Stabilization of POD-ROMs David Wells Virginia Tech/Rensselaer - PowerPoint PPT Presentation

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Stabilization of POD-ROMs David Wells Virginia Tech/Rensselaer Polytechnic Institute Wednesday, August 5, 2015 1 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Overview deal.II : A Powerful System for Model Reduction 2 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Overview deal.II : A Powerful System for Model Reduction 1. POD 2. POD with deal.II 3. Filtering and ROM 4. REG-ROMs 5. Conclusions & Future Work 2 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Collaborators Some of my collaborators: Volker John, (WIAS), Traian Iliescu (VT), Swetlana Giere (WIAS), Zhu Wang (SC), Xuping Xie (VT) A special thanks to Abner Salgado (UT) for writing step-35 . 3 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions VT to RPI 4 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Why deal.II ? 1. ROMs are usually expressed as finite element methods 5 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Why deal.II ? 1. ROMs are usually expressed as finite element methods 2. Community is nice 5 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Why deal.II ? 1. ROMs are usually expressed as finite element methods 2. Community is nice 3. Great documentation 5 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Why deal.II ? 1. ROMs are usually expressed as finite element methods 2. Community is nice 3. Great documentation [drwells@archway dealii-dev]$ cloc ./include 378 text files. 378 unique files. 2 files ignored. http://cloc.sourceforge.net v 1.64 T=1.42 s (264.3 files/s, 180140.8 lines/s) ------------------------------------------------------------------------------- Language files blank comment code ------------------------------------------------------------------------------- C/C++ Header 375 34261 113105 108829 CMake 1 4 23 18 ------------------------------------------------------------------------------- SUM: 376 34265 113128 108847 ------------------------------------------------------------------------------- 5 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Why deal.II ? 1. LAPACK support ( geev , getrf , getrs ) 2. HDF5 and XDMF support 3. C++11 support 6 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions The Navier-Stokes Equations u − 1 � u t + � u · ∇ � Re ∆ � u + ∇ p = 0, (1) ∇ · � u = 0 1. Specified (parabolic) inflow 2. � u × � n = 0 outflow 3. deal.II step 35 [1, 2] 4. Fractional step method 5. About 600, 000 DoFs, Re = 100 7 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions The Navier-Stokes Equations Goal: Preserve large structures and phase portraits. 8 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions The Navier-Stokes Equations y -velocity contours at t = 10 . There is a circular cylinder near the inflow on the left. 9 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Proper Orthogonal Decomposition (POD) Given a set of data with high dimensionality, what is the best (under some norm) approximation to the data for a given rank r ? 10 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions What are POD-derived basis functions? Deriving POD basis functions is a linear procedure. Let Y denote the “snapshot” matrix [5] and M = LL T denote the mass matrix. 11 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions What are POD-derived basis functions? Deriving POD basis functions is a linear procedure. Let Y denote the “snapshot” matrix [5] and M = LL T denote the mass matrix. ESV T = SVD ( L T Y ) → Φ = ( L T ) − 1 E (2) 11 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions What are POD-derived basis functions? Deriving POD basis functions is a linear procedure. Let Y denote the “snapshot” matrix [5] and M = LL T denote the mass matrix. ESV T = SVD ( L T Y ) → Φ = ( L T ) − 1 E (2) N − 1 Y T MY ν i = λ i ν i → ϕ i = ∑ y n ν i ( n ) (3) n = 0 11 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions The Method of Snapshots 1. Does either the method of snapshots or the reduced order matrices suffer a loss of accuracy from inaccurate inner product calculations? 12 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions The Method of Snapshots 1. Does either the method of snapshots or the reduced order matrices suffer a loss of accuracy from inaccurate inner product calculations? 2. Do the POD vectors calculated by the method of snapshots recover the POD interpolation error equations? 12 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions From vector.templates.h : // this is the main working loop for all vector sums using the templated 1 // operation above. it accumulates the sums using a block-wise summation 2 // algorithm with post-update. this blocked algorithm has been proposed in 3 // a similar form by Castaldo, Whaley and Chronopoulos (SIAM 4 // J. Sci. Comput. 31, 1156-1174, 2008) and we use the smallest possible 5 // block size, 2. Sometimes it is referred to as pairwise summation. The 6 // worst case error made by this algorithm is on the order O(eps * 7 // log2(vec_size)), whereas a naive summation is O(eps * vec_size). Even 8 13 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions The Method of Snapshots 0 20 40 60 80 100 0.0 0 2.5 20 5.0 7.5 40 10.0 60 12.5 15.0 80 17.5 100 20.0 Magnitudes of entries in the POD mass matrix. 14 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Interpolation Errors 2 � � r 1 − 1 R − 1 R − 1 � � ∑ σ 2 ∑ ∑ � � u n − � � ϕ i � � r 1 u n , � ϕ i � � i � � i = r 1 n = 0 i = 0 � 2 182753.567915 182753.570693 4 164311.705302 164311.712343 6 156296.758264 156296.757146 8 148780.806184 148780.808336 10 141653.502162 141653.507387 20 114794.313701 114794.326822 40 83565.3337824 83565.3313631 60 65667.1960201 65667.1963493 80 53841.1631402 53841.1635371 100 45045.8004678 45045.8035251 15 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Introduction Regularized models imply the use of a filter. 16 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions The POD Projection Filter u r ∈ X r , the POD projection seeks For a fixed r 1 < r and a given � u r ) ∈ X r 1 such that F ( � ( F ( � u r ) , � ϕ j ) = ( � ϕ j ) , ∀ j = 0, · · · , r 1 − 1. u r , � (4) 17 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions The POD Projection Filter u r ∈ X r , the POD projection seeks For a fixed r 1 < r and a given � u r ) ∈ X r 1 such that F ( � ( F ( � u r ) , � ϕ j ) = ( � ϕ j ) , ∀ j = 0, · · · , r 1 − 1. u r , � (4) Doesn’t work so well, see [6]. 17 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions The POD Differential Filter The POD differential filter is defined as follows: let δ be the radius u r ∈ X r , find F ( u r ) ∈ X r of the POD differential filter. For a given � such that � � ( I − δ 2 ∆ ) F ( u r ) , � = ( � ϕ j ) , ∀ j = 0, · · · , r − 1. (5) ϕ j u r , � 18 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions What does the differential filter do? Figure: The first POD vector for the NSE experiment. 19 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions What does the differential filter do? Figure: The filtered first POD vector for the NSE experiment, δ = 0.5 . 20 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions What does the differential filter do? Figure: The fifth POD vector for the NSE experiment. 21 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions What does the differential filter do? Figure: The filtered fifth POD vector for the NSE experiment, δ = 0.5 . 22 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Overview Considered REG-ROMs: 1. Leray regularization [4] 2. Evolve-then-filter [3] 23 / 45

Introduction Numerical Experiment POD Filtering & ROM REG-ROMs Conclusions Overview Considered REG-ROMs: 1. Leray regularization [4] 2. Evolve-then-filter [3] REG-ROMs are not: 1. consistent with the original PDE 23 / 45