Publications

25. M. Kärcher, S. Boyaval, M.A. Grepl, and K. Veroy. Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation. Optimization and Engineering, 19(3), pp 663-695, 2018.         preprint available   |   doi:10.1007/s11081-018-9389-2

    We propose a certified reduced basis approach for the strong- and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the given observational data to estimate only the unknown initial condition of the model, the weak-constraint 4D-Var formulation additionally provides an estimate for the model error and thus can deal with imperfect models. Since the model error is a distributed function in both space and time, the 4D-Var formulation leads to a large-scale optimization problem for every given parameter instance of the PDE model. To solve the problem efficiently, various reduced order approaches have therefore been proposed in the recent past. Here, we employ the reduced basis method to generate reduced order approximations for the state, adjoint, initial condition, and model error. Our main contribution is the development of efficiently computable a posteriori upper bounds for the error of the reduced basis approximation with respect to the underlying high-dimensional 4D-Var problem. Numerical results are conducted to test the validity of our approach.

    
    
24. M. Kärcher, Z. Tokoutsi, M.A. Grepl, and K. Veroy. Certified Reduced Basis Methods for Parametrized Distributed Optimal Control Problems. Journal of Scientific Computing, 75(1), pp 276-307, 2018.         preprint available   |   doi:10.1007/s10915-017-0539-z

    In this paper, we consider the efficient and reliable solution of distributed optimal control problems governed by parametrized elliptic partial differential equations. The reduced basis method is used as a low-dimensional surrogate model to solve the optimal control problem. To this end, we introduce reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable. We also propose two different error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. The reduced basis optimal control problem and associated a posteriori error bounds can be efficiently evaluated in an offline–online computational procedure, thus making our approach relevant in the many-query or real-time context. We compare our bounds with a previously proposed bound based on the Banach-Necas-Babuska (BNB) theory and present numerical results for two model problems: a Graetz flow problem and a heat transfer problem. Finally, we also apply and test the performance of our newly proposed bound on a hyperthermia treatment planning problem.

    
    
23. C. Kuppe et al. Inverse correlation between vascular endothelial growth factor back-filtration and capillary filtration pressures. Nephrology Dialysis Transplantation, 33(9), pp 1514 - 1525, 2018.     doi: 10.1093/ndt/gfy057

    Background: Vascular endothelial growth factor A (VEGF) is an essential growth factor during glomerular development and postnatal homeostasis. VEGF is secreted in high amounts by podocytes into the primary urine, back-filtered across the glomerular capillary wall to act on endothelial cells. So far it has been assumed that VEGF back-filtration is driven at a constant rate exclusively by diffusion. Methods: In the present work, glomerular VEGF back-filtration was investigated in vivo using a novel extended model based on endothelial fenestrations as surrogate marker for local VEGF concentrations. Single nephron glomerular filtration rate (SNGFR) and/or local filtration flux were manipulated by partial renal mass ablation, tubular ablation, and in transgenic mouse models of systemic or podocytic VEGF overexpression or reduction. Results: Our study shows positive correlations between VEGF back-filtration and SNGFR as well as effective filtration rate under physiological conditions along individual glomerular capillaries in rodents and humans. Conclusion: Our results suggest that an additional force drives VEGF back-filtration, potentially regulated by SNGFR.

    
    
22. E. Qian, M.A. Grepl, K. Veroy, and K. Willcox: A Certified Trust Region Reduced Basis Approach to PDE-constrained Optimization.
    SIAM Journal of Scientific Computing, 39(5), pp 434– 460, 2017.         preprint available   |   doi:10.1137/16M1081981

    Parameter optimization problems constrained by partial differential equations (PDEs) appear in many science and engineering applications. Solving these optimization problems may require a prohibitively large number of computationally expensive PDE solves, especially if the dimension of the design space is large. It is therefore advantageous to replace expensive high-dimensional PDE solvers (e.g., finite element) with lower-dimension surrogate models. In this paper, the reduced basis (RB) model reduction method is used in conjunction with a trust region optimization framework to accelerate PDE-constrained parameter optimization. Novel a posteriori error bounds on the RB cost and cost gradient for quadratic cost functionals (e.g., least squares) are presented, and used to guarantee convergence to the optimum of the high-fidelity model. The proposed certified RB trust region approach uses high-fidelity solves to update the RB model only if the approximation is no longer sufficiently accurate, reducing the number of full-fidelity solves required. We consider problems governed by elliptic and parabolic PDEs and present numerical results for a thermal fin model problem in which we are able to reduce the number of full solves necessary for the optimization by up to 86%.

    
    
21. R. O'Connor and M.A. Grepl: Offline Error Bounds for the Reduced Basis Method. In Model Reduction of Parametrized Systems , MS&A book series, volume 17, pp 121-136, 2017         preprint available   |   doi:10.1007/978-3-319-58786-8_8

    The reduced basis method is a model order reduction technique that is specifically designed for parameter-dependent systems. Due to an offline-online computational decomposition, the method is particularly suitable for the many-query or real-time contexts. Furthermore, it provides rigorous and efficiently evaluable a posteriori error bounds, which are used offline in the greedy algorithm to construct the reduced basis spaces and may be used online to certify the accuracy of the reduced basis approximation. Unfortunately, in real-time applications a posteriori error bounds are of limited use. First, if the reduced basis approximation is not accurate enough, it is generally impossible to go back to the offline stage and refine the reduced model; and second, the greedy algorithm guarantees a desired accuracy only over the finite parameter training set and not over all points in the admissible parameter domain. Here, we propose an extension or ``add-on'' to the standard greedy algorithm that allows us to evaluate bounds over the entire domain, given information for only a finite number of points. Our approach employs sensitivity information at a finite number of points to bound the error and may thus be used to guarantee a certain error tolerance over the entire parameter domain during the offline stage. We focus on an elliptic problem and provide numerical results for a thermal block model problem to validate our approach.

    
    
20. E. Bader, M.A. Grepl, and K. Veroy. A Certified Reduced Basis Approach for Parametrized Optimal Control Problems with Two-Sided Control Constraints. In Model Reduction of Parametrized Systems , MS&A book series, volume 17, pp 37-54, 2017         preprint available   |   doi:10.1007/978-3-319-58786-8_3

    In this paper, we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by elliptic partial differential equations. We consider the standard linear-quadratic problem setting with distributed control and two-sided control constraints, which play an important role in many industrial and economical applications. For this problem class, we propose two different reduced basis approximations and associated error estimation procedures. In our first approach, we directly consider the resulting optimality system, introduce suitable reduced basis approximations for the state, adjoint, control, and Lagrange multipliers, and use a projection approach to bound the error in the reduced optimal control. For our second approach, we first reformulate the optimal control problem using two slack variables, we then develop a reduced basis approximation for both slack problems by suitably restricting the solution space, and derive error bounds for the slack based optimal control. We discuss benefits and drawbacks of both approaches and substantiate the comparison by presenting numerical results for a model problem.

    
    
19. E. Bader, M.A. Grepl, and S. Müller: A Static Condensation Reduced Basis Element Approach for the Reynolds Lubrication Equation.
    Com. in Computational Physics, Vol. 21, 1, pp. 126-148, 2017         preprint available   |   doi: 10.4208/cicp.130815.180416a

    In this paper, we propose a Static Condensation Reduced Basis Element (SCRBE) approach for the Reynolds Lubrication Equation (RLE). The SCRBE method is a computational tool that allows to efficiently analyze parametrized structures which can be decomposed into a large number of similar components. Here, we extend the methodology to allow for a more general domain decomposition, a typical example being a checkerboard-pattern assembled from similar components. To this end, we extend the formulation and associated a posteriori error bound procedure. Our motivation comes from the analysis of the pressure distribution in plain journal bearings governed by the RLE. However, the SCRBE approach presented is not limited to bearings and the RLE, but directly extends to other component-based systems. We show numerical results for plain bearings to demonstrate the validity of the proposed approach.

    
    
18. E. Bader, M. Kärcher, M.A. Grepl, and K. Veroy: Certified Reduced Basis Methods for Parametrized Distributed Optimal Control Problems with Control Constraints.
    SIAM Journal of Scientific Computing 38 ( 6), pp. 3921-3946, 2016.       |   doi:10.1137/16M1059898

    In this paper, we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by scalara coercive elliptic partial differential equations. We consider the standard linear-quadratic problem setting with distributed control and unilateral control constraints. For this problem class, we propose two different reduced basis approximations and associated error estimation procedures. In our first approach, we directly consider the resulting optimality system, introduce suitable reduced basis approximations for the state, adjoint, control, and Lagrange multipliers, and use a projection approach to bound the error in the reduced optimal control. For our second approach, we first reformulate the optimal control problem using a slack variable, we then develop a reduced basis approximation for the slack problem by suitably restricting the solution space, and derive error bounds for the slack based optimal control. We discuss benefits and drawbacks of both approaches and substantiate the comparison by presenting numerical results for several model problems.

    
    
17. B. Ahmedov, M. Herty, and M.A. Grepl. Certified Reduced Order Methods for Optimal Treatment Planning. M3AS: Mathematical Models and Methods in Applied Sciences, 26(4), 2016, pp. 699-727. 2016.         preprint available (revised version)   |   doi:10.1142/S0218202516500159

    We study numerical methods for inverse problems arising in cancer therapy treatment under uncertainty. The interest is on efficient and reliable numerical methods that allow to determine the influence of possible unknown parameters on the treatment plan for cancer therapy. The Boltzmann transport equation is used to model the evolution of charged particles in tissue. A mixed variational framework is presented and existence and uniqueness of a weak solution is established. The inverse problem is discretized using a low--dimensional reduced basis formulation based on a P1-FE discretization. We derive a posteriori bounds for the error in the reduced basis solution of the optimal control problem with respect to the solution of the P1-FE discretization. Numerical results in slab geometry are presented to confirm the validity of our approach.

    
    
16. M. Kärcher and M.A. Grepl. A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems. ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), pp. 1615-1638.         preprint available (revised version)   |   doi:10.1051/m2an/2014012

    We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving control constraints and that, even in the presence of control constraints, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. We also propose two greedy sampling procedures to construct the reduced basis space. Numerical results are presented to confirm the validity of our approach.

    
    
15. M. Kärcher and M.A. Grepl. A Certified Reduced Basis Method for Parametrized Elliptic Optimal Control Problems. ESAIM: Control, Optimisation and Calculus of Variations, 20(02), April 2014, pp 416-441.         preprint available   |   doi:10.1051/cocv/2013069

    In this paper, we employ the reduced basis method as a surrogate model for the solution of linear-quadratic optimal control problems governed by parametrized elliptic partial differential equations. We present a posteriori error estimation and dual procedures that provide rigorous bounds for the error in several quantities of interest: the optimal control, the cost functional, and general linear output functionals of the control, state, and adjoint variables. We show that, based on the assumption of affine parameter dependence, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. Numerical results are presented to confirm the validity of our approach.

    
    
14. M.A. Grepl. Reduced Basis Methods for Parametrized Optimal Control Problems. Oberwolfach Reports , 10(3), 2013, pp. 2179–2257. Multiscale and High-Dimensional Problems.     doi:10.4171/OWR/2013/39
    
    
13. J.L. Eftang, M.A. Grepl, A.T. Patera, and E. Rønquist. Approximation of Parametric Derivatives by the Empirical Interpolation Method. Foundations of Computational Mathematics, 13(5), 2013, pp. 763-787.         preprint available | doi:10.1007/s10208-012-9125-9

    We introduce a general a priori convergence result for the approximation of parametric derivatives of parametrized functions. We consider the best approximations to parametric derivatives in a sequence of approximation spaces generated by a general approximation scheme, and we show that these approximations are convergent provided that the best approximation to the function itself is convergent. We also provide estimates for the convergence rates. We present numerical results with spaces generated by a particular approximation scheme—the Empirical Interpolation Method—to confirm the validity of the general theory.

    
    
12. V. Tenten et al. Albumin is recycled from the primary urine by tubular transcytosis. Journal of the American Society of Nephrology, 24(12):1966-80, Dec 2013.     doi: 10.1681/ASN.2013010018
    
    
11. R. Hausmann, M.A. Grepl, V. Knecht, and M.J. Möller. The glomerular filtration barrier function - new concepts. Current Opinion in Nephrology and Hypertension, 21(4):441-449, July 2012.     doi: 10.1097/MNH.0b013e328354a28e

    In this review, we examine our novel model for glomerular filtration in more detail. We outline the physical mechanisms by which electrokinetic effects (streaming potentials) are generated. We investigate the potential impact of the electrical field on the passage of albumin across the glomerular filtration barrier. We review the mathematical heteroporous model including electrical effects and analyse a selection of experimental studies for indications that electrical effects influence glomerular permeability significantly.

    
    
10. M.A. Grepl. Model Order Reduction of Parametrized Nonlinear Reaction-Diffusion Systems. Computers and Chemical Engineering 43(2012), pp. 33-44.         preprint available | doi:10.1016/j.compchemeng.2012.03.013

    We present a model order reduction technique for parametrized nonlinear reaction–diffusion systems. In our approach we combine the reduced basis method – a computational framework for rapid evaluation of functional outputs associated with the solution of parametrized partial differential equations – with the empirical interpolation method – a tool to construct “affine” coefficient-function approximations of nonlinear parameter dependent functions. We develop an efficient offline–online computational procedure for the evaluation of the reduced basis approximation: in the offline stage, we generate the reduced basis space; in the online stage, given a new parameter value, we calculate the reduced basis output. The operation count for the online stage depends only on the dimension of the reduced order model and the parametric complexity of the problem. The method is thus ideally suited for the many-query or real-time contexts. We present numerical results for a non-isothermal reaction–diffusion model to confirm and test our approach.

    
    
9. M. Rasty and M.A. Grepl. Efficient Reduced Basis Solution of Quadratically Nonlinear Diffusion Equations. Proceedings of 7th Vienna Conference on Mathematical Modelling -- MATHMOD 2012, 2012.         preprint available | doi:10.3182/20120215-3-AT-3016.00125

   We present reduced basis approximations and associated a posteriori error estimation procedures for a steady quadratically nonlinear diffusion equation. We develop an efficient computational procedure for the evaluation of the approximation and bound. The method is thus ideally suited for many-query or real-time applications. Numerical results are presented to confirm the rigor, sharpness and fast convergence of our approach.

   
   
8. D. Klindworth, M.A. Grepl, and G.Vossen. Certified Reduced Basis Methods for Parametrized Parabolic Partial Differential Equations with Non-Affine Source Terms. Computer Methods in Applied Mechanics and Engineering 209-212 (2012), pp.144-155.         preprint available (revised August 2011) | doi:10.1016/j.cma.2011.10.010

   We present rigorous a posteriori output error bounds for reduced basis approximations of parametrized parabolic partial differential equations with non-affine source terms. The method employs the empirical interpolation method in order to construct affine coefficient-function approximations of the non-affine parametrized functions. Our a posteriori error bounds take both error contributions explicitly into account—the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation. To this end, we employ recently developed rigorous error bounds for the empirical interpolation method and develop error estimation and primal–dual formulations to provide rigorous bounds for the error in specific outputs of interest. We present an efficient offline–online computational procedure for the calculation of the reduced basis approximation and associated error bound. The method is thus ideally suited for many-query or real-time contexts. As a specific motivational example we consider a three-dimensional mathematical model of a welding process. Our numerical results show that we obtain efficient and reliable mathematical models which may be gainfully employed in manufacturing and product development.

   
   
7. M.A. Grepl. Certified Reduced Basis Methods for Nonaffine Linear Time-Varying and Nonlinear Parabolic Partial Differential Equations. M3AS: Mathematical Models and Methods in Applied Sciences, 2012, 22(3), 40 pages.         preprint available | doi:10.1142/S0218202511500151

   We present reduced basis approximations and associated a posteriori error bounds for parabolic partial differential equations involving (i) a nonaffine dependence on the parameter and (ii ) a nonlinear dependence on the field variable. The method employs the Empirical Interpolation Method in order to construct "affine" coefficient-function approximations of the "nonaffine" (or nonlinear) parametrized functions. We consider linear time-invariant as well as linear time-varying nonaffine functions and introduce a new sampling approach to generate the function approximation space for the latter case. Our a posteriori error bounds take both error contributions explicitly into account — the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation. We show that these bounds are rigorous upper bounds for the approximation error under certain conditions on the function interpolation, thus addressing the demand for certainty of the approximation. As regards efficiency, we develop an offline–online computational procedure for the calculation of the reduced basis approximation and associated error bound. The method is thus ideally suited for the many-query or real-time contexts. Numerical results are presented to confirm and test our approach.

   
   
6. M.A. Grepl and K. Veroy. A level set reduced basis approach to parameter estimation. C. R. Acad. Sci. Paris, Ser. I 349 (2011), pp.1229-1232.         preprint available | doi:10.1016/j.crma.2011.10.020

   We introduce an efficient level set framework to parameter estimation problems governed by parametrized partial differential equations. The main ingredients are: (i) an “admissible region” approach to parameter estimation; (ii) the certified reduced basis method for efficient and reliable solution of parametrized partial differential equations; and (iii) a parameter-space level set method for construction of the admissible region. The method can handle nonconvex and multiply connected regions. Numerical results for two examples in design and inverse problems illustrate the versatility of the approach.

   
   
5. M.A. Grepl and M. Kärcher. Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. C. R. Acad. Sci. Paris, Ser. I 349, 2011, pp. 873-877.         preprint available | doi:10.1016/j.crma.2011.07.010

   We employ the reduced basis method as a surrogate model for the solution of optimal control problems governed by parametrized partial differential equations (PDEs) and develop rigorous a posteriori error bounds for the error in the optimal control and the associated error in the cost functional. The proposed bounds can be efficiently evaluated in an offline–online computational procedure. We present numerical results that confirm the validity of our approach.

   
   
4. J.L. Eftang, M.A. Grepl, and A.T. Patera. A Posteriori Error Bounds for the Empirical Interpolation Method. C. R. Acad. Sci. Paris, Ser. I 348, 2010, pp. 575-579.         preprint available | doi:10.1016/j.crma.2010.03.004

   We present rigorous a posteriori error bounds for the Empirical Interpolation Method (EIM). The essential ingredients are (i) analytical upper bounds for the parametric derivatives of the function to be approximated, (ii) the EIM “Lebesgue constant,” and (iii) information concerning the EIM approximation error at a finite set of points in parameter space. The bound is computed “off-line” and is valid over the entire parameter domain; it is thus readily employed in (say) the “on-line” reduced basis context. We present numerical results that confirm the validity of our approach.

   
   
3. M.A. Grepl, Y. Maday, N.C. Nguyen, and A.T. Patera. Efficient Reduced-Basis Treatment of Nonaffine and Nonlinear Partial Differential Equations. M2AN Mathematical Modelling and Numerical Analysis, 2007, 41(3), pp. 575-605.        preprint available | doi:10.1051/m2an:2007031

   In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numerical results are presented to assess our approach.

   
   
2. M.A. Grepl, N.C. Nguyen, K. Veroy, A.T. Patera, and G.R. Liu. Certified Rapid Solution of Partial Differential Equations for Real-Time Parameter Estimation and Optimization. In Real-Time PDE-Constrained Optimization , (L.T. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, and B. van Bloemen Waanders, eds.), SIAM Computational Science and Engineering Book Series, 2007, pp. 197-215.         preprint available | doi:10.1137/1.9780898718935.ch10

   This paper reviews the RB methodology for noncoercive (Helmholtz) elliptic equations, the Navier-Stokes equations, and parabolic equations, from the perspective of real-time and reliable computation. It includes an example of real-time robust parameter estimation in the context of nondestructive evaluation of a crack in an elastic medium. We can rapidly deduce — from the response of the medium to tuned harmonic forcing — the "possibility" set of all crack lengths consistent with experimental measurements. No regularization or unverifiable prior assumptions are required: well-posedness is manifested in the shrinking of our "possibility" set as the experimental error tends to zero. The key computational enabler is the RB output prediction and associated a posteriori error estimator.

   
   
1. M.A. Grepl and A.T. Patera. A Posteriori Error Bounds for Reduced-Basis Approximations of Parametrized Parabolic Partial Differential Equations. M2AN Mathematical Modelling and Numerical Analysis, 2005, 39(1), pp. 157-181.         full text available | doi:10.1051/m2an:2005006

   In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space W N spanned by solutions of the governing partial differential equation at N selected points in parameter-time space – and develop a new greedy adaptive procedure to “optimally” construct the parameter-time sample set. Second, we propose error estimation and adjoint procedures that provide rigorous and sharp bounds for the error in specific outputs of interest: the estimates serve a priori to construct our samples, and a posteriori to confirm fidelity. Third, based on the assumption of affine parameter dependence, we develop offline-online computational procedures: in the offline stage, we generate the reduced-basis space; in the online stage, given a new parameter value, we calculate the reduced-basis output and associated error bound. The operation count for the online stage depends only on N (typically small) and the parametric complexity of the problem; the method is thus ideally suited for repeated, rapid, reliable evaluation of input-output relationships in the many-query or real-time contexts.

   
   

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