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Model order reduction for Bayesian approach to inverse problems
Asia Pacific Journal on Computational Engineering volume 1, Article number: 2 (2014)
Abstract
This work presents an approach to solve inverse problems in the application of water quality management in reservoir systems. One such application is contaminant cleanup, which is challenging because tasks such as inferring the contaminant location and its distribution require large computational efforts and data storage requirements. In addition, real systems contain uncertain parameters such as wind velocity; these uncertainties must be accounted for in the inference problem. The approach developed here uses the combination of a reducedorder model and a Bayesian inference formulation to rapidly determine contaminant locations given sparse measurements of contaminant concentration. The system is modelled by the coupled NavierStokes equations and convectiondiffusion transport equations. The Galerkin finite element method provides an approximate numerical solutionthe ’full model’, which cannot be solved in realtime. The proper orthogonal decomposition and Galerkin projection technique are applied to obtain a reducedorder model that approximates the full model. The Bayesian formulation of the inverse problem is solved using a Markov chain Monte Carlo method for a variety of source locations in the domain. Numerical results show that applying the reducedorder model to the source inversion problem yields a speedup in computational time by a factor of approximately 32 with acceptable accuracy in comparison with the full model. Application of the inference strategy shows the potential effectiveness of this computational modeling approach for managing water quality.
Background
Hydrodynamic processes such as contaminant transport in lakes and reservoirs have a direct impact on water quality. The contaminants will appear, spread out, and decrease in concentration, etc. because of some processes such as convection, diffusion, time rate release of contaminants, and distance of travel. To simulate such processes, a coupled system of partial differential equations (PDEs) including the NavierStokes equations (NSEs) and contaminant transport equations needs to be solved. A better understanding of these processes is important in managing water resources effectively.
The direct or forward problems compute the distribution of contaminant directly from given input information such as contaminant location, contaminant properties, fluid flow properties, boundary conditions, initial conditions, etc. On the contrary, the inverse problems infer the unknown physical parameters, boundary conditions, initial conditions, or geometry given a set of measured data. These known data can be obtained experimental or computational. In realistic applications, data are not perfect because of error due to sensor noise. In addition, the model may contain some uncertain parameters such as wind velocity. Thus, inverse problems are generally stochastic problems.
A Bayesian inference approach to inverse problems is one way to deal with the stochastic problem. This approach takes into account both the information of the model parameters with uncertainty and the inaccuracy of data in terms of a probability distribution [1, 2]. The Bayesian approach provides a general framework for the formulation of wide variety of problems such as climate modeling [3], contaminant transport modeling [4–6], and heat transfer [7]. Under the Bayesian framework, simulated solutions need to be evaluated repeatedly over different samples of the input parameters. There are available sampling strategies associated with Bayesian computation such as Markov chain Monte Carlo (MCMC) methods [8–11]. However, if we use traditional PDE discretization methods, such as finite element or finite volume methods, the resulting numerical models describing the system will be very large and expensive to solve.
This paper presents an efficient computational approach to solve the statistical inverse problem. The approach uses the combination of a reducedorder model and a Bayesian inference formulation. The Galerkin finite element method [12] provides an approximate numerical solution  the ‘full model’ or ‘forward model’. The MCMC method is applied to solve the Bayesian inverse problems. In particular, we are interested in inferring an arbitrary source location in a timedependent convectiondiffusion transport equation, given a velocity field and a set of measured data of the concentration field at sparse spatial and temporal locations. We obtain the velocity field by solving the NavierStokes equations.
Solution of largescale inverse problems can be accelerated by applying model order reduction [13–15]. The idea is to project the largescale governing equations onto the subspace spanned by a reducedspace basis, yielding a loworder dynamical system. Proper orthogonal decomposition (POD) is the most popular method to find the reduced basis for a given set of data. The snapshot POD method, which was proposed by Sirovich [16], provide an efficient way for determining the POD basis vectors. Snapshots are solutions of a numerical simulation at selected times or choices of the parameters. The choice of snapshots should ensure that the resulting POD basis captures the most important characteristics of the system.
Since the convection term in contaminant transport equations is a velocitydependent term, we cannot apply the standard POD reduction framework. The evaluation of the velocitydependent convection term in the reduced system still depends on the full finite element dimension and has the same complexity as the fullorder system. Thus, a system with velocitydependent convection term needs an additional treatment to obtain an efficient reducedorder model.
This paper is outlined as follows. In the section ‘Methods’, we introduce the mathematical model, the finite element approximate technique, and the Bayesian inference formulation and its solution using an MCMC method. In section ‘Model order reduction’, we describe the model order reduction technique and the treatment of the velocitydependent convection term. In the section ‘Results and discussion’, we use the numerical example to demonstrate the solution of the statistical inverse problem and the reducedorder model performance. We provide some concluding remarks in the final section.
Methods
In this section, the mathematical model, finite element approximation technique, Bayesian inference formulation, and MCMC method are briefly introduced.
Transport model
Consider the fluid flow through a physical domain $\mathcal{D}\subset {\mathbb{R}}^{d},d=1,2,3$ with boundary Γ. The contaminant concentration c:≡c(x,t;θ) satisfies the dimensionless parabolic PDEs, boundary conditions, and initial conditions as follows:
where $\mathbf{x}\in \mathcal{D}$ denotes the spatial coordinates, t∈ [ t_{0},t_{ f }] denotes time, and c_{0} is the given initial condition. The inlet boundary Γ_{ D } is subjected to a Dirichlet condition, while the remainder of the boundary Γ_{ N }=Γ∖Γ_{ D } satisfies Neumann conditions. The external source f(x,t;θ) in Equation 1 is used here as a input parameter, where $\theta \in \mathcal{D}$ is the source location. In this study, we suppose that the contaminant source drives the transport process, given a velocity field $\mathbf{u}\in {\mathbb{R}}^{d}$. Thus, the Péclet number $\text{Pe}=\frac{\parallel \mathbf{u}\parallel \mathrm{L}}{\kappa}$, where κ is the diffusivity and L is the characteristic length. The velocity field is a function of x and t, u(x,t), which can be obtained by solving the NavierStokes equations, as follows:
where Re is the Reynolds number, p is the pressure field, and f_{ c } is the body force. So, for any given f and u, one can solve for the solution c(x,t;θ).
Finite element approximations
The finite element method [12] associated with the stabilized secondorder fractionalstep method is applied to solve the incompressible NavierStokes equations (5 to 8). The detailed discussion and derivation of this method is beyond the scope of this paper. For more details, please refer to [17, 18].
We also use the finite element method for spatial discretization of the contaminant transport equations in Equations 1 to 4. This leads to the semidiscrete equations
with the output of contaminant solution at sensor locations in the domain
Here, $\mathbf{c}\left(t\right)\in {\mathbb{R}}^{N}$ is the discretized approximation of c(x,t) and contains N state unknowns where N is the number of grid points in the spatial discretization. $\stackrel{\u0307}{\mathbf{c}}$ is the derivative of c with respect to time. The vector $\mathbf{y}\in {\mathbb{R}}^{{N}_{o}}$ contains the N_{ o } outputs of the system related to the state through the matrix $\mathbf{B}\in {\mathbb{R}}^{{N}_{o}\times N},{N}_{o}\ll N$. The matrices $\mathbf{M},K,C\left(u\right)\in {\mathbb{R}}^{N\times N}$ and the vector $\mathbf{F}(t;\theta )\in {\mathbb{R}}^{N}$ are defined in the finite element space as follows:
Here ϕ_{ i } with i=1,⋯,N is the finite element basis and $\theta \in {\mathbb{R}}^{d}$ is the coordinate vector of the source location.
Bayesian inference to inverse problems
The Bayesian formulation of an inference problem is derived from Bayes’ theorem [1]:
where $\mathbf{y}\in {\mathbb{R}}^{{N}_{o}}$ is the observed data and $\theta \in {\mathbb{R}}^{d}\subseteq \mathcal{D}$ is the vector of input parameters. p(θ) denotes the prior distribution of the parameters, p(yθ) is the likelihood function, and p(θy) is the posterior distribution of parameters given the observed data. p(y) is the prior predictive distribution of y and may be set to an unknown constant that is not needed in the MCMC solution method. In this case, we write Bayes’s formula as
A simple model for the likelihood function can be obtained from the following relationship,
where η is the noise which we assume to be white Gaussian noise η∼N(0,σ^{2}I). The inputoutput relative in Equations 1 to 4 is denoted as the forward model G(θ). Subsequently, the likelihood can be written as
There are many approaches to define the prior information, such as Gaussian Markov random fields or uniform distribution models, etc. In this work, we assume that our only prior information on the source location is given by the bounds on the domain. Thus, using the principle of maximum entropy [19], we take our prior to be a uniform distribution. Therefore, Equation 17 becomes
Here, K is the number of time steps over which we collect the output data.
Markov chain Monte Carlo for posterior sampling
MCMC simulation provides a sampling strategy from a proposal distribution to the target distribution using the Markov chain mechanism. Hastings introduced the MetropolisHastings (MH) algorithm [10], which is a generalized version of the Metropolis algorithm [8, 9], in which we can apply both symmetric and asymmetric proposal distributions. In this work, the MH algorithm is used to solve the Bayesian inverse problem. The MH algorithm is summarized as follows:

1.
Initialize the chain θ ^{0} and set n=0

2.
Repeat

n=n+1

Generate a proposal point θ^{∗}∼q(θ^{∗}θ)

Generate U from a uniform U(0,1) distribution

Update the state to θ^{n+1} as
$$\begin{array}{l}{\theta}^{n+1}=\left\{\begin{array}{cc}{\theta}^{\ast},& \text{if}\phantom{\rule{2.77626pt}{0ex}}\beta <\mathrm{U}\\ {\theta}^{n},& \text{otherwise}\end{array}\right.\end{array}$$

3.
Until n=N _{mcmc}→ stop.
Here, β is the acceptancerejection ratio, given by
N_{mcmc} is the total number of samples, π(θ) is the target distribution, q(θ^{∗}θ) is a proposal distribution, and U is a random number.
The discretized model in the form of ordinary differential equations (ODEs) (Equations 9 to 11) may have very large dimensions and be expensive to solve. The MCMC method requires evaluating repeatedly the solution of the forward model (many thousands or even millions of times). Hence, these simulations can be a computationally expensive undertaking. In such situations, the reducedorder model is needed to approximate the largescale model, which allows efficient simulations.
Model order reduction
This section presents the model order reduction framework. This includes the reduction via projection, the proper orthogonal decomposition method, the additional treatment for the velocitydependent convection term, and the error estimation.
Reduction via projection
A reducedorder system approximating the ODEs (9 to 11) can be obtained by approximating the full state vector c as a linear combination of m basis vectors as follows:
where ${\mathbf{c}}_{r}\in {\mathbb{R}}^{m}$ is the reduced order state and $V=\phantom{\rule{2.77626pt}{0ex}}[\phantom{\rule{0.3em}{0ex}}{v}_{1}\phantom{\rule{2.77626pt}{0ex}}{v}_{2}\phantom{\rule{2.77626pt}{0ex}}\cdots {v}_{m}]\in {\mathbb{R}}^{N\times m}$ is an orthonormal basis, i.e., V^{T}V=I. The loworder model can be derived by projecting the governing system (9 to 11) onto the reduced space formed by the column span of basis V. This yields
where
The model reduction task is to find a suitable basis V so that m≪N and the reducedorder model yields accurate results. This study will consider POD as the method to compute the basis.
Proper orthogonal decomposition
POD provides a method to compute the reducedorder basis V and construct the loworder system. Here, we briefly describe the general POD method (more details may be found in [16]).
Supposed that we have the snapshot matrix $X\in {\mathbb{R}}^{N\times Q}$, where Q is the number of snapshot solutions, which are built from the instantaneous solution c^{s}(t_{ k }) of the forward model, corresponding to s=1,⋯,S random input source locations, picked up at k=1,⋯,T different time steps (so that Q=S T). The POD basis vectors are the m left singular vectors of X corresponding to the largest singular values (m≤Q). Let σ_{ i },i=1,2,⋯,Q be the singular values of X in decreasing order. We choose the number of POD basis vectors to retain in the reducedorder model as m≤Q so that
where ε_{ E }(%) is the required amount of energy, typically taken to be 99% or higher.
After obtaining the POD basis vectors for the contaminant, we have defined our reducedorder system. However, as mentioned earlier, the dimension of the system of Equations 24 to 26 is reduced only in state (concentration). The reduced convection matrix (C_{ r }(u)) in Equation 29 still depends on the full dimension of the velocity field as in Equation 14. This means that we have to recompute the full convection matrix (C(u)) at each timestep before projecting it onto the reducedspace basis to obtain the reduced convection matrix (C_{ r }(u)). Hence, we need an additional treatment to avoid this computational cost.
Linear expansion techniques for velocitydependent term
We proceed by representing the velocity field as a linear combination of POD velocity basis vectors. The velocitydependent convection matrix can then be expanded based on this velocity expansion. In particular, let u(x,t) be a given velocity field and $\left\{\right.\mathbf{u}(\mathbf{x},{t}_{k})k=1T$ be a set of ‘snapshots’ of velocities which are obtained from Equations 5 to 8 and taken over T time steps. The velocity field is decomposed as
where ${\mathbf{u}}_{m}\left(\mathbf{x}\right)=\frac{1}{T}\sum _{k=1}^{T}\mathbf{u}(\mathbf{x},{t}^{k})$ is the mean velocity field and u^{′}(x,t) is the fluctuating velocity field. The fluctuating velocity field is then represented by
where Ψ_{ k }(x) is the k th POD velocity basis and α_{ k }(t) is the corresponding time dependent amplitude. We now consider the expansion of the velocity field as
where N_{ u }≪T is the number of POD basis vectors use to represent the velocity.
Galerkin projection of the timedependent velocity term in Equation 1 has the form as
The discrete form of Equation 37 then becomes
Here, C_{ r m }(u_{ m }(x)) and C_{ r k }(Ψ_{ k }(x)),k=1,⋯,N_{ u } are the reducedorder forms of the fullorder convection matrices C(u_{ m }(x)) and C(Ψ_{ k }(x)), respectively. They are computed only once in the offline step. Thus, the first case of the reducedorder model (with Equation 29) is only reduced in state (concentration), but the second case (with Equation 38) is now reduced in both state (concentration) and parameter (velocity).
Error estimation
In order to estimate the efficiency of the reduced model relative to the full model, we use the relative errors of state solutions and relative errors of outputs. These errors are defined as follows:
Here, c(t^{k};θ),c_{ r }(t^{k};θ),1≤k≤T are the full and reduced solutions, while y(t^{k};θ) and y_{ r }(t^{k};θ),1≤k≤T are the full and reduced outputs of interest, respectively.
Results and discussion
We consider a numerical example based on a 2D mathematical model. The velocity field in the reservoir is obtained by solving a 2D laterally averaged NavierStokes model. The Bayesian formulation of the inverse problem is then solved to determine an uncertain contaminant source location. We compare the effectiveness of solving the inverse problem using both the full model and reduced model techniques.
Model setup
The physical domain is illustrated in Figure 1a, which represents a simplified model of a 2D laterally averaged reservoir system. The reservoir system includes a main reservoir section and the river connections. We assume that the contaminant is present within the main reservoir section and that the contaminant transport processes are mainly affected by the inflow and wind velocity. We assume that other factors (such as heat exchange, etc.) have little influence on the processes. The spatial domain is discretized with a finite element mesh, as shown in Figure 1b. The total number of grid points is N=1,377 and the total number of elements is N_{ E }=2,560. The simulation is run from initial time t_{0}=0 to final time t_{ f }=40, with timestep size Δ t=0.08. Thus, the number of time steps is T=500.
A timedependent velocity field is obtained from the 2D laterally averaged system, which is given in Equations 5 to 8, where the body force f_{ c }= [f_{ c x }; f_{ c z }]^{T}= [ 0; g]^{T} with g the gravitational acceleration.
The boundary conditions are set up as follows (refer to Figure 1a for the boundary definitions):
The velocity on remaining boundaries are set to zero. Here, V_{ a } is the wind speed at 10 m above the water surface. In this example, we assumed that V_{ a }=2 m/s for the entire simulation time. The Reynolds number is set to Re=5.0e^{5}, and a mixing length turbulence model is used [20]. The full model system is given in Equations 9 to 10. The CrankNicolson method [21] is used to discretize the system in time.
The setup for the contaminant transport part of the problem is as follows. We use a contaminant source that is defined as the superposition of Gaussian sources, each one active on the time interval t_{0k}∈ [ t_{0},t_{off}] and centered at ${\theta}_{k}\in \mathcal{D}$, with strength h_{ k } and width σ_{ s k }:
Here, to simplify the problem, we choose the number of sources to be n_{ s }=1, located at θ_{1}=(x_{ c },z_{ c }), with strength h_{1}=0.2 and width σ_{s 1}=0.05. The active time of the source is t_{01}∈ [ 0,t_{off}] with t_{off}=10. The inflow boundary and other solid boundaries satisfy a homogeneous Dirichlet condition on the contaminant concentration; the outflow boundaries and free surface boundary satisfy a homogeneous Neumann condition. The diffusivity coefficient is assumed to be constant, κ=0.005. Thus, the Péclet number $\text{Pe}=\frac{\parallel \mathbf{u}\parallel \mathrm{L}}{\kappa}=100$, where the length of the inflow section is used as the characteristic length L=0.5. The contaminant concentration is assumed to be zero at initial time t_{0}=0.
With this setup, we have specified our forward problem with the input parameter $\theta \in {\mathbb{R}}^{2}$ in the range Θ:= [ 0, 2]× [ −2, 0]. Figure 2 shows the contaminant solution c(x,t;θ) of the forward model with θ_{1}=(0.5,−0.5) at specific times. The contaminant field grows while the source is active. After the shutoff time of the source, the contaminant moves away, spreads out, and decreases in concentration due to convection and diffusion, until it flows out of the domain.
The outputs of interest are the values of contaminant solution c(x,t;θ) at selected sensor locations in the computational domain. These sensors are located on a 4×4 uniform grid covering the reservoir domain as shown in Figure 1b.
Model order reduction
To perform the model order reduction, we need to obtain the POD expansion of velocity field.
Figure 3a shows all eigenvalues of the velocity field. As the number of eigenvalues increases, the significant of eigenvalues decreases, respectively. Figure 3b shows the relative errors of representing the velocity field with various numbers of POD velocity modes. The goal of this step is to find a suitable N_{ u } that can provide a fast computation of the velocitydependent convection term and to ensure that the relative errors are acceptable. In this example, we choose the number of terms in the velocity expansion to be N_{ u }=12 yields the relative error around 5×10^{−3}.
Figure 4 illustrates the first four POD basis vectors of the velocity field. Clearly, they represent the most important characteristics of the velocity field.
A comparison between the original velocity and the approximate velocity representation at final simulating time is given in Figure 5. We observe that the approximate velocity profiles at x=0.5 (Figures 5a,b) and x=1.0 (Figures 5c,d) are able to represent well the characteristics of the original velocity profiles.
To generate the snapshots needed for the POD basis to represent the contaminant concentration, we choose S location samples in the computational domain. In this example, we generate random input locations $\theta \in \mathcal{D}\subset {\mathbb{R}}^{2}$ for source term f(x,t;θ).
Table 1 shows the relative errors and computational time ratio of the reducedorder models (ROM) for different sample sizes of snapshot matrices. Here, we use the notation ROM1 to denote the reduced model mentioned in the section ‘Proper orthogonal decomposition’ and ROM2 for the one mentioned in the section ‘Linear expansion techniques for velocitydependent term’. The POD basis is chosen based on the energy as in Equation 33, with ε_{ E }=99.999%. We observe that while the relative errors in the outputs are similar for both cases, the ratio of computational time differs significantly. The case ROM2 shows speedups from 15 to 111 times, while the case ROM1 is in fact more expensive to solve than the full forward model (t_{ f u l l }≈285 seconds)^{1}. This is because the first case consumes a lot of time in recomputing the velocitydependent convection matrix and then projecting it onto the reduced space at each time step. We choose as our reducedorder model the case ROM2 with sample size S=30.
Figure 6a presents the eigenvalues of contaminant snapshot matrix for the sample test case S=30. We observe that POD eigenvalues of the contaminant field are decayed very fast, and they are significantly important only at the first few hundred values. The first four POD contaminant basis vectors are shown in Figure 7.
Figure 6b shows relative errors of ROMs for different numbers of POD basis vectors. With the energy taken to 99.99999%, the ROM resulted in 434 POD basis vectors with relative error around 4×10^{−3}. In the tradeoff between accuracy results and computational efforts (please refer to Table 1), we can choose our ROM with 99.999% of energy captured which resulting in the size of m=257 POD basis vectors and an acceptably small error.
Figure 8a shows the outputs of the full model (y) and ROM2 (y_{ r }) with size m=257 at selected sensor locations. The selected sensors (sensor 6, 7, 10, and 11) capture more information of the contaminant than the others for the case studied. We observe that ROM2 is able to capture well the behavior of the full model at the sensor locations. This reducedorder model will be utilized as the forward solver in solving the inverse problem.
Inverse problems
MCMC is now used to solve the inverse problem for a variety of source locations using the ROM2 solver above. By accounting for both measured noise and uncertain information in model parameters, Bayesian inference to inverse problem leads to a wellposed problem resulting in a posterior distribution of the unknown [1, 2]. Once obtaining the posterior distribution sample, all statistical questions related to the unknown could be answered with sample averages.
To represent the behavior of uncertain variables such as wind velocity, we generate noisy data at sensor locations. The ‘real’ data (y_{real}) is generated synthetically by adding noise to the ideal data (y_{ideal}) as in Equation 18. The noise is assumed to be Gaussian η∼N(0,σ^{2}I). Figure 8b shows the noisy data at sensor locations with σ=0.3.
We use the snapshots with S=30 samples above to generate several reducedorder models with different energies. To estimate the relative error of the inverse problem solutions (ε_{ θ }), we choose the solution corresponding to the largest order as a ‘truth’ solution. We do the MCMC simulation with the starting point θ_{ini}=(0.821; −0.955)^{T}. The total number of MCMC samples is set to N_{mcmc}=5000. The initial burnin period is set to N_{burnin}=500. After this stage, data is saved to compute summary statistics of source locations.
Table 2 shows the results of the inverse source problems with different numbers of POD basis. For the case with m=257, the computational time is around 24 h to obtain the solution. If we use the full forward solver, with the speedup factor as given in Table 1, the computational time is estimated around 773 h or approximately 32 days.
Figure 9a shows the credible regions of the source locations θ. In this figure, both a pairwise scatter plot and 1D marginal distributions are displayed. The dash line (in θ_{1} axis) and dashdot line (in θ_{2} axis) show the probability density function of each parameter while the solid blue regions show the 95% and 99% credible regions in which the parameters appeared the most (the blue dots).
To assess the convergence of the MCMC approach, we generate four different initial points for the MH sampler. Figure 9b shows the contour plot of the posterior probability density function (the contour plot corresponds to the result of Init1) and the historical trajectories of MCMC samplers for each starting point. We observe that for all four initial points, the MCMC simulations converge to the same final solution. However, depending on the position of the initial points, the computational time varies, as given in Table 3.
Conclusion
This study has applied successfully the combination of a model order reduction technique based on the POD and a Bayesian inference approach to solve an inverse problem that seeks to identify an uncertain contaminant location. Applying an additional POD expansion to approximate the velocity results in a reducedorder model in both state (concentration) and parameter (velocity). The resulting reducedorder model is efficient for solution of the forward problem. Solution of the Bayesian formulation of the inverse problem using the reducedorder solver is much more rapid than using the full model, yielding the probability density of the source location in reasonable computational times. The computational time of using a reducedorder model with m=257 degrees of freedom is about a factor of 32 times lower than using the full model with size N=1,377. This reduction is important in realtime water quality management applications because it reduces time cost and storage requirements.
Endnote
^{1} The simulations were performed on a personal computer (PC) with processor Intel(R) Core(TM)2 Duo CPU E8200 @2.66GHz 2.66GHz, RAM 3.25GB, 32bit Operating System.
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Nguyen, N., Cheong Khoo, B. & Willcox, K. Model order reduction for Bayesian approach to inverse problems. Asia Pac. J. Comput. Engin. 1, 2 (2014). https://doi.org/10.1186/2196116612
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Keywords
 Bayesian
 Convectiondiffusion equation
 NavierStokes equations
 Markov chain Monte Carlo
 Inverse problem
 Proper orthogonal decomposition
 Reducedorder model