Model order reduction for Bayesian approach to inverse problems
© Nguyen et al.; licensee Springer. 2014
Received: 9 October 2013
Accepted: 23 November 2013
Published: 29 April 2014
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 reduced-order model and a Bayesian inference formulation to rapidly determine contaminant locations given sparse measurements of contaminant concentration. The system is modelled by the coupled Navier-Stokes equations and convection-diffusion transport equations. The Galerkin finite element method provides an approximate numerical solution-the ’full model’, which cannot be solved in real-time. The proper orthogonal decomposition and Galerkin projection technique are applied to obtain a reduced-order 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 reduced-order model to the source inversion problem yields a speed-up 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.
KeywordsBayesian Convection-diffusion equation Navier-Stokes equations Markov chain Monte Carlo Inverse problem Proper orthogonal decomposition Reduced-order model
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 Navier-Stokes 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 , contaminant transport modeling [4–6], and heat transfer . 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 reduced-order model and a Bayesian inference formulation. The Galerkin finite element method  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 time-dependent convection-diffusion 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 Navier-Stokes equations.
Solution of large-scale inverse problems can be accelerated by applying model order reduction [13–15]. The idea is to project the large-scale governing equations onto the subspace spanned by a reduced-space basis, yielding a low-order 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 , 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 velocity-dependent term, we cannot apply the standard POD reduction framework. The evaluation of the velocity-dependent convection term in the reduced system still depends on the full finite element dimension and has the same complexity as the full-order system. Thus, a system with velocity-dependent convection term needs an additional treatment to obtain an efficient reduced-order 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 velocity-dependent convection term. In the section ‘Results and discussion’, we use the numerical example to demonstrate the solution of the statistical inverse problem and the reduced-order model performance. We provide some concluding remarks in the final section.
In this section, the mathematical model, finite element approximation technique, Bayesian inference formulation, and MCMC method are briefly introduced.
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  associated with the stabilized second-order fractional-step method is applied to solve the incompressible Navier-Stokes 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].
Here ϕ i with i=1,⋯,N is the finite element basis and is the coordinate vector of the source location.
Bayesian inference to inverse problems
Here, K is the number of time steps over which we collect the output data.
Markov chain Monte Carlo for posterior sampling
Initialize the chain θ 0 and set n=0
Generate a proposal point θ∗∼q(θ∗|θ)
Generate U from a uniform U(0,1) distribution
Update the state to θn+1 as
Until n=N mcmc→ stop.
Nmcmc 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 reduced-order model is needed to approximate the large-scale 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 velocity-dependent convection term, and the error estimation.
Reduction via projection
The model reduction task is to find a suitable basis V so that m≪N and the reduced-order 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 reduced-order basis V and construct the low-order system. Here, we briefly describe the general POD method (more details may be found in ).
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 reduced-order 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 re-compute the full convection matrix (C(u)) at each time-step before projecting it onto the reduced-space 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 velocity-dependent term
where N u ≪T is the number of POD basis vectors use to represent the velocity.
Here, C r m (u m (x)) and C r k (Ψ k (x)),k=1,⋯,N u are the reduced-order forms of the full-order 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 reduced-order 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).
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 Navier-Stokes 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.
A time-dependent 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 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=5.0e5, and a mixing length turbulence model is used . The full model system is given in Equations 9 to 10. The Crank-Nicolson method  is used to discretize the system in time.
Here, to simplify the problem, we choose the number of sources to be n s =1, located at θ1=(x c ,z c ), with strength h1=0.2 and width σs 1=0.05. The active time of the source is t01∈ [ 0,toff] with toff=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 , 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 t0=0.
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.
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 for source term f(x,t;θ).
The properties of various reduced-order models
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 trade-off 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.
MCMC is now used to solve the inverse problem for a variety of source locations using the ROM-2 solver above. By accounting for both measured noise and uncertain information in model parameters, Bayesian inference to inverse problem leads to a well-posed 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 (yreal) is generated synthetically by adding noise to the ideal data (yideal) as in Equation 18. The noise is assumed to be Gaussian η∼N(0,σ2I). 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 reduced-order 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 θini=(0.821; −0.955) T . The total number of MCMC samples is set to Nmcmc=5000. The initial burn-in period is set to Nburnin=500. After this stage, data is saved to compute summary statistics of source locations.
Estimated inverse solutions for different numbers of POD basis
ε E (%)
Estimated inverse solutions for different MCMC starting points
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 reduced-order model in both state (concentration) and parameter (velocity). The resulting reduced-order model is efficient for solution of the forward problem. Solution of the Bayesian formulation of the inverse problem using the reduced-order 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 reduced-order 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 real-time water quality management applications because it reduces time cost and storage requirements.
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, 32-bit Operating System.
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