sensitivity study is carried out in order to see how the change of different parameters leads to change of the breakthrough curves. Such sensitivity studies are often first stage for optimization or parameter identification procedures. The dependence of the average outlet concentration from Peclet number is shown on Fig.\ref{f-6}a. Smaller $\mathrm{Pe}$ means larger diffusion, what is well illustrated by the spreading of the outlet concentration.
Note that the amount of the deposited concentration is given by $1- c_{out}(t)$.

\begin{figure}[htp]
  \begin{center}
 	\includegraphics[width=0.49\linewidth] {6-1.png}
 	\includegraphics[width=0.49\linewidth] {6-2.png} \\
	\makebox[0.49\linewidth][c] {a} \makebox[0.49\linewidth][c] {b}
 	\includegraphics[width=0.49\linewidth] {6-3.png}
 	\includegraphics[width=0.49\linewidth] {6-4.png} \\
	\makebox[0.49\linewidth][c] {c} \makebox[0.49\linewidth][c] {d}
	\caption{Outlet concentration: a --- dependence from  $\mathrm{Pe}$,
            b --- dependence from $\mathrm{Da}_a$, c --- dependence from $\mathrm{Da}_d$, d --- dependence from $\mathrm{M}$}
	\label{f-6}
  \end{center}
\end{figure}

The rate of adsorption is characterized by $\mathrm{Da}_a$. The influence of this parameter on the outlet concentration is illustrated on Fig.\ref{f-6}b. Increasing $\mathrm{Da}_a$, as expected, leads to more intensive adsorption and larger amount of the deposited substance. The influence of the parameter $\mathrm{Da}_d$ on the outflow concentration is illustrated on Fig.\ref{f-6}c. In the current paper we restrict our considerations to the case of small Pe and Damkoler numbers, the study for other regimes will be presented in forthcoming papers.

Henry isotherm usually describes well the initial stages of the adsorption. In cases when only a limited mass can be adsorbed at a surface, Langmuir isotherm should be used to reflect the decay of the adsorption rate close to the saturation. In this case an additional parameter appears, namely $\mathrm{M}$ (see (\ref{25})).
The influence of $\mathrm{M}$ on the average output concentration is shown on Fig.\ref{f-6}d.

\clearpage
\section{Numerical solution of the inverse problem}

Consider an inverse problem for determining unknown adsorption rate (so called parameter identification problem), using information about the dynamics of the average outlet concentration $\widetilde{c}_{out}(t)$ (the latter is usually easy to measure in experiments). The starting point is monitoring of the difference (residual) between the measured $\widetilde{c}_{out}(t)$ and the computed  $c_{out}(t)$ average outflow concentration for different values of the parameters $\mathrm{Da}_a$ and $\mathrm{Da}_d$ in Henry isotherm Eq.(\ref{24}).

\subsection{Direct computation of the residual functional}

Let us consider the problem for identifying the adsorption and desorption rates in Langmuir isotherm, $\mathrm{Da}_a, \mathrm{Da}_d$, respectively, using given measurement results $\widetilde{c}_{out}(t)$. The functional of the residual is given by
\begin{equation}\label{31}
 J(\mathrm{Da}_a, \mathrm{Da}_d) = \int_{0}^{T} (c_{out}(t) - \widetilde{c}(t))^2 d t,
\end{equation}
where $c_{out}(t)$ is given by (\ref{30}) after solving the dimensionless form of (\ref{3}) together with the respective boundary conditions, and after solving (\ref{20})--(\ref{23}). We use the so called synthetic approach to emulate measurements data. At a preliminary stage, the direct problem is solved for  selected fixed values of ($\widetilde{\mathrm{Da}}_a, \ \widetilde{\mathrm{Da}}_d$) (called later exact solution of the identification procedure, or exact parameters, here $(0.005,0.05)$),and the computed $c_{out}(t)$ is used as measurement data in the case of no noise, i.e., in this case $\widetilde{c}(t)=c_{out}(t)$.

First of all, having in mind that this is a relatively simple parameter identification problem (only two parameters need to be identified, the direct problem can be fast and efficiently solved numerically), a simple strategy for parameter identification will be considered here. Consider starting feasible set $G$ defined as
\[
 0 \leq \mathrm{Da}_a \leq 0.01,
 \quad 0 \leq  \mathrm{Da}_d \leq 0.1 .
\]
Computations are performed on uniform grid for the parameters $\mathrm{Da}_a, \ \mathrm{Da}_d$ with $51\times 51$ nodes, and the direct problem is solved for each pair of parameters from this grid. Square root from the residual functional
is shown on Fig.\ref{f-7}a. Here and below the isolines are visualized using the library {\it matplotlib}. The isolines of the functional are drawn with step $0.02$. In this case the minimum of the functional is zero, $\min J(\mathrm{Da}_a, \mathrm{Da}_d) = 0$, and it is achieved at point $(0.005,0.05)$). Recall that this point was used to emulate by computations the measurement data.  In this particular case this point belongs to the parameter grid on which we seek the solution, therefore $\min J(\mathrm{Da}_a, \mathrm{Da}_d) = 0$ in this case.

\begin{figure}[htp]
  \begin{center}
 	\includegraphics[width=0.75\linewidth] {7-1.png} \\
	a \\
  	\includegraphics[width=0.75\linewidth] {0_01_191120.png}  \\
	b \\
 	\includegraphics[width=0.75\linewidth] {0_05_191120.png} \\
	c \\
	\caption{Square root of the residual functional ($J^{1/2}(\mathrm{Da}_a, \mathrm{Da}_d)$): a --- exact data ($\delta = 0$),
                  b --- $\delta = 0.01$,  c --- $\delta = 0.05$}
	\label{f-7}
  \end{center}
\end{figure}

\subsection{The influence of the noise in the measured data}

In the reality the measurement data are often noisy, therefore it is important to understand how the noise influences the parameter identification. Suppose that the measurement are carried out at each time step. Instead of (\ref{30}), let us consider noisy measurement:
\[
  \widetilde{c}(t^n) = c_{out}(t^n; \widetilde{\mathrm{Da}}_a, \widetilde{\mathrm{Da}}_d) +
  \delta \sigma(t^n),
  \quad n = 1,2, ..., N,
  \quad N \tau = T .
\]
Here the parameter $\delta$ quantifies the amplitude of the noise, while $\sigma(t^n)$ is a random variable, uniformly distributed on the interval $[-1,1]$.

In the case of noisy measurement data, instead of trying to identify one point in the parameters space (what is the most often considered case in parameter identification problems), one should identify a set in the parameters space, for which the value of the residual functional is below prescribed tolerance. Having in mind the dispersion of the uniformly distributed random variable $\sigma$, we define the set of the admissible parameters by the inequality
\begin{equation}\label{33}
J(\mathrm{Da}_a, \mathrm{Da}_d) \leq  \gamma \delta^2 \frac{T}{3} ,
\end{equation}
where $\gamma > 1$ is a numerical parameter.
Here and below in order to avoid misunderstanding, we will call {\it admissible set, or resultimg admissible set, or final admissible set} the set which is identified as a result of the identification, thus making difference with {\it feasible set or starting feasible set}, which is the parameter set $G$ which is input for our identification procedure.

In the above a priori estimate the definition of the admissible set explicitly depends on the time interval during which the measurements are done. Obviously, it may be reasonable to ignore the measurement data before the breakthrough, as well as to ignore the measurement data after an equilibrium is achieved. The question about the minimum required information for an identification procedure, e.g., with which time interval it is reasonable to work, is not discussed here. Relatively large time interval for the measurements is considered here, see Fig.\ref{f-8}.

One realization of the noisy data for the the average outlet concentration is shown on Fig.\ref{f-8}a for the case $\delta = 0.01$. Square root of the residual functional for this case is visualized by color on Fig.\ref{f-7}b, some isolines are also plotted. In this case, due to the noise, the minimum of the functional is not equal to zero, it is equal to $0.0361$. The point at which the functional takes its minimum is marked with $\times$ on the figure. The exact parameters are marked with $\star$ on the figure.
The red square on the picture, centered around the star (which almost coincides with the $\times$ in this case), is a drawing of $(0.99\times1.01)\mathrm{Da}_a,  (0.99\times1.01)\mathrm{Da}_d$ isolines of the residual. That is, within this square the error in identifying the exact parameters is less than $1\%$. Furthermore, the first plotted isoline for the residual $J$, is computed from (\ref{33}) for $\gamma=1.02625$.

Similar results for significantly higher amplitude of the noise, ($\delta = 0.05$, Fig.\ref{f-8}b) are shown on Fig.\ref{f-7}c. In this case, the minimum of the functional is equal to $0.1840$. The red square in this case indicates $5\%$ error, to be in the same range as the noise. In this case the first plotted isoline for the residual $J$ is also calculated with $\gamma=1.02625$. We see that the minimum of the functional in this case is achieved in a point which is relatively far from the exact value, indicating that one measurement can not be enough to identify the parameters accurately in the case of very noisy data. This is manifestation of the known fact that more Monte Carlo simulations (simulations with different noise realizations, see the next paragraph) have to be done for larger amplitude of the noise.

\begin{figure}[htp]
  \begin{center}
 	\includegraphics[width=0.49\linewidth] {8-1.png}
 	\includegraphics[width=0.49\linewidth] {8-2.png} \\
	\makebox[0.49\linewidth][c] {a} \makebox[0.49\linewidth][c] {b}
	\caption{Average outflow concentration: a --- $\delta = 0.01$, b --- $\delta = 0.05$}
	\label{f-8}
  \end{center}
\end{figure}

\begin{comment}
The calculated range of parameters with different noise levels are shown on Fig.\ref{f-9} for one realization of noise with $\delta=0.01$ and for one realization of noise with $\delta=0.05$. The shown isolines are computed, with the first isoline selected from (\ref{33}) by choosing $\gamma=1.02625$. Obviously, larger amplitude of the noise leads to larger set of admissible parameters due to the flattening of the residual functional. The plotted squares have the same meaning as above. It can be seen that exact solution (denoted by $\star$) and the point at which the minimum of the functional is achieved for this particular noise realization (denoted by $\times$), are very close when $\delta=0.01$. This is not the case for $\delta=0.05$. This is manifestation of the known fact that more Monte Carlo simulations (simulations with different noise realizations, see the next paragraph) have to be done for larger amplitude of the noise.
\begin{figure}[htp]
  \begin{center}
    \includegraphics[scale = 0.3] {Adm_set_191120.png}
	\caption{Admissible set of adsorption parameters for different noise amplitudes $\delta$: $\delta = 0.01$ --- red,
    $0.05$ --- blue}
	\label{f-9}
  \end{center}
\end{figure}
\end{comment}

To investigate the role of the stochasticity in the noise, simulations with ten different realizations of the noise are done for each of the noise levels $\delta = 0.01$ and $\delta = 0.05$. The resulted admissible sets are shown on Fig.\ref{adm001} and Fig.\ref{adm005}. For a better visualization, here one isoline is plotted for each realization, and relatively large $\gamma=1.21$ is selected for this isoline. The points in which the minima of the functional are achieved are close to the exact point for $\delta=0.01$ and therefore they are not plotted. For $\delta=0.05$ they are denoted with $\times$ on the figure. The overlapping area is shrinking, indicating that the accuracy of the parameter identification can be improved by performing a number of simulations for different realizations of the noise. It should be noted that in \cite{maday2015pbdw} the authors have investigated the role of the noisy data in connection with another parameter identification problem. They have proven that under certain conditions on the noise (which are satisfied also in our case), within a Monte Carlo procedure (working with sufficiently large set of measurement data), the exact parameters can be identified. However, very large number of measurements may be needed in the case of large variance. Such an approach is important in the case when the parameters have to be identified with high accuracy. However, in many industrial problems the goal is to identify the parameters with an accuracy which is of the order of the noise. For such problems, it makes sense to identify admissible sets of parameters, instead of trying to identify the exact parameters.

It should be noted that it is not granted that the computed final admissible sets for two different realizations of the noise will overlap. If this will be the case with some of the computed final admissible sets, a subset of overlapping admissible sets has to be chosen in order to select the shrinked area. We leave the discussion on Monte Carlo and other approaches for more accurate parameter identification in the case of noisy data for a future article.

\begin{figure}[htp]
  \begin{center}
    \includegraphics[width=1\linewidth]{adm_fill_0_01.png}
	\caption{Admissible set of adsorption parameters for ten realizations for noise amplitude $\delta = 0.01$}
	\label{adm001}
  \end{center}
\end{figure}

\begin{figure}[htp]
  \begin{center}
    \includegraphics[width=1\linewidth]{adm_mins_0_05.png}
	\caption{Admissible set of adsorption parameters for ten realizations for noise amplitude $\delta = 0.05$}
	\label{adm005}
  \end{center}
\end{figure}

\subsection{Statistical parameter identification}

The problem for identifying adsorption parameters in this case is formulated as follows.
Find resulting (final) admissible set $\mathrm{Da}_a, \mathrm{Da}_d$ satisfying (\ref{33}) from a starting set of admissible parameters $G$.
This problem can be solved by different local, and in general case, global optimization approaches \cite{horst2013handbook,nocedal2006numerical}.

Recall, that the standard approach for solving parameter identification problems for PDEs \cite{samarskii2007numerical,kern2016numerical} does not target at determining an admissible set of parameters, instead, it targets at determining a particular point in the parameter space. Usually gradient methods are used as a part of such approaches. The evaluation of the admissible set of parameters for such deterministic approaches can be based on iterative minimization of the residual functional starting from different initial values, the so called multistart method.

As alternative we consider here stochastic methods and their variants based on deterministic sampling of points. A stochastic approach for global optimization in its simplest
form consists only of a random search and it is called Pure Random Search
\cite{Zhigljavsky2008}. In this case the residual functional
is evaluated at randomly chosen points belonging to the feasible (initial) parameter set. Such an stochastic approach is more efficient than the computation of the residual function on uniform grid in the feasible set.

Consider the case when the starting feasible set $G$ is defined in a simple way:
\[
 0 \leq \mathrm{Da}_a \leq \overline{\mathrm{Da}}_a,
 \quad  0 \leq \mathrm{Da}_d \leq \overline{\mathrm{Da}}_d .
\]
To generate of uniformly distributed points $D$ from $G$, Sobol sequences \cite{sobol1976uniformly,sobol1979systematic} are used. The software implementation is based on the sensitivity analysis library SALib \cite{Herman2017}. Such an approach is widely used in applied problems for multicriteria parameter identification (see., e.g. \cite{sobol1981choosing}).

\begin{figure}[htp]
  \begin{center}
  	\includegraphics[width=0.75\linewidth] {150_points_only_with_exact.png} \\
	a \\
  	 	\includegraphics[width=0.75\linewidth] {600_points_only_with_exact.png} \\
	b \\
	\caption{Deterministic sampling of points, plot of points for which the inequality (\ref{33}) is satisfied, $\gamma=1.02625$ ($\delta = 0.01$ --- red, $0.03$ --- green,  $0.05$ --- blue): a --- $N = 150$,
 	b ---

 $N = 600$}
	\label{f-10}
  \end{center}
\end{figure}

The result of the statistical identification for
$N = 150$ sampling points is shown on Fig.\ref{f-10}a. Similar results for
$N = 600$ sampling points is shown on Fig.\ref{f-10}b
In both cases we are plotting the points from the random set for which the value of the residual satisfies (\ref{33}) with $\gamma=1.02625$.
It can be seen that for small number of the sampling points it is difficult to reconstruct reasonable final admissible set because only few points satisfy (\ref{33}). More points are available for larger amplitude of the noise because of the flattening of the functional, but at the same time the accuracy is worse in this case.

\subsection{Multistage parameter identification}

In the considered here case, when only two parameters need to be identified and we deal with convex functional, one can consider multistage parameter identification (could also be called predictor-refinement parameter identification).  Instead of increasing the number of the samples, one can consider a multistage method, consecutively shrinking the starting feasible set $G$. For example, the results (see Fig.\ref{f-7}a) obtained with $N = 150$ samples for the original initial feasible set $G$
\[
0. \leq \mathrm{Da}_a \leq 0.01,
\quad  0. \leq \mathrm{Da}_d \leq 0.1 .
\]
can be evaluated and a new, smaller starting feasible set can be defined for the second stage of the identification procedure, e.g.;
\[
 0.002 \leq \mathrm{Da}_a \leq 0.009,
 \quad  0.03 \leq \mathrm{Da}_d \leq 0.07 .
\]
The simulation with $N = 150$ sample in the reduced domain at the second stage of the identification procedure, give good accuracy in determining the final admissible set, see Fig.\ref{f-11}.
\begin{figure}[htp]
  \begin{center}
    \includegraphics[scale = 0.3] {11.png}
	\caption{Shrinked domain of feasible parameters for $N = 150$: $\delta = 0.01$ --- red, $0.03$ --- green,  $0.05$ --- blue}
	\label{f-11}
  \end{center}
\end{figure}

Additional peculiarity of the considered here problem is the fact that different adsorption parameters have different influence on the solution at different time intervals. For example, results from Fig.\ref{f-6} indicate that the dynamic of the outflow concentration is different on the subintervals left and right from $T = 15$. This makes reasonable the consideration of the residual functional (\ref{31}) at a smaller time interval, namely till $T = 15$. Fig.\ref{f-12} shows admissible set computed at different noise amplitude in this case. The comparison with simulations done on $(0,T)$ for $T = 40$ (see Fig.\ref{f-10}a) shows that with relatively small number of samples we have identified one of the two parameters ($\mathrm{Da}_d$ in this case) with a reasonable accuracy. We can use this as a starting point for a next stage, considering smaller set $G$, (for which
$0.03 \leq \mathrm{Da}_d \leq 0.06$).

\begin{figure}[htp]
  \begin{center}
    \includegraphics[scale = 0.3] {12.png}
	\caption{Usage of input data till $T = 15$ with $N = 150$: $\delta = 0.01$ --- red, $0.03$ --- green,  $0.05$ --- blue}
	\label{f-12}
  \end{center}
\end{figure}

\section{Conclusion}
Deterministic and stochastic, single stage and multistage algorithms for identification of unknown adsorption and desorption rates in Langmuir isotherm are presented in conjunction with pore scale simulation of reactive flow. Breakthrough curves are the extra data needed for the solution of the parameter identification problem. Exact and noisy data are considered. In the latter case their impact on the identification procedure is discussed.

\begin{enumerate}
 \item
The 2D mathematical model of the direct problem includes steady state Stokes equations, and convection--diffusion equation supplemented with Robin type boundary conditions accounting for adsorption and desorption. Henry and Langmuir isotherms describe the kinetics. A simple pore scale geometry described by periodic arrangement of cylindrical obstacles, is considered for illustration of the identification procedure. The key dimensionless parameters are specified. The approach is applicable for wide range of microgeometries and process parameters.

 \item
The numerical solution is based on triangular grids and FEM with Taylor and Hood elements. Computations on series of refined grids are performed to confirm the convergence.

 \item
Mass transport is simulated for as given velocity field (one way coupling). The numerical solution is based on FEM with piecewise linear elements. Crank-Nikolson scheme is used in the time discretization. Sensitivity studies are carried out to investigate the influence of different parameters on the reactive transport through the porous media.

 \item
Deterministic and stochastic, single stage and multistage identification procedures are presented and tested. The influence of the noise in the data on the accuracy of the identification procedure is discussed. It should be pointed that in the real problem one can not identify single values for the seek parameters, instead, final admissible sets of parameters are identified.
\end{enumerate}

\section*{Acknowledgements}

VVG acknowledges the support by Mega-grant of the Russian Federation Government, N 14.Y26.31.0013. PNV acknowledges the support by Mega-grant of the Russian Federation Government, N 14.Y26.31.0013 and by the RFBR, N 17-01-00689.

redpajama_set_name: RedPajamaArXiv


Joe and I rode his motorcycle to Blissfield. Michigan. We then went to Hidden Lake Gardens in Tipton, Michigan where we went on a hike. We ate hot dogs in Techumseh, had coffee in Milan, and came back to Blissfield for dinner at an old horse stable, turned into a nice restaurant. It was a wonderful day of memory making! I absolutely love riding with him on the back of his motorcycle.


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The roadmap to driving better business outcomes
Research indicates that there is massive ROI in investing in data and analytics that support decision-making on managing risks and leveraging opportunities. According to Forrester, high-growth B2B businesses invest in brand when facing economic uncertainty, especially in brand purpose, in times of climate change, a pandemic, social injustice and economic threats.
Taking cues from the financial services sector, which is a mature industry in terms of data and analytics, getting one or two decisions right on how to hedge risk or embrace an opportunity early on could pay for many years of data analytics subscriptions to come. Isolating relevant signals from the data noise and closing the insight-to-action gap requires vision and collaboration across organizations – as well as high-velocity information that is essential to decision-making.
"Organizations can have all the data in the world, but it takes people to interpret the stories from the data and bring about creative differentiation to drive informed decision-making and business outcomes," Boyer says. "This requires balancing human intuition and data intellect, enabled by technology and quality data."
"Given the amount of noise across channels, it's critical for organizations to re-evaluate data signals with an eye on getting to know their customers better, improving the customer experience through personalization and innovating the next winning product and service.
"Having access to real-time data and the right tools further allows organizations to take action in a customer's moment of need or make predictions about future outcomes with the mindset that these efforts are continuous works in progress."
Organizations across the globe can find opportunity in understanding and using values as foundations for their brand strategy. When mountains of data leave them feeling overwhelmed, devising a roadmap for improvement that balances human intuition with data intellect is an effective way to drive better business outcomes.
The 5 main consumer segments and their corresponding key values, according to GfK:
Discoverers: Freedom, creativity, curiosity, open-mindedness, learning, knowledge, social tolerance, equality, internationalism
Indulgers: Enjoying life, leisure, pleasure, material security
Elementals: Helpfulness, modesty, thrift, simplicity, respecting ancestors
Mindfuls: Spirituality, working hard
Aspirers: Social recognition, status
By Julie CooperFull Bio
Julie Cooper is an American journalist on Sydney's stunning Northern Beaches. When she's not writing inspiring executive profiles for top leaders from around the globe at The CEO Magazine, you can find her at the barre, drinking a green juice and telling people she's Canadian to avoid awkward talks about politics.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Add audio player to active admin panel I have a Blog that accepts comments with audios files attached with Rails 6.
Is it possible to show the audios and listen to them in the active admin dashboard?
I have
#/models/comment.rb
class Comment < ApplicationRecord
  belongs_to :post
  belongs_to :user

  has_one_attached :audio_file
end

#/admin/comments.rb
ActiveAdmin.register Comment do
  actions :all, except: [:edit, :new]
  permit_params :audio_file, :post_id, :created_at, :deleted_by
end

I have try adding
  index do
    selectable_column
    id_column
    column :audio_file
    column :challenge_id
    column :created_at
    column :deleted_by
    actions
  end

But it just add the text audio_file.
How can I add the audio_tag with the audio_file attached?

A: Try to show the audio tag with rails audio_tag helper(https://apidock.com/rails/ActionView/Helpers/AssetTagHelper/audio_tag) like below.
index do
    selectable_column
    id_column
    column :audio_file do |comment|
      if comment.audio_file.attached?
        audio_tag(comment.audio_file, autoplay: true, controls: true)
      else
        # Do whatever needed
      end
    end
    column :challenge_id
    column :created_at
    column :deleted_by
    actions
  end

redpajama_set_name: RedPajamaStackExchange


"Wonder Woman" fans can expect a fully formed female superhero for DC's sequel. With this year's "Wonder Woman" film mostly serving as an origin story for the character, fans did not really have the opportunity to watch the Amazonian heroine fully formed until right near the end.
Speaking briefly about the upcoming sequel with ET Online, director Patty Jenkins has promised that Gal Gadot's character will be a 'full blown' Wonder Woman in the follow-up, one more akin to the one seen in "Batman v Superman: Dawn of Justice" and the upcoming "Justice League".
"The greatest thing about making this [the first Wonder Woman] movie was the fact that you're really building to the Wonder Woman that we all love, but not until the end of the movie.
It is still not clear what period the next "Wonder Woman" movie will take place in, though rumors of a 1980s Cold War setting with Diana taking on the Soviets has been around for a while. Wonder Woman has worldwide box office earnings exceeding $800 million and an impressive 92 percent rating on Rotten Tomatoes.
Wonder Woman stars Gal Gadot (The Fast and the Furious films, Batman v Superman: Dawn of Justice), Chris Pine (Star Trek), Robin Wright (The Girl with the Dragon Tattoo, Netflix's House of Cards), Danny Huston (Clash of the Titans, X-Men Origins: Wolverine), Ewen Bremner (Exodus: Gods and Kings, Snowpiercer), Saïd Taghmaoui (American Hustle), Elena Anaya (The Skin I Live In), Connie Nielsen (Gladiator, Nymphomaniac: Vol. I) and Lucy Davis (Shaun of the Dead).
What do you hope to see out of the sequel to Wonder Woman? What did you think of Wonder Woman's origin story? Let us know in the comments section.


text:
"""
Settings for OSF OAuth2 demo client
"""

import os

DEBUG = True
SECRET_KEY = "Never commit a secret key to production. For our purposes, though, the key just needs to exist"
SESSION_COOKIE_NAME = 'OAUTH_DEMO'

API_BASE_URL = 'http://localhost:8000/v2/'

######## OAUTH2 CLIENT SETTINGS ##########
# Describe the application as registered on the OSF
# NOTE: If the callback URL is not registered (or does not match one that is registered), you will get a
#   "successful login" screen from CAS rather than a confirmation page. This is not intuitive, so make sure the
#   callback URL you provide below exactly matches the one used by your application.

# This file contains fake locally generated data; never commit production secret data to Github. Use env vars instead
CALLBACK_URL = 'http://localhost:5001/callback'  # Must exactly match the URL registered with app
CLIENT_ID = os.environ.get('OSF_OAUTH_CLIENT_ID',
                           'b90cf426f99849988d648a0a68987a22')
CLIENT_SECRET = os.environ.get('OSF_OAUTH_CLIENT_SECRET',
                               '7R0XjBt6VvzGhaQxkLDuAptyiDhTCXLvFSIheQUi')

# Server URLs (from the API documentation)
AUTH_BASE_URL = 'https://localhost:8443/oauth2/authorize'
TOKEN_REQUEST_URL = 'https://localhost:8443/oauth2/token'
TOKEN_REFRESH_URL = TOKEN_REQUEST_URL

# If in debug mode, allow oauth client to work without using and/or checking HTTPS
REQUIRE_HTTPS = not DEBUG

redpajama_set_name: RedPajamaGithub


Receipt Bank can make your bookkeeping faster, easier and more efficient.
Many businesses lose money every year because they fail to claim expenses they are eligible for. And with pockets or draws full of paper, it's no wonder receipts can get lost or forgotten. At DTA Chartered Accountants, we are aware of how costly this can be and that is exactly why we have partnered with Receipt Bank.
Receipt Bank is a unique data extraction solution that is easy and simple to use. This award-winning software converts all your bills, invoices and receipts into data.
All you have to do is use the Receipt Bank app to take photos of your receipts and invoices. You can then email them in or upload them straight to your account. The app extracts all the relevant information and even organises it for you. Storing it secure online. No more time wasted entering data!
In addition to saving time and effort, Receipt Bank also integrates seamlessly with accounting systems like Xero. Making your bookkeeping even easier.
You can login to your account at any time to access your expenses data. Receipt Bank allows you to export the data to an excel spreadsheet, download as a PFD or publish it directly to Xero.
As Receipt Bank is a cloud based system, there is no need to keep your physical documents. So, you can have a clearout and create some space in the office.
DTA & Co can also manage all your expenses direct from your Receipt Bank account. This allows us to quickly and efficiently add your expenses information to your annual accounts, VAT returns and any other required documents. As well as offer business advice based on real time information.
To get started with Receipt Bank, contact us today and we will set you up with your own Receipt Bank account.
Get in touch today and get the best cloud accounting partner right away.
Let us help you take out the hassle from accounting.
Our accountancy fees are always agreed in advance, meaning you can budget with confidence.


CCPL supports adults with English as a second language in 100 Mile House and surrounding communities.
Those with English as a Second Language are encouraged to improve their English language skills by attending the weekly Conversation Circles held Tuesday mornings. Participants discuss current issues, participate in presentations by community members, enjoy cooking classes, become familiar with the community through field trips and complete grammar worksheets. Participants enjoy a less structured learning environment while making new friends and becoming more comfortable within their new community.
The One-to-One Tutoring Program provides a more structured learning environment which matches a learner with a volunteer tutor. The weekly meetings are learner directed and focus on developing the language skills most important to the learner. Areas of interest can be: improving listening, speaking, reading or writing skills; increasing vocabulary; and learning new job specific terms.
Classes are also available, as needed, to assist those preparing to take an English Language test for immigration or citizenship purposes.


Christmas time is finally upon us. It is definitely my favorite time of the year, and I am sure many of us feel the same way. As a believer, I especially love the story of the birth of our Savior, Jesus Christ.
As time goes on, I think that I appreciate the backstories about this event which is the central theme of our lives and even of time itself. The unfolding of the events from the very beginning reveals a special gift that is available to all of us as a hidden Christmas gift if we will simply receive it. That is the gift of faith.
When the young virgin Mary was visited by the angel Gabriel, she was troubled by this miraculous experience, mostly because of Gabriel's salutation. "Rejoice, highly favored one, the Lord is with you: blessed are you among women!" Luke 1:28. He continued, "Do not be afraid Mary, for you have found favor with God. And behold, you will conceive in your womb and bring forth a Son, and shall call His name Jesus. He will be great, and will be called the Son of the Highest, and the Lord God will give Him the throne of His father David. And He will reign over the house of Jacob forever, and of His kingdom there will be no end." Luke 1:30-33.
Mary was engaged to be wed to Joseph. In the Jewish custom, this was the same as being married, but they had not yet consummated the marriage by joining together physically. That would happen only after the official wedding feast.
Mary told this to the angel and he answered her, "The Holy Spirit will come upon you, and the power of the Highest will overshadow you. Therefore, also that Holy One who is to be born will be called the Son of God. For with God, nothing will be impossible." Luke 1:35-37.
"Then Mary said, 'Behold the maidservant of the Lord! Let it be to me according to your word.'" Luke 1:38. This faith answer that Mary gave to the angel Gabriel is fascinating. She was telling him that she was willing for everything that he had spoken to come to pass in her life. But what caused her to have such faith in the angel's words?
The answer is in his statement, "For with God, nothing will be impossible." As a minister and as a student of the God's word, I have learned to use a few Greek word study tools. As a graduate of Rhema Bible Training College, I was quite familiar with the word "rhema", which means, "spoken word", with the idea of it being a word spoken to you personally.
One day I was looking through my word study concordance and observing the various ways that the KJV translators had translated this word. I looked down the list of scriptures in order and noticed that most of the time they chose the word, "word", and many times the word, "saying" was chosen. But all of a sudden I noticed it was translated as "nothing". It was from this passage in Luke's gospel, 1:37. Which means literally, "No word of God will be impossible." The word "rhema" had the Greek prefix which negated the word, now rendering it as "no word" but translated by the translators as "nothing".
Or I Corinthians 8:9, which says, "For you know the grace of our Lord Jesus Christ, that though He was rich, yet for your sakes He became poor, that you through His poverty might become rich." Jesus was certainly rich in Heaven before He chose to take on human flesh to become our Savior. But even while on Earth, He led a traveling mega-ministry around, providing for all of His disciples—70 of them, not just the 12, and obviously their families that they left in order to follow Jesus. There was so much provision in the money pouch that Judas was constantly stealing from it and no one noticed it but Jesus.
This is the gift that God wants us to receive this Christmas. His Word is His bond. Take Him at His Word.
Jesus taught His disciples a faith lesson after Peter noticed that a fig tree that Jesus had cursed withered from the roots. Jesus immediately said in Mark 11:22, "Have faith in God." When we look at the literal translation its says, "Have God's faith." You see, we don't have faith of our own until we believe God's Word, and then His faith comes into our heart. What Jesus was telling His disciples was that were being offered the gift of God's faith to use as their own. He is saying the same thing to us today.
I had the opportunity to travel to Nazareth in Israel back in 1977. My friends and I were shopping through the markets that lined the streets when I got separated from them and was lost for about an hour.
Immediately the priest began to weep. Then I began to weep, and we just both stood there weeping and worshiping God for His goodness, His grace, and His mercy, and blessing us to allow us to be standing in such a holy place.
Eventually my friends showed up at the Church of the Annunciation, and I was able to come back home to the USA, but it would have been okay if I had to stay there till Jesus came back for me.
Receive the gift of God's faith this Christmas. Believe that all of His promises, His Words, are for you and your loved ones, so that you might receive the abundant life that Jesus died to give you.
Join us every Thursday night at 7:00pm for Healing Services at Life Christian University, 410 E. Chapman Rd. Lutz, Fl. LCU classes are offered on Monday nights from 7:00 to 10:00pm. Call at 813-909-9720 or log on at www.lcus.edu.


text:
require 'liquid/weighted_selector'

describe WeightedSelector do

  describe "#empty?" do

    it do
      should be_empty
    end

    context do

      before do
        subject.add 'a', 0
      end

      it do
        should_not be_empty
      end

    end
  end

  describe '#pick_one' do

    context do

      its(:pick_one){ should == nil }

    end

    context do

      before do
        subject.add 'a', 2
        subject.add 'b', 2
        subject.add 'c', 0
      end

      it do
        ['a', 'b'].should include subject.pick_one
      end

    end

    context do

      before do
        subject.add 'a', 2
        subject.add 'b', 2
        subject.add 'c', 0
        subject.delete 'a'
      end

      its(:pick_one){ should == 'b' }
      its(:pick_one_with_index){ should == ['b', 0] }

    end

    context do

      before do
        Kernel.stub(:rand).and_return(1.00)
        subject.add 'a', 0.1
      end

      its(:pick_one){ should == 'a' }

    end
  end
end


Compliance functions have struggled these last years with tough regulatory tide. They are now mature to face new challenges. Still, they need to comply with new regulations, but with a stronger governance on projects. Besides, Compliance costs are still very high for the business, for whom the pace of revenues is slowing down. Innovation then is an opportunity to relieve Compliance functions, and to adopt a smarter predictive approach.


Located on the market place, this is a key to a number of The Village project's objectives, particularly in the areas of research, education, and community resilience.
The building has been designed to incorporate a numbers of meeting spaces as well as residential research facilities, a library and flexible office spaces.
Three storey timber post and beam frame, to allow flexible, open-plan layout of ground floor spaces. A social/ cultural space jointly owned by members for their own administrative, educational, recreational and leisure activities. Since inception, community facilities have been central to the concept of building a sustainable community together. In addition to providing space for these core activities is envisaged that the new spaces will allow for educational course delivery, conferences, local groups activities, academic research, and workspace for an array of specialist spin-off entities.
There are 2 live-work units adjoining which are flexible; whether occupied independently or by an Artist in Residence, for instance. Planning Permission extended to 2017. Fire Safety & Disability Access certificates have been granted in 2010. The proposal is ready to go upon funding being secured.


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We had another busy Saturday at our house with our first TKD Tournament of the season. My "boys" still absolutely awesome, and I was a very proud wife and mama!
2nd winning, a Silver Medal!
Hi... I'm linking to your Church Supper.
Thanks for hosting Krista, your little man is just the cutest! Congrats on the silver medal.
Thanks so much for the feature Krista and for hosting this great party! Have a wonderful Sunday!
Mmm, that pasta looks amazing!! How did I miss that last week?? Thanks for hosting and have a blessed week, Krista!!
Thanks for hosting Krista! Have a wonderful week!
Thanks for hosting - all linked up. Enjoy the rest of your weekend.
Thanks so much for hosting Krista!! Have a great week!


text:
./start_all.sh

sleep 1
./show_cluster.sh

sleep 2
../output/bin/ins_cli --ins_cmd=register --flagfile=nexus.flag --ins_key=zhangkai --ins_value=123456
#../output/bin/ins_cli --ins_cmd=register --flagfile=nexus.flag --ins_key=fxsjy --ins_value=123456

sleep 1
nohup ../output/bin/ins_cli --ins_cmd=login --flagfile=nexus.flag --ins_key=zhangkai --ins_value=123456 <op1.txt &>out1.log &
nohup ../output/bin/ins_cli --ins_cmd=login --flagfile=nexus.flag --ins_key=zhangkai --ins_value=123456 <op2.txt &>out2.log &


text:
Q: Undefined index with inversedBy in Doctrine I have an issue with my query if when I have this following setup.
I have an entity
<?php
class EntityB
{
     /**
      * @ORM\ManyToOne(targetEntity="EntityA", inversedBy="someKeyWord")
      * @ORM\JoinColumn(name="entityA_id", referencedColumnName="id")
      */
     private $entityA;
}

Now when you query and get an array as result, it works
// EntityBRepository.php
public function querySomeStuff($isArray=false)
{
    $b = $entityManager->createQuery('
        SELECT partial eA.{id, title}
        FROM EntityB eB
        LEFT JOIN eB.entityA eA
    ');

    if ($isArray) {
        $r = $b->getArrayResult();
    } else {
        $r = $b->getResult();
    }

    if (count($r)) {
        return $r[0];
    } else {
        return null;
    }
}

But if I do the same function call but pass the 2nd parameter as false, meaning, I don't want to get the result as an array, it crashes saying Notice: Undefined index: someKeyWord.
This is related to doctrine inversedBy mapping with partial left join query.


Rieter Components Agent Award 2016: Novibra Boskovice s.r.o.
Saurabh Sales Pvt. Ltd. achieved this appreciation for the professional representation of our products and corporate values, the executed promotional activities and the achievement of the sales targets set for the year 2016.
The precious award was handed over to Mr. R. N . Sodhi, owner of Saurabh Sales, by Mr. Shandheep , Senior Sales Manager Novibra, and Mr. Sundaram, Head Marketing & Sales Rieter India, during the exhibition ITMACH.