Fast solution of the linearized Poisson\u2013Boltzmann equation with nonaffine parametrized boundary conditions using the reduced basis method

The Poisson-Boltzmann equation (PBE) is a nonlinear elliptic parametrized partial differential equation that is ubiquitous in biomolecular modeling. It determines a dimensionless electrostatic potential around a biomolecule immersed in an ionic solution \cite{Chen}. For a monovalent electrolyte (i.e. a symmetric 1:1 ionic solution) it is given by $$\label{PBE} -\vec{\nabla}.(\epsilon(x)\vec{\nabla}u(x)) + \bar{\kappa}^2(x) \sinh(u(x)) = \frac{4\pi e^2}{k_B T}\sum_{i=1}^{N_m}z_i\delta(x-x_i) \quad \textrm{in} \quad \Omega \in \mathbb{R}^3, $$ $$\label{eq:Dirichlet} u(x) = g(x) \quad \textrm{on} \quad \partial{\Omega}, $$ where $\epsilon(x)$ and $\bar{k}^2(x)$ are discontinuous functions at the interface between the charged biomolecule and the solvent, respectively. $\delta(x-x_i)$ is the Dirac delta distribution at point $x_i$ . In this study, we treat the PBE as an interface problem by employing the recently developed range-separated tensor format as a solution decomposition technique \cite{BKK_RS:16}. This is aimed at separating efficiently the singular part of the solution, which is associated with $\delta(x-x_i)$ , from the regular (or smooth) part. It avoids building numerical approximations to the highly singular part because its analytical solution, in the form of $u_{\textrm{s}}(x) = \alpha \sum_{i=1}^{Nm}z_i/ \lvert x-x_i \rvert$ exists, hence increasing the overall accuracy of the PBE solution.\\ On the other hand, numerical computation of \eqref{PBE} yields a high-fidelity full order model (FOM) with dimension of $\mathcal{O}(10^5)$ $\sim$ $\mathcal{O}(10^6)$ , which is computationally expensive to solve on modern computer architectures for parameters with varying values, for example, the ionic strength, $I \in \bar{k}^2(x)$ . Reduced basis methods are able to circumvent this issue by constructing a highly accurate yet small-sized reduced order model (ROM) which inherits all of the parametric properties of the original FOM \cite{morRozHP08}. This greatly reduces the computational complexity of the system, thereby enabling fast simulations in a many-query context. We show numerical results where the RBM reduces the model order by a factor of approximately $350,000$ and computational time by $7,000$ at an accuracy of $\mathcal{O}(10^{-8})$ . This shows the potential of the RBM to be incorporated in the available software packages, for example, the adaptive Poisson-Boltzmann software (APBS).

For applications of numerical reservoir simulators such as production optimization or uncertainty assessment, hundreds or thousands of reservoir simulation runs must be performed. Such huge computational cost is a major problem in petroleum engineering, and reduced-order model (ROM) based on proper orthogonal decomposition (POD) has been intensively studied in recent decade to overcome it. While POD-based methods have been shown to be efficient compared to full order model (FOM), it is not as efficient when it is applied to a typical large dimensional nonlinear reservoir system. One of the reasons for the inefficiency is that in a typical nonlinear reservoir system ROM based only on POD is still dependent on the dimension of FOM. This is due to the fact that to compute the reduced nonlinear term in mass balance equation of a reservoir system, one must first reconstruct the full-order state solution such as pressure and saturation, evaluate the full-order nonlinear term before projecting it onto a reduced subspace. Therefore, we developed a reduced-order reservoir model based on a discrete empirical interpolation method (DEIM) to approximate nonlinear phase potential terms so that the repeated online evaluations of the ROM in Newton iteration are independent of full-order dimension. The independence comes from the fact that DEIM just needs to evaluate the nonlinear term only at interpolation indices that represent grid blocks that are important in terms of preserving the continuity properties of the mass balance equation. A case study was carried out to investigate the performance of DEIM compared POD. Although the testing schedule of well controls is far apart from the training schedule of well controls, close matches are obtained. Thus, the ROM using DEIM is expected to enable the practical application of reservoir simulator, such as production optimization in which many simulation runs must be performed, in a reasonable time frame by significantly relieving the required numerical effort.

To speed-up the solution of parametrized differential problems, reduced order models (ROMs) have been developed over the years, including projection-based ROMs such as the reduced-basis (RB) method, deep learning-based ROMs, as well as surrogate models obtained through machine learning techniques. Thanks to its physics-based structure, ensured by the use of a Galerkin projection of the full order model (FOM) onto a linear low-dimensional subspace, the Galerkin-RB method yields approximations that fulfill the differential problem at hand. However, to make the assembling of the ROM independent of the FOM dimension, intrusive and expensive hyper-reduction techniques, such as the discrete empirical interpolation method (DEIM), are usually required, thus making this strategy less feasible for problems characterized by (high-order polynomial or nonpolynomial) nonlinearities. To overcome this bottleneck, we propose a novel strategy for learning nonlinear ROM operators using deep neural networks (DNNs). The resulting hyper-reduced order model enhanced by DNNs, to which we refer to as Deep-HyROMnet, is then a physics-based model, still relying on the RB method approach, however employing a DNN architecture to approximate reduced residual vectors and Jacobian matrices once a Galerkin projection has been performed. Numerical results dealing with fast simulations in nonlinear structural mechanics show that Deep-HyROMnets are orders of magnitude faster than POD-Galerkin-DEIM ROMs, still ensuring the same level of accuracy.

Affine Decompositions of Parametric Stochastic Processes for Application within Reduced Basis Methods

The use of model-based numerical simulations of wave propagation in rooms for engineering applications requires that acoustic conditions for multiple parameters are evaluated iteratively, which is computationally expensive. We present a reduced basis method (RBM) to achieve a computational cost reduction relative to a traditional full-order model (FOM) for wave-based room acoustic simulations with parametrized boundaries. The FOM solver is based on the spectral-element method; however, other numerical methods could be applied. The RBM reduces the computational burden by solving the problem in a low-dimensional subspace for parametrized frequency-independent and frequency-dependent boundary conditions. The problem is formulated in the Laplace domain, which ensures the stability of the reduced-order model (ROM). We study the potential of the proposed RBM in terms of computational efficiency, accuracy, and storage requirements, and we show that the RBM leads to 100-fold speedups for a two-dimensional case and 1000-fold speedups for a three-dimensional case with an upper frequency of 2 and 1 kHz, respectively. While the FOM simulations needed to construct the ROM are expensive, we demonstrate that the ROM has the potential of being 3 orders of magnitude faster than the FOM when four different boundary conditions are simulated per room surface.

<abstract><p>The numerical simulation of several virtual scenarios arising in cardiac mechanics poses a computational challenge that can be alleviated if traditional full-order models (FOMs) are replaced by reduced order models (ROMs). For example, in the case of problems involving a vector of input parameters related, e.g., to material coefficients, projection-based ROMs provide mathematically rigorous physics-driven surrogate ROMs. In this work we demonstrate how, once trained, ROMs yield extremely accurate predictions (according to a prescribed tolerance) – yet cheaper than the ones provided by FOMs – of the structural deformation of the left ventricular tissue over an entire heartbeat, and of related output quantities of interest, such as the pressure-volume loop, for any desired input parameter values within a prescribed parameter range. However, the construction of ROM approximations for time-dependent cardiac mechanics is not straightforward, because of the highly nonlinear and multiscale nature of the problem, and almost never addressed. Our approach relies on the reduced basis method for parameterized partial differential equations. This technique performs a Galerkin projection onto a low-dimensional space for the displacement variable; the reduced space is built from a set of solution snapshots – obtained for different input parameter values and time instances – of the high-fidelity FOM, through the proper orthogonal decomposition technique. Then, suitable hyper-reduction techniques, such as the Discrete Empirical Interpolation Method, are exploited to efficiently handle nonlinear and parameter-dependent terms. In this work we show how a fast and reliable approximation of the time-dependent cardiac mechanical model can be achieved by a projection-based ROM, taking into account both passive and active mechanics for the left ventricle providing all the building blocks of the methodology, and highlighting those challenging aspects that are still open.</p></abstract>

An efficient reduced order model for nonlinear transient porous media flow with time-varying injection rates

Conventional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) may incur in severe limitations when dealing with nonlinear time-dependent parametrized PDEs, as these are strongly anchored to the assumption of modal linear superimposition they are based on. For problems featuring coherent structures that propagate over time such as transport, wave, or convection-dominated phenomena, the RB method may yield inefficient reduced order models (ROMs) when very high levels of accuracy are required. To overcome this limitation, in this work, we propose a new nonlinear approach to set ROMs by exploiting deep learning (DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM, both the nonlinear trial manifold (corresponding to the set of basis functions in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to the projection stage in a linear ROM) are learned in a non-intrusive way by relying on DL algorithms; the latter are trained on a set of full order model (FOM) solutions obtained for different parameter values. We show how to construct a DL-ROM for both linear and nonlinear time-dependent parametrized PDEs. Moreover, we assess its accuracy and efficiency on different parametrized PDE problems. Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to efficiently approximate the solution of parametrized PDEs, especially in cases for which a huge number of POD modes would have been necessary to achieve the same degree of accuracy.

This paper presents a methodology that combines the hybrid snapshot simulation (Bai and Wang in Int J Comput Methods 18:2050029, 2020) and the discrete empirical interpolation method (DEIM) to reduce the computational cost of constructing a DEIM-based nonlinear reduced order model (ROM) while preserving its accuracy. In distinct contrast to its traditional counterpart, the time span of the hybrid snapshot simulation is divided into multiple intervals, and a majority of the intervals are simulated by local ROMs, while a fraction by the full order model (FOM). To tackle the challenge associated with limited FOM data for ROM-DEIM construction, a new approach is proposed to reconstruct and enrich the snapshot data of nonlinear model terms using solution data in ROM intervals. A formulation and procedure based on the cell-centered finite volume method (FVM) scheme are also developed to take into account two types of nonlinearities in ROM-DEIM: the componentwise and the transport-related, non-componentwise. A global ROM-DEIM is produced immediately at the end of the snapshot simulation and can be used to accelerate online simulation for different scenarios. The proposed methodology demonstrates excellent computational performance. Specifically, the computational time of the snapshot simulation is reduced by \(\sim\) 50% without compromising ROM-DEIM accuracy (relative error less than 0.8% compared with FOM). The results prove feasibility of combining hybrid snapshot simulation and DEIM for robust and efficient ROM-DEIM development.

This paper presents a methodology that combines the hybrid snapshot simulation (Bai and Wang in Int J Comput Methods 18:2050029, 2020) and the discrete empirical interpolation method (DEIM) to reduce the computational cost of constructing a DEIM-based nonlinear reduced order model (ROM) while preserving its accuracy. In distinct contrast to its traditional counterpart, the time span of the hybrid snapshot simulation is divided into multiple intervals, and a majority of the intervals are simulated by local ROMs, while a fraction by the full order model (FOM). To tackle the challenge associated with limited FOM data for ROM-DEIM construction, a new approach is proposed to reconstruct and enrich the snapshot data of nonlinear model terms using solution data in ROM intervals. A formulation and procedure based on the cell-centered finite volume method (FVM) scheme are also developed to take into account two types of nonlinearities in ROM-DEIM: the componentwise and the transport-related, non-componentwise. A global ROM-DEIM is produced immediately at the end of the snapshot simulation and can be used to accelerate online simulation for different scenarios. The proposed methodology demonstrates excellent computational performance. Specifically, the computational time of the snapshot simulation is reduced by 50% without compromising ROM-DEIM accuracy (relative error less than 0.8% compared with FOM). The results prove feasibility of combining hybrid snapshot simulation and DEIM for robust and efficient ROM-DEIM development.

Many industrial chemical processes are complex, multi-phase and large scale in nature. These processes are characterized by various nonlinear physiochemical effects and fluid flows. Such processes often show coexistence of fast and slow dynamics during their time evolutions. The increasing demand for a flexible operation of a complex process, a pressing need to improve the product quality, an increasing energy cost and tightening environmental regulations make it rewarding to automate a large scale manufacturing process. Mathematical tools used for process modeling, simulation and control are useful to meet these challenges. Towards this purpose, development of process models, either from the first principles (conservation laws) i.e. the rigorous models or the input-output data based models constitute an important step. Both types of models have their own advantages and pitfalls. Rigorous process models can approximate the process behavior reasonably well. The ability to extrapolate the rigorous process models and the physical interpretation of their states make them more attractive for the automation purpose over the input-output data based identified models. Therefore, the use of rigorous process models and rigorous model based predictive control (R-MPC) for the purpose of online control and optimization of a process is very promising. However, due to several limitations e.g. slow computation speed and the high modeling efforts, it becomes difficult to employ the rigorous models in practise. This thesis work aims to develop a methodology which will result in smaller, less complex and computationally efficient process models from the rigorous process models which can be used in real time for online control and dynamic optimization of the industrial processes. Such methodology is commonly referred to as a methodology of Model (order) Reduction. Model order reduction aims at removing the model redundancy from the rigorous process models. The model order reduction methods that are investigated in this thesis, are applied to two benchmark examples, an industrial glass manufacturing process and a tubular reactor. The complex, nonlinear, multi-phase fluid flow that is observed in a glass manufacturing process offers multiple challenges to any model reduction technique. Often, the rigorous first principle models of these benchmark examples are implemented in a discretized form of partial differential equations and their solutions are computed using the Computational Fluid Dynamics (CFD) numerical tools. Although these models are reliable representations of the underlying process, computation of their dynamic solutions require a significant computation efforts in the form of CPU power and simulation time. The glass manufacturing process involves a large furnace whose walls wear out due to the high process temperature and aggressive nature of the molten glass. It is shown here that the wearing of a glass furnace walls result in change of flow patterns of the molten glass inside the furnace. Therefore it is also desired from the reduced order model to approximate the process behavior under the influence of changes in the process parameters. In this thesis the problem of change in flow patterns as result of changes in the geometric parameter is treated as a bifurcation phenomenon. Such bifurcations exhibited by the full order model are detected using a novel framework of reduced order models and hybrid detection mechanisms. The reduced order models are obtained using the methods explained in the subsequent paragraphs. The model reduction techniques investigated in this thesis are based on the concept of Proper Orthogonal Decompositions (POD) of the process measurements or the simulation data. The POD method of model reduction involves spectral decomposition of system solutions and results into arranging the spatio-temporal data in an order of increasing importance. The spectral decomposition results into spatial and temporal patterns. Spatial patterns are often known as POD basis while the temporal patterns are known as the POD modal coefficients. Dominant spatio-temporal patterns are then chosen to construct the most relevant lower dimensional subspace. The subsequent step involves a Galerkin projection of the governing equations of a full order first principle model on the resulting lower dimensional subspace. This thesis can be viewed as a contribution towards developing the databased nonlinear model reduction technique for large scale processes. The major contribution of this thesis is presented in the form of two novel identification based approaches to model order reduction. The methods proposed here are based on the state information of a full order model and result into linear and nonlinear reduced order models. Similar to the POD method explained in the previous paragraph, the first step of the proposed identification based methods involve spectral decomposition. The second step is different and does not involve the Galerkin projection of the equation residuals. Instead, the second step involves identification of reduced order models to approximate the evolution of POD modal coefficients. Towards this purpose, two different methods are presented. The first method involves identification of locally valid linear models to represent the dynamic behavior of the modal coefficients. Global behavior is then represented by ‘blending’ the local models. The second method involves direct identification of the nonlinear models to represent dynamic evolution of the model coefficients. In the first proposed model reduction method, the POD modal coefficients, are treated as outputs of an unknown reduced order model that is to be identified. Using the tools from the field of system identification, a blackbox reduced order model is then identified as a linear map between the plant inputs and the modal coefficients. Using this method, multiple local reduced LTI models corresponding to various working points of the process are identified. The working points cover the nonlinear operation range of the process which describes the global process behavior. These reduced LTI models are then blended into a single Reduced Order-Linear Parameter Varying (ROLPV) model. The weighted blending is based on nonlinear splines whose coefficients are estimated using the state information of the full order model. Along with the process nonlinearity, the nonlinearity arising due to the wear of the furnace wall is also approximated using the RO-LPV modeling framework. The second model reduction method that is proposed in this thesis allows approximation of a full order nonlinear model by various (linear or nonlinear) model structures. It is observed in this thesis, that, for certain class of full order models, the POD modal coefficients can be viewed as the states of the reduced order model. This knowledge is further used to approximate the dynamic behavior of the POD modal coefficients. In particular, reduced order nonlinear models in the form of tensorial (multi-variable polynomial) systems are identified. In the view of these nonlinear tensorial models, the stability and dissipativity of these models is investigated. During the identification of the reduced order models, the physical interpretation of the states of the full order rigorous model is preserved. Due to the smaller dimension and the reduced complexity, the reduced order models are computationally very efficient. The smaller computation time allows them to be used for online control and optimization of the process plant. The possibility of inferring reduced order models from the state information of a full order model alone i.e. the possibility to infer the reduced order models in the absence of access to the governing equations of a full order model (as observed for many commercial software packages) make the methods presented here attractive. The resulting reduced order models need further system theoretic analysis in order to estimate the model quality with respect to their usage in an online controller setting.

The motivation in this article is to foster and conduct an in-depth investigation on reduced-order modeling (ROM) techniques via proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) in conjunction with high-order Legendre spectral element method (SEM) applied for time-dependent linear and nonlinear heat conduction problems. This approach significantly relieves the CPU burden, yet preserves the numerical accuracy of high-order schemes; this is especially pronounced for linear transient problems. For nonlinear cases, we use DEIM to save CPU time for nonlinear counterparts. For this purpose, we first obtain training data by solving the system of full-order models (FOM) using snapshot techniques to construct the POD basis. The well-known generalized single step single solve (GSSSS-1) framework is next employed for the first-order system for the time discretization. Numerical results for both linear and nonlinear heat conduction problems such as in a nuclear reactor, etc., show excellent agreement between FOM solutions and ROM solutions, and the CPU advantages via POD ROM techniques are also prominent for high-order Legendre SEM.

Although model order reduction techniques based on searching for a solution in a low-rank subspace are researched well for the case of linear differential equations, it is still questionable if such model order reduction techniques would work well for nonlinear PDEs. In this work, model order reduction via POD-DEIM (Proper Orthogonal Decomposition via Discrete Empirical Interpolation) method is applied to a particular nonlinear parametric PDE that is used for the modeling of electric machines. The idea of the POD-DEIM algorithm is to use statistical data about ‘typical solutions’ that correspond to ‘typical’ parameter values, to approximate solutions for other parameter values. Practical POD-DEIM application to the particular PDE has met a number of difficulties, and several improvements to the selection of initial approximation, selection of interpolation nodes, selection of interpolation basis and handling moving physical entities were made to make the method to work. These improvements, along with some numerical experiments, are presented.

.We present a new surrogate modeling technique for efficient approximation of input-output maps governed by parametrized PDEs. The model is hierarchical as it is built on a full order model, reduced order model (ROM), and machine learning (ML) model chain. The model is adaptive in the sense that the ROM and ML model are adapted on the fly during a sequence of parametric requests to the model. To allow for a certification of the model hierarchy, as well as to control the adaptation process, we employ rigorous a posteriori error estimates for the ROM and ML models. In particular, we provide an example of an ML-based model that allows for rigorous analytical quality statements. We demonstrate the efficiency of the modeling chain on a Monte Carlo and a parameter-optimization example. Here, the ROM is instantiated by Reduced Basis methods, and the ML model is given by a neural network or by a kernel model using vectorial kernel orthogonal greedy algorithms.Keywordsreduced order modelsmachine learningcertified surrogate modelingMSC codes65N3065M6068T07

In this work we derive a parametric reduced-order model (ROM) for the unsteady three-dimensional incompressible Navier–Stokes equations without additional pre-processing on the reduced-order subspaces. Concerning the high-fidelity, full-order model, we start from a streamline-upwind Petrov–Galerkin stabilized finite element discretization of the equations using P1−P1 elements for velocity and pressure, respectively. We rely on Galerkin projection of the discretized equations onto reduced basis subspaces for the velocity and the pressure, respectively, obtained through Proper Orthogonal Decomposition on a dataset of snapshots of the full-order model. Both nonlinear and nonaffinely parametrized algebraic operators of the reduced-order system of nonlinear equations, including the projection of the stabilization terms, are efficiently assembled exploiting the Discrete Empirical Interpolation Method (DEIM), and its matrix version (MDEIM), thus obtaining an efficient offline–online computational splitting. We apply the proposed method to (i) a two-dimensional lid-driven cavity flow problem, considering the Reynolds number as parameter, and (ii) a three-dimensional pulsatile flow in stenotic vessels characterized by geometric and physiological parameter variations. We numerically show that the projection of the stabilization terms on the reduced basis subspace and their reconstruction using (M)DEIM allows to obtain a stable ROM with coupled velocity and pressure solutions, without any need for enriching the reduced velocity space, or further stabilizing the ROM. Additionally, we demonstrate that our implementation allows to compute the ROM solution about 20 times faster than the full order model.