PUBLICATIONS. Water Resources Research. Probabilistic collocation method for strongly nonlinear problems: 3. Transform by time

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1 PUBLICATIONS Water Resources Research TECHNICAL REPORTS: METHODS./5WR774 Key Points: UQ methods may give bad results due to strong nonlinearity and non- Gaussianity A new method is developed in case of strong nonlinearity and non- Gaussianity The ttpcm yields accurate statistics and produces reasonable realizations Supporting Information: Supporting Information S Correspondence to: D. Zhang, dz@pku.edu.cn Citation: Liao, Q., and D. Zhang (6), Probabilistic collocation method for strongly nonlinear problems: 3. Transform by time, Water Resour. Res., 5, , doi:./ 5WR774. Received 3 JUN 5 Accepted 5 FEB 6 Accepted article online 7 FEB 6 Published online 6 MAR 6 VC 6. American Geophysical Union. All Rights Reserved. Probabilistic collocation method for strongly nonlinear problems: 3. Transform by time Qinzhuo Liao and Dongiao Zhang Department of Energy Resources Engineering, Stanford University, Stanford, California, USA, ERE & BIC-ESAT, College of Engineering, Peking University, Beijing, China Abstract The probabilistic collocation method (PCM) has drawn wide attention for stochastic analysis recently. Its results may become inaccurate in case of a strongly nonlinear relation between random parameters and model responses. To tackle this problem, we proposed a location-based transformed PCM (TPCM) and a displacement-based transformed PCM (dtpcm) in previous parts of this series. Making use of the transform between response and space, the above two methods, however, have certain limitations. In this study, we introduce a time-based transformed PCM (ttpcm) employing the transform between response and time. We conduct numerical eperiments to investigate its performance in uncertainty quantification. The results show that the ttpcm greatly improves the accuracy of the traditional PCM in a costeffective manner and is more general and convenient than the TPCM/dTPCM.. Introduction In the past decades, stochastic computation and uncertainty quantification have attracted a great deal of interest [Ghanem and Spanos, 99; Zhang, ; Babuska et al., 7; Xiu, ; Le Ma^ıtre and Knio, ]. The Monte Carlo (MC) method and the quasi-monte Carlo (QMC) method, albeit robust and fleible, require large computational effort to maintain statistical accuracy [Caflisch, 998; Ballio and Guadagnini, 4; Lu and Zhang, 4; Lemieu, 9; Chang et al., 5]. As an efficient alternative, the probabilistic collocation method (PCM) has been etensively utilized for uncertainty quantification and sensitivity analysis [Tatang et al., 997; Mathelin and Hussaini, 3; Li and Zhang, 7; Li et al., 9; Fajraoui et al., ; Ciriello et al., 3; Formaggia et al., 3; Dai et al., 3; Shi et al., 3]. The PCM relies on the regularity of the model responses with regard to the parameters; thus, its accuracy deteriorates quickly as the regularity decreases, and its convergence is not guaranteed in systems with unsmooth dependence on input parameters [Crestau et al., 9; Zhang et al., ]. To address this issue, part of this series [Liao and Zhang, 3] introduced a location-based transformed PCM (TPCM). Instead of directly approimating the response values sð; t; nþ at locations and time t for parameters n, we determine the locations ðs; t; nþ where the responses take possible values of s at time t for parameters n, then approimate these locations, and finally recover an approimation of sð; t; nþ from ðs; t; nþ. This approach increases accuracy significantly since ðs; t; nþ are much easier to approimate by polynomials than sð; t; nþ. Part of this series [Liao and Zhang, 4] presented a displacement-based transformed PCM (dtpcm), which etended the idea of transform from one-dimensional (-D) media and two-dimensional or three-dimensional (-D or 3-D) homogeneous media to -D and 3-D heterogeneous media. The dtpcm considers the model responses in space at a fied time from different random parameters as images and analyzes the deformation of these images by displacement, using digital image processing and scattered data interpolation techniques. In the dtpcm, the topology of the geometric shapes of the images has to be preserved, otherwise, the estimated deformation could be inaccurate, and the approimation of the statistics would be poor. For eample, in subsurface flow and transport problems, if in some realizations the solute is within the model space while in others the solute is transported outside the boundaries, then the images of the solute concentration in different realizations are not similar, and the concentration uncertainty estimated by the dtpcm could be inaccurate, as will be illustrated in section 4.. In this work, we use the arrival time of a given response value as an alternative variable to represent the model response, i.e., analyzing the response in a temporal domain. Since we are no longer dealing with LIAO AND ZHANG TIME-TRANSFORMED PCM 366

2 ./5WR774 the images in the space domain, the topology restriction is no longer present. Through solute transport eamples, we show that the ttpcm is more accurate than the PCM and requires lower computational burden compared to the MC method. As the last part of this series, we summarize the advantages and disadvantages of three (i.e., location-based, displacement-based, and time-based) transformed methods in the conclusion and point out that the ttpcm is more universal and convenient than the TPCM and the dtpcm.. Probabilistic Collocation Method (PCM) Consider a stochastic partial differential equation: Lsð; t; nþfð; t; nþ5 () where L is a differential operator, s is model response, f is a known function, is physical location, t is time, and n5ðn ; n ;...; n N Þ T is a parameter vector with N components, each being a random variable. We approimate s by the polynomial chaos epansion (PCE) [Ghanem and Spanos, 99] (see also section S of the supporting information) in truncated forms as: ^sð; t; nþ XQ i5 a i ð; tþw i ðnþ ; Q5ðNMÞ!=ðN!M!Þ () where a i ð; tþ are the PCE coefficients, and W i are the multivariate orthogonal polynomials with total degree M, with respect to the joint probability density function of the parameter qðnþ, i.e., hw i W j i Ð W i ðnþw j ðnþqðnþdn5c i d ij, where d ij is the Kronecker delta (d ij 5; i5j; and d ij 5; i 6¼ j) and the normalization constants c i are known in practice. To compute ^sð; t; nþ, we solve equation () at some selected collocation points n j ; j5; ;...; P obtaining the corresponding solutions sð; t; n j Þ. Then we set ^sð; t; n j Þ5sð; t; n j Þ and obtain: X Q i5 a i ð; tþw i ðn j Þ5sð; t; n j Þ; j5; ;...; P (3) The PCE coefficients can be computed by spectral projection [Xiu, ; Le Ma^ıtre and Knio, ] as: ð a i ð; tþ5 sð; t; nþw i ðnþqðnþdn ð W i ðnþw i ðnþqðnþdn X P j5 sð; t; n jþw i ðn j Þw j hw i i (4) P j5 where n j ; w j is a set of nodes and associated weights, such that the summation approimates the integral. Finally, we may estimate the statistical moments and probability density functions (PDFs) with a sampling approach based on the surrogate model in equation (). Note that the moments of s may also be calculated directly, e.g., mean and variance: ð hsð; tþi5 sð; t; nþqðnþdn XP sð; t; n j Þw j j5 ð (5) r s ð; tþ5 ðsð; t; nþhsð; tþiþ qðnþdn XP s ð; t; n j Þw j hsð; tþi In groundwater flow, the uncertain parameters (e.g., log conductivity) are usually described as random fields ehibiting spatial dependence. In this work, we represent the input fields by the Karhunen-Loeve epansion (KLE) [Ghanem and Spanos, 99; Zhang and Lu, 4] for given covariance functions of the input fields. Another important issue is the selection of the collocation points, since the set of nodes and associated weights as in equation (4) should be a good cubature rule. Here we use Smolyak sparse grid algorithm [Xiu and Hesthaven, 5], which is an effective choice and widely used. Readers are referred to section S of the supporting information for more details about KLE and sparse grids. j5 LIAO AND ZHANG TIME-TRANSFORMED PCM 367

3 ./5WR Time-Based Transformed Probabilistic Collocation Method (ttpcm) In the ttpcm, we use the same collocation points as in the PCM. Then we transform the corresponding responses to the arrival times. Note that the ttpcm is essentially devised for advection-dominated problems (e.g., solute concentration, saturation, and water cut), not for dispersion-dominated problems (e.g., hydraulic head and pressure), since the latter ones are usually approimated well by the PCM (see section S3 of the supporting information for more details). In case that the maimum/minimum (over time) of the response values varies in different realizations, we should first rescale the responses, in order to ensure there eists an arrival time for any response value. For eample, if the initially released solute concentration or the dispersivity changes between realizations, the maimum value (over time) of the concentration at a fied location would change as well, and the ttpcm proceeds by recording the time when a certain percent of the solute has arrived. Define a rescaled response as: s r ð; t; n j Þ sð; t; n jþs min ð; n j Þ s ma ð; n j Þs min ð; n j Þ s min ð; n j Þ5min sð; t; n jþ td s ma ð; n j Þ5ma sð; t; n jþ td (6) where the scaling factors s min and s ma are the minimum and maimum value in the time domain D, respectively. Considering a given (relative) response value (e.g., ; :; :;...; :), we search for its arrival time: tð; s; n j Þ fjs t r ð; t; n j Þ5sg (7) which is performed by interpolation of time with respect to response. In this study, we use piecewise cubic Hermite interpolation [Fritsch and Carlson, 98], which can be implemented by the function interp in MATLAB while setting the interpolation method to pchip. Interested readers may refer to Kahaner et al. [989] for other choices. It is worth noting that if there are multiple arrival times (e.g., in the solute transport cases with an instantaneous release at one location initially, there are two arrival times for a given concentration level: one is during the period in which the concentration increases from zero to the maimum value, the other is during the period in which the concentration decreases from the maimum value to zero), we may analyze these times individually. Essentially, the proposed method relies on the similarity of the response profile in time. Readers are referred to section S of the supporting information for further discussion in more complicated situations. The arrival time is then approimated by the PCE with dimension N and total order M similar to equation (): ^tð; s; n j Þ5 XQ i5 b i ð; sþw i ðn j Þ ; Q5ðNMÞ!=N!=M! (8) where b i ð; sþ are the PCE coefficients for the arrival time. We set ^tð; s; n j Þ5tð; s; n j Þ and obtain: X Q i5 b i ð; sþw i ðn j Þ5tð; s; n j Þ; j5; ;...; P (9) The coefficients b i ð; sþ can be computed through a spectral projection as: ð X tð; s; nþw i ðnþqðnþdn P b i ð; sþ5 ð tð; s; n j5 jþw i ðn j Þw j hw W i ðnþw i ðnþqðnþdn i i () The scaling factors s min and s ma in equation (6) are treated similarly. Then we generate a sufficient number of arrival time samples ^tð; s; nþ, as well as the scaling factor samples ^s min ð; nþ and ^s ma ð; nþ, using a MC/ QMC sampling for fied /s and random n. Net, the arrival time samples ^tð; s; nþ are transformed back to obtain the response samples s r ð; t; nþ, by searching for the response value at a given time as: LIAO AND ZHANG TIME-TRANSFORMED PCM 368

4 ./5WR774 (a).5 (b).5 (c).5 c t=.5 c t=.5 c t= c t= (d) c =.3,t= (e).5.5 MC (e5) PCM (3) ξ (f) PDF 6 4 MC (e5) PCM (3) c =.3,t= c =.3 (g).5.5 (h) t =.3,c=.5 3 MC (e5) ttpcm (3) (i) PDF.5.5 MC (e5) ttpcm (3).5 4 t 4 4 ξ 3 t =.3,c=.5 Figure. Illustration of the MC method, the PCM, and the ttpcm in the -D case: (a) 5 model runs at a given time in the MC method; (b) 5 realizations generated in the PCM; (c) 5 realizations generated in the ttpcm; (d) three model runs at a given time for the PCM; (e) response as a function of parameter at a fied location and a given time; (f) PDF of the response; (g) three model runs at a fied location for the ttpcm; (h) arrival time as a function of parameter at a fied location and a certain concentration value; (i) PDF of the arrival time. c denotes solute concentration, denotes location, t denotes time, n denotes random parameter, and means under certain conditions (e.g., c t5 means concentration at time t 5 day). s r ð; t; nþ fj^tð; s s; nþ5tg () which is performed by interpolation of response over time, similar to the forward transform in equation (7). Note that the responses are rescaled, and they have to be restored to the original state according to equation (6) as: ^sð; t; nþ5^s min ð; nþs r ð; t; nþ½^s ma ð; nþ^s min ð; nþš () These response samples are used to obtain the statistics such as moments and PDFs. In summary, it can be seen that compared to the PCM, the ttpcm has two additional steps, i.e., forward transform from the response to the arrival time (equations (6) and (7)) and back transform from the arrival time to the response (equations () and ()), and the ttpcm analyzes the arrival time instead of the response by polynomials (equations (8) ()). 4. Numerical Eamples We first use a -D eample to demonstrate the implementation process and performance of the ttpcm, and then show the applicability of the ttpcm in higher-dimensional problems through another -D eample. In this work, we used the MC/QMC method as a benchmark to verify the PCM/tTPCM. In practice, we may use a higher-order PCE to verify the lower-order PCEs. LIAO AND ZHANG TIME-TRANSFORMED PCM 369

5 ./5WR774 (a) MC (e5) PCM (3) ttpcm (3) (b).5 <c> σ c (c). (d).5 E[(c <c>) 3 ].5.5 E[(c <c>) 4 ] Figure. Statistical moments of solute concentration from different methods at t 5 day: (a) mean, (b) variance, (c) third moment, and (d) fourth moment. E denotes epectation with respect to the random event. 4.. One-Dimensional Eample With a Random Variable We consider a -D homogeneous water-saturated porous medium, in which a steady state flow has been established. The domain length is L 5 m, and the difference of the hydraulic heads between the left (a) (b) mean error number of model runs variance error number of model runs (c) third moment error (d) fourth moment error MC QMC PCM ttpcm TPCM number of model runs number of model runs Figure 3. Moment errors of concentration using different methods at t 5 day: (a) mean error, (b) variance error, (c) third moment error, and (d) fourth moment error. LIAO AND ZHANG TIME-TRANSFORMED PCM 37

6 ./5WR774 (a) y (m) (b) y (m) (c) y (m) 5 3 (m) 5 3 (m) 5 3 (m).4. ( 5L/) and the right ( 5 L/) boundaries is fied at Dh 5.3 m. The porosity is constant at / 5., and the log conductivity (in m/day) follows a normal distribution as ln(k) 5.3n, where n N(,). Thus, the pore velocity v5ðkdhþ= ð/lþ is also random. Consider Dirichlet boundary conditions and the initial condition of stepfunction-like ( c ; < c; ð Þ5 ; cðl=; tþ5c ; t > cl=; ð tþ5; t > (3) where D L 5a L v is the dispersion coefficient, and a L is dispersivity. Since the domain is large enough, the boundary conditions can be approimated by cð; tþ5c ; cð; tþ5; t >. Thus, the analytical solution [Bear, 97] is: cð; tþ5 c erfc vt p ffiffiffiffiffiffiffi ; D L t where erfcðþ5 ð (4) pffiffiffi e s ds p Assuming that c 5g/Landa L 5:5 m, the concentration results are analyzed between 5 mand 5 mattimet 5 day. The MC method with, model runs (5 of which are shown in Figure a) serves as a benchmark. In the PCM, since the log conductivity is modeled as a Gaussian random variable, we use Gauss-Hermite quadrature points. In particular, considering a second-order polynomial approimation, three model runs are conducted on the selected collocation points (Figure d). The model response is then analyzed by constructing a polynomial of the random parameter (Figure e). The MC reference (solid line) indicates strong nonlinearity, which cannot be approimated accurately by a second-order polynomial (dashed line) through three collocation points (circles). Therefore, the true concentration PDF from the MC method, which is bimodal, cannot be captured by the PCM (Figure f). In addition, the realizations randomly generated from the PCE have nonphysical oscillatory values (Figure b). As for the ttpcm, the same collocation points are used, but the concentration is observed in the time domain (Figure g). Note that to determine the arrival time of a given response value, ideally, we need to run the deterministic model long enough until this value is reached in all realizations (more details are discussed in section S of the supporting information). Then we search for the arrival time of a given concentration value, and analyze it as a polynomial of the random parameter (Figure h). It can be seen that the reference from the MC method (solid line) is easier to approimate (compared to Figure e) and thus can be approimated accurately by a second-order polynomial (dash-dotted line) through three collocation points (triangles). Therefore, the arrival time PDF from the ttpcm resembles that from the MC method (Figure i). Net, we randomly generate arrival time samples from surrogate polynomials and then transform them back to obtain the concentration realizations (Figure c), which are physically meaningful and successfully reproduce the results in Figure a Figure 4. Solute concentrations computed from three realizations of log conductivity (on three collocation points): (a) concentration with regard to a homogeneous media, (b) concentration image inside the boundary, and (c) concentration image partially outside the boundary. If the dtpcm was implemented, the estimated displacement between subplots (a) and (b) could be accurate, but that between subplots (a) and (c) would be inaccurate. LIAO AND ZHANG TIME-TRANSFORMED PCM 37

7 ./5WR774 (a).5 <c> (b) (m) σ c MC () PCM (4) PCM (88) PCM (3) ttpcm (4) ttpcm (88) MC () PCM (4) PCM (88) PCM (3) ttpcm (4) ttpcm (88) (m) Figure 5. Moments of solute concentration at (, y 5 5 m) obtained from the MC method with, realizations as a benchmark and from the PCM and the ttpcm with different numbers of collocation points: (a) mean and (b) variance. Figure evaluates the first four moments of concentration since the true concentration PDF is non-gaussian as observed in Figure f. We can see that the higher-order moment is intrinsically more difficult to estimate than the lower-order moment and the ttpcm is more accurate than the PCM for all moments. To further analyze the accuracies of the PCM and the ttpcm, we define a rootmean-square error as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð error5 ðy estimation y reference Þ d (5) where y is the concentration moment. We obtain a reference from the QMC method based on Sobol sequences [Caflisch, 998; Lemieu, 9] with, model runs. Figure 3 shows the moment errors using different methods. We can see that the convergence rates of the four moments from the MC method are close to O(n.5 ), and those from the QMC method are close to O(n. ), where n is the number of model runs. While the PCM outperforms the MC and QMC methods, the ttpcm converges even faster. For eample, to reach an error threshold of 4 for all moments, the MC method needs over 5 model runs, the QMC method needs about 3 model runs, the PCM needs model runs, whereas the ttpcm only needs 5 model runs. We also test the TPCM in this case, which is similar to the ttpcm in accuracy. Further discussions on the effects of running time and time step in the ttpcm, as well as the effect of dispersivity are included in section S3 of the supporting information. 4.. Two-Dimensional Eample With a Random Field The solute transport model under advection and dispersion is [Bear, tþ 5r D ij ð; tþrcð; tþ vð; tþrcð; tþ where is location, t is time, c is the solute concentration, D ij is the hydrodynamic dispersion tensor, v is the pore water velocity, computed by vð; tþ5uð; tþ=/, where / is the porosity; u is the Darcy velocity, computed by uðþ5kðþrhðþ, where KðÞ is the hydraulic conductivity, and hðþ is the hydraulic head. The hydrodynamic dispersion tensor is: D ij 5ða L a T v i v j Þ jvð; tþj a Tjvð; tþjd ij D d d ij (7) where a L is the longitudinal dispersivity, a T is the transverse dispersivity, jvð; tþj is the magnitude of the pore velocity, and D d is the molecular diffusion coefficient. Consider a groundwater flow in a -D rectangular space, whose left and right sides have constant heads ( and m, respectively), while the top and the bottom boundaries are impervious. The domain is divided into 3 4 grid blocks with each block being m. Assume /5:, a L 5: m,anda T 5: m. The log conductivity (in m/d) is represented by a stationary Gaussian random field with a mean of, and a LIAO AND ZHANG TIME-TRANSFORMED PCM 37

8 ./5WR774 (a) moment error (b) moment error number of model runs covariance of C ln K ð ; Þ5r ln K ep ðj j=g jy y j=g y Þ, where r ln K 5:5 and g 5g y 55 m. An instantaneous line source ( 5 5 m, y m) with unit concentration ( g/l) is released initially. The concentration at time t 5 days is analyzed. In the ttpcm, the model is run up to 3 days with atimestepof.day. In the MC method, we carried out, realizations by direct model runs as a reference, and verified that the number of samples is sufficient (not shown here for brevity). In the PCM and the ttpcm, the first N 5 random dimensions in the KLE are retained. The number of collocation points is 4, 88, and 3, for level,, and 3 sparse grid, respectively. We point out that the dtpcm is not applicable for this eample. The reason is that the solute is transported outside the right boundary in some realizations (Figure 4), and the topology of concentration images is not preserved, hence the displacement etracted from the concentration images is inaccurate, resulting in unreliable estimated concentration statistics. Figure 5 shows the mean and variance of concentration along the line (, y 5 5m).We can see that the PCM results ehibit multimodality and overestimate the variance in most locations. In particular, the mean estimation using a level sparse grid with 4 collocation points is not even guaranteed to remain nonnegative. Increasing the sparse grid level improves the accuracy to a certain etent, but a level 3 sparse grid with 3, collocation points is still far from enough. On the other hand, the ttpcm results with 4 points already match the MC results quite well, and the ttpcm results with 88 points are almost eactly the same as those from the MC method. Figure 6a compares the moment errors from the PCM and the ttpcm with respect to the number of model runs. It is clear that the ttpcm is much more accurate than the PCM for both mean and variance. As for the computational effort, the CPU time taken by the MC simulations is about 44 min on a 3.3 GHz processor. In the PCM, it took about, 37, and 575 min for the level,, and 3 sparse grid, respectively. In the ttpcm, it took about 7 and min for the level and sparse grid, respectively, including the transform process. It can be seen that for the same level of sparse grid, the ttpcm costs about three times as much as does the PCM since we run the simulator up to 3 days in the ttpcm other than days in the PCM. Nevertheless, for the same computational effort, the ttpcm is still much more accurate than the PCM, as shown in Figure 6b. 5. Discussions and Conclusions PCM <c> PCM σ c ttpcm <c> ttpcm σ c 3 CPU time (mins) Figure 6. Moment error of the PCM and the ttpcm with respect to: (a) number of model runs and (b) computational effort. In uncertainty quantification, the traditional probabilistic collocation method (PCM) approimates model responses directly using polynomials. Under strongly nonlinear conditions, it requires a high-level sparse grid and high-order polynomials with a large number of collocation points to produce physical realizations and accurate statistics. To address this drawback, we proposed a time-based transformed PCM (ttpcm) in this work, by introducing a transform process between the output response and the arrival time. Although the ttpcm usually has to run the model longer than the PCM, we showed in the numerical eamples that the ttpcm ehibits better accuracy and efficiency than does the PCM due to the regularity gained LIAO AND ZHANG TIME-TRANSFORMED PCM 373

9 ./5WR774 from the transform. The idea of transform can also be etended to Kalman filter approaches as presented in Liao and Zhang [5] and surrogate-based Bayesian inference [e.g., Zhang et al., 5] for data assimilation. The ttpcm is more general and convenient than the location/displacement-based transformed PCM (TPCM/dTPCM) [Liao and Zhang, 3, 4]. The TPCM is only applicable to one-dimensional (-D) models and some two-dimensional or three-dimensional (-D or 3-D) homogeneous media; whereas, the dtpcm and the ttpcm are suitable for -D to 3-D general models. As discussed in section, the dtpcm considers model responses as images, in which the topology of the geometric shapes has to be preserved. However, the ttpcm does not have this limitation, since the response is analyzed in the temporal domain instead of the space domain. In addition, the ttpcm is more convenient as it only requires -D interpolations in time, compared to the dtpcm that requires digital image processing techniques and -D or 3-D scattered data interpolations in space. In some cases where both the dtpcm and the ttpcm are valid, the computational efficiencies depend on the type of the quantity of interest. If we are interested in the model outputs in the whole space at a few time steps, the dtpcm would be more suitable. In contrast, if we are interested in the outputs in only a few locations but at many time steps, the ttpcm would be more efficient. Acknowledgments This work is partially funded by the National Natural Science Foundation of China (grant U64), the Public Project of the Ministry of Land and Resources of China (grant 63), the National Science and Technology Major Project of China (grant ZX59-6 and ZX55), and the National Key Technology R&D Program of China (grant BAC4B). 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