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Model Reduction for Dynamical Systems - Lecture 1-

Otto-von- G uericke Universität Magdeburg Faculty of Mathematics Summer term 2017. Model Reduction for Dynamical Systems - Lecture 1-. Peter Benner and Lihong Feng. Max Planck Institute for Dynamics of Complex Technical Systems

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Model Reduction for Dynamical Systems - Lecture 1-

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  1. Otto-von-Guericke Universität Magdeburg FacultyofMathematics Summer term 2017 Model ReductionforDynamical Systems -Lecture 1- Peter Benner andLihong Feng Max Planck Institute for Dynamics ofComplex Technical Systems ComputationalMethods in Systems andControlTheory Magdeburg, Germany benner@mpi-magdeburg.mpg.defeng@mpi-magdeburg.mpg.de http://www.mpi-magdeburg.mpg.de/csc/teaching/17ss/mor

  2. Lecture: Peter Benner andLihong Feng Exercise: Petar Mlinaric Lectureandexercises/homeworkinformation: http://www.mpi-magdeburg.mpg.de/csc/teaching/17ss/mor Ask questions. Take notes when necessary, and only when necessary.

  3. What is this Lecture considering? Not Not Not Not But:

  4. Motivation Large-scale dynamical systems are omnipresent in science and technology. Example 1. Copper interconnect pattern(IBM) Picture from Encarta http://encarta.msn.com/media_461519585/pentium_microprocessor.html Picture from [N.P can der Meijs’01] Signal integrity: noise, delay. MOR Lihong Feng

  5. Motivation The interconnect can be modelled by a mathematical model: With O(106) ordinary differential equations. • Example 2 • A Gyroscope is a device for measuring or • maintaining orientation and has been used • in various automobiles (aviation, shipping • and defense). • Thedesign of the device is verified by • modelling and simulation. Pic. by Jan Lienemann and C. Moosmann, IMTEK MOR Lihong Feng

  6. Motivation • The butterfly Gyroscope can be modeled by partial differential equations (PDEs). • Using finite element method, the PDE is discretized into (in space) ODEs: With O(105) ordinary differential equations. is the vector of parameters. Multi-query, real-time simulation? MOR Lihong Feng

  7. Motivation Example 3: Simulated Moving Bed Simulated Moving Bed Process • Fine chemical • Petrochemical SMB chromatography • Pharmaceutical • Food MOR Lihong Feng

  8. Motivation porous adsorbent Mathematical modeling of SMB process PDE system (couple NColcolumns)) Initial and boundary conditions: MOR Lihong Feng

  9. Motivation Mathematical modeling of SMB process PDE system (couple NCol columns) spatial discretization DAE system A complex system of nonlinear, parametric DAEs: : operating conditions with proper initial conditions. MOR Lihong Feng

  10. Analytical solution of the LTI System LTI Syatem: Multiplying on both sides of yields which implies, Its integration from 0 to t yields,

  11. Analytical solution of the LTI System Thus we have (1) Because the inverse of is and , (1) implies (2) • It is impossible to plot the waveform of x(t) by hand, we need computersto compute x(t) numerically and plot x(t) at many samples of time. • It is difficult to compute x(t) by following the analytical formulation in (2) if A is very large. We need to solve the LTI system numerically with some numerical methods, like backward Euler, ...etc. • If the system is very large, then MOR is necessary!

  12. Basic Idea of MOR Reduced order model Original model (discretized) Goal: MOR Lihong Feng

  13. Basic Idea of MOR MOR Original model Reduce order model (ROM) Replace the original model with the reduced model. The reduced model is then integrated with other components in the device or circuit; or is integrated into a process. The reduced model is repeatedly used in design analysis. MOR Lihong Feng

  14. Basic Idea of MOR Example: Optimization for SMB s.t. • SMB model (PDEs) • Cyclic steady state (CSS) constraints • Product purity constraints: • Operational constraints on p ROM-based optimization ROMs SMB MOR Lihong Feng

  15. Basic Idea of MOR Axial concentration profiles of the full-order DAE model with order of 672. Axial concentration profiles reproduced by POD-based ROM (with reduced order of only 2). MOR Lihong Feng

  16. Basic Idea of MOR Example: Layout of a switch with four microbeams MOR Lihong Feng

  17. Basic Idea of MOR The microbeam is replaced by the ROM The schematicswitch MOR Lihong Feng

  18. Basic Idea of MOR ROM of the microbeam Enlarged microbeam Part MOR Lihong Feng

  19. Projection based MOR Original model (discretized) Reduced order model Find a subspace which includes the trajectory of x, use the projection of x in the subspace to approximate x. x Let: Px range(V) MOR Lihong Feng

  20. Projection based MOR Basic Idea of MOR Petrov-Galerkinprojection: MOR Lihong Feng

  21. Conclusion Basic Idea of MOR The question: How to construct the ROMs (W, V) for large-scale complex systems? Answer: The lecture will provide many solutions. MOR Lihong Feng

  22. Outline of the Lecture Lecture 1: Introduction Lecture 2-5: Mathematical basics Lecture 6: Balanced truncation method for linear time invariant systems. Lecture 7: Moment-matching and rational interpolation methods for linear time invariant systems. Lecture 8: Krylov subspace based method for nonlinear systems and POD method for nonlinear systems. Lecture 9: Krylov subspace based method for linear parametric systems. Lecture 10: POD and reduced basis method for nonlinear parametric systems. Notice: time and location changes can be found at: http://www.mpi-magdeburg.mpg.de/csc/teaching/17ss/mor MOR Lihong Feng

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