Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space

A model-constrained adaptive sampling methodology is proposed for the reduction of large-scale systems with high-dimensional parametric input spaces. Our model reduction method uses a reduced basis approach, which requires the computation of high-fidelity solutions at a number of sample points throughout the parametric input space. A key challenge that must be addressed in the optimization, control, and probabilistic settings is the need for the reduced models to capture variation over this parametric input space, which, for many applications, will be of high dimension. We pose the task of determining appropriate sample points as a PDE-constrained optimization problem, which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters. The methodology is demonstrated using examples with parametric input spaces of dimension 11 and 21, which describe thermal analysis and design of a heat conduction fin, and compared with statistically based sampling methods. For these examples, the model-constrained adaptive sampling leads to reduced models that, for a given basis size, have error several orders of magnitude smaller than that obtained using the other methods.

[1]  W. Rudin.  Principles of mathematical analysis , 1964 .

[2]  T. Manteuffel,et al.  FIRST-ORDER SYSTEM LEAST SQUARES FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS : PART II , 1994 .

[3]  Homer F. Walker,et al.  Choosing the Forcing Terms in an Inexact Newton Method , 1996, SIAM J. Sci. Comput..

[4]  Pavel B. Bochev,et al.  Finite Element Methods of Least-Squares Type , 1998, SIAM Rev..

[5]  B. Jiang.  The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics , 1998 .

[6]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[7]  K. Kunisch,et al.  Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition , 1999 .

[8]  Thomas F. Coleman,et al.  A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems , 1999, SIAM J. Sci. Comput..

[9]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[10]  E. Sachs,et al.  Trust-region proper orthogonal decomposition for flow control , 2000 .

[11]  Richard J. Beckman,et al.  A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code , 2000, Technometrics.

[12]  E. Dowell,et al.  A Parametric Analysis of Reduced Order Models of Potential Flows in Turbomachinery Using Proper Orthogonal Decomposition , 2001 .

[13]  K. Afanasiev,et al.  Adaptive Control Of A Wake Flow Using Proper Orthogonal Decomposition1 , 2001 .

[14]  H. Tran,et al.  Modeling and control of physical processes using proper orthogonal decomposition , 2001 .

[15]  D. Rovas,et al.  A blackbox reduced-basis output bound method for noncoercive linear problems , 2002 .

[16]  D. Rovas,et al.  Reduced--Basis Output Bound Methods for Parametrized Partial Differential Equations , 2002 .

[17]  D. Rovas,et al.  A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations , 2003 .

[18]  Stefan Körkel,et al.  Numerical Methods for Nonlinear Experimental Design , 2003, HPSC.

[19]  Jacob K. White,et al.  A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models , 2004, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[20]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[21]  J. Burkardt,et al.  Initial Evaluation of Pure and “ Latinized ” Centroidal Voronoi Tessellation for Non-Uniform Statistical Sampling * , 2004 .

[22]  A. Patera,et al.  A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations , 2005 .

[23]  Luca Daniel,et al.  Parameterized model order reduction of nonlinear dynamical systems , 2005, ICCAD-2005. IEEE/ACM International Conference on Computer-Aided Design, 2005..

[24]  M. Hinze,et al.  Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control , 2005 .

[25]  M. Grepl.  Reduced-basis approximation a posteriori error estimation for parabolic partial differential equations , 2005 .

[26]  A. Patera,et al.  Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds , 2005 .

[27]  A. Antoulas,et al.  A Rational Krylov Iteration for Optimal H 2 Model Reduction , 2006 .

[28]  Bui-Thanh Tan,et al.  Model-Constrained Optimization Methods for Reduction of Parameterized Large-Scale Systems , 2007 .

[29]  G. Rozza,et al.  On the stability of the reduced basis method for Stokes equations in parametrized domains , 2007 .

[30]  M. Gunzburger,et al.  Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data , 2007 .

[31]  Alexandre Megretski,et al.  A Quasi-Convex Optimization Approach to Parameterized Model Order Reduction , 2005, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.