a connection between time domain model order reduction
play

A connection between time domain model order reduction and moment - PowerPoint PPT Presentation

Oct 05, 2022 •588 likes •1.14k views

A connection between time domain model order reduction and moment matching Manuela Hund joint with Jens Saak September 2, 2016 Partners: Introduction Model order reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  1. A connection between time domain model order reduction and moment matching Manuela Hund joint with Jens Saak September 2, 2016 Partners:

  2. Introduction Model order reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear time-invariant (LTI) system x ( t ) = ˙ x ( t ) + B u ( t ) E A y ( t ) = x ( t ) C M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 2/24

  3. Introduction Model order reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear time-invariant (LTI) system x ( t ) = ˙ x ( t ) + B u ( t ) E A y ( t ) = x ( t ) C E r = V T EV ∈ R m × m A r = V T AV ∈ R m × m B r = V T B ∈ R m × p C r = CV ∈ R q × m M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 2/24

  4. Introduction Model order reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear time-invariant (LTI) system x ( t ) = ˙ x ( t ) + B u ( t ) E A y ( t ) = x ( t ) C E r = V T EV ∈ R m × m A r = V T AV ∈ R m × m B r = V T B ∈ R m × p C r = CV ∈ R q × m Reduced LTI system: E r x r ( t ) = A r x r ( t )+ B r u ( t ) ˙ y r ( t ) = C r x r ( t ) M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 2/24

  5. Time domain MOR based on orthogonal polynomials Basic idea [ JIANG/CHEN 2012 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Input Single-Output (SISO) system ( p = q = 1): E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) . M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 3/24

  6. Time domain MOR based on orthogonal polynomials Basic idea [ JIANG/CHEN 2012 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Input Single-Output (SISO) system ( p = q = 1): E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) . Approximation of state and input: ( ∀ i : v i ∈ R n , w i ∈ R , g i : [ t 0 , t f ] → R ) m − 1 m − 1 � � x ( t ) ≈ x m ( t ) = v i g i ( t ) , u ( t ) ≈ u m ( t ) = w i ˙ g i ( t ) . i =0 i =1 Artificial initial condition: m − 1 � x 0 = x ( t 0 ) ≈ x m ( t 0 ) = v i g i ( t 0 ) . i =0 M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 3/24

  7. Time domain MOR based on orthogonal polynomials Restriction [ HUND 2015 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Input Single-Output (SISO) system ( p = q = 1): E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) . Approximation of state and input: ( ∀ i : v i ∈ R n , w i ∈ R , g i : [ t 0 , t f ] → R ) m m � � x ( t ) ≈ x m +1 ( t ) = v i g i ( t ) , u ( t ) ≈ u m +1 ( t ) = w i ˙ g i ( t ) . i =1 i =1 Fixed initial condition:   0 .  ∈ R n . . x 0 = x ( t 0 ) =   .  0 M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 3/24

  8. Time domain MOR based on orthogonal polynomials Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem 1 (Differential recurrence formula) [ HUND 2015 ] For three sequenced orthogonal polynomials g i ( t ), where i ∈ N 0 , it holds: g n ( t ) = α n ˙ g n +1 ( t ) + β n ˙ g n ( t ) + γ n ˙ g n − 1 ( t ) , n = 1 , 2 , . . . , where α n , β n , γ n are diffential recurrence coefficients. M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 4/24

  9. Time domain MOR based on orthogonal polynomials Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem 1 (Differential recurrence formula) [ HUND 2015 ] For three sequenced orthogonal polynomials g i ( t ), where i ∈ N 0 , it holds: g n ( t ) = α n ˙ g n +1 ( t ) + β n ˙ g n ( t ) + γ n ˙ g n − 1 ( t ) , n = 1 , 2 , . . . , where α n , β n , γ n are diffential recurrence coefficients. application of differential recurrence formula in x m +1 ( t ) ⇒ application to state equation leads to expressions depending on g i ( t ) ˙ M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 4/24

  10. Time domain MOR based on orthogonal polynomials Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem 1 (Differential recurrence formula) [ HUND 2015 ] For three sequenced orthogonal polynomials g i ( t ), where i ∈ N 0 , it holds: g n ( t ) = α n ˙ g n +1 ( t ) + β n ˙ g n ( t ) + γ n ˙ g n − 1 ( t ) , n = 1 , 2 , . . . , where α n , β n , γ n are diffential recurrence coefficients. application of differential recurrence formula in x m +1 ( t ) ⇒ application to state equation leads to expressions depending on g i ( t ) ˙ coefficient comparison leads to a linear system of equations Hv = f , where H ∈ R mn × mn , v ∈ R mn × 1 , f ∈ R mn × 1 M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 4/24

  11. Time domain MOR based on orthogonal polynomials Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem 1 (Differential recurrence formula) [ HUND 2015 ] For three sequenced orthogonal polynomials g i ( t ), where i ∈ N 0 , it holds: g n ( t ) = α n ˙ g n +1 ( t ) + β n ˙ g n ( t ) + γ n ˙ g n − 1 ( t ) , n = 1 , 2 , . . . , where α n , β n , γ n are diffential recurrence coefficients. application of differential recurrence formula in x m +1 ( t ) ⇒ application to state equation leads to expressions depending on g i ( t ) ˙ coefficient comparison leads to a linear system of equations Hv = f , where H ∈ R mn × mn , v ∈ R mn × 1 , f ∈ R mn × 1 determine projection matrix V by orthogonalization of span { v 1 , . . . , v m } M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 4/24

  12. Time domain MOR based on orthogonal polynomials Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   E − β 1 A − γ 2 A 0 0 · · · · · · · · · . ...  .  − α 1 A E − β 2 A − γ 3 A .     . ...   .  0 − α 2 A E − β 3 A − γ 4 A .    . .  ... ... ... ... ...  . . H = ,   . .     . ... ... ... ...  .  . 0     . ... ... ...   .  . − γ m A  0 0 − α m − 1 A E − β m A · · · · · · · · ·   v 1 .   . v =  , .  v m   Bw 1 .   . f =  . .  Bw m M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 5/24

  13. Time domain MOR based on orthogonal polynomials Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   E − β 1 A − γ 2 A 0 0 · · · · · · · · · . ...  .  − α 1 A E − β 2 A − γ 3 A .     . ...   .  0 − α 2 A E − β 3 A − γ 4 A .    . .  ... ... ... ... ...  . . H = ,   . .     . ... ... ... ...  .  . 0     . ... ... ...   .  . − γ m A  0 0 − α m − 1 A E − β m A · · · · · · · · ·   v 1 .   . v =  , .  v m   Bw 1 .   . f =  . .  Bw m M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 5/24

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Krylov Model-Order Reduction Techniques for Time- and Frequency-Domain Wavefjeld Problems Rob

Krylov Model-Order Reduction Techniques for Time- and Frequency-Domain Wavefjeld Problems Rob

Krylov Model-Order Reduction Techniques for Time- and Frequency-Domain Wavefjeld Problems Rob Remis Delft University of Technology November 6 10, 2017 ICERM Brown University 1 Acknowledgment This is joint work with Mikhail

457 views • 45 slides
Generalised FEs: Domain Decomposition, Optimal Local Approximation & Model Order Reduction

Generalised FEs: Domain Decomposition, Optimal Local Approximation & Model Order Reduction

Generalised FEs: Domain Decomposition, Optimal Local Approximation & Model Order Reduction Robert Scheichl Institute of Applied Mathematics & Interdisciplinary Centre for Scientific Computing Universit at Heidelberg Collaborators:

1.07k views • 105 slides
Model Order Reduction of Model Order Reduction of Parameterized Interconnect Networks

Model Order Reduction of Model Order Reduction of Parameterized Interconnect Networks

Model Order Reduction of Model Order Reduction of Parameterized Interconnect Networks Parameterized Interconnect Networks via a Two- -Directional Directional Arnoldi Arnoldi Process Process via a Two Yung-Ta Li joint work with Zhaojun

509 views • 33 slides
(Randomized) Localized Model Order Reduction Kathrin Smetana (University of Twente) March 24,

(Randomized) Localized Model Order Reduction Kathrin Smetana (University of Twente) March 24,

(Randomized) Localized Model Order Reduction Kathrin Smetana (University of Twente) March 24, 2020 ICERM Workshop Algorithms for Dimension and Complexity Reduction K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March

1.23k views • 88 slides
Reduced order method combined with domain decomposition method IHP Workshop: Recent developments

Reduced order method combined with domain decomposition method IHP Workshop: Recent developments

Reduced order method combined with domain decomposition method IHP Workshop: Recent developments in numerical methods for model reduction Davide Baroli, Stphane Bordas, Lars Beex Jack Hale IHP Workshop 1 / 22 Outline 1 Motivation 2

575 views • 22 slides
Model Order Reduction of Higher Order Systems Joint work with Peter Benner and Philip Saltenberger

Model Order Reduction of Higher Order Systems Joint work with Peter Benner and Philip Saltenberger

Model Order Reduction of Higher Order Systems Joint work with Peter Benner and Philip Saltenberger Heike Fabender, ICERM, February 2020 Institute for Numerical Analysis, TU Braunschweig Introduction Approach 1 Linearizations Example Robot

1.31k views • 118 slides
Domain-Specific Reduction of Language Model Databases: Overcoming Chatbot Implementation

Domain-Specific Reduction of Language Model Databases: Overcoming Chatbot Implementation

Domain-Specific Reduction of Language Model Databases: Overcoming Chatbot Implementation Obstacles Nicholas J. Kaimakis, Dan M. Davis Samuel Breck, & Benjamin D. Nye HPC-Education Institute for Creative Technologies (ICT) and USC Univ.

235 views • 20 slides
The Frequency Domain Time domain methods: regress present on past; capture dynamics in

The Frequency Domain Time domain methods: regress present on past; capture dynamics in

The Frequency Domain Time domain methods: regress present on past; capture dynamics in terms of velocity (first order), accel- eration (second order), inertia, etc. Frequency domain methods: regress present on periodic sines and

585 views • 10 slides
On the KalmanYakubovichPopov Lemma and its Application in Model Order Reduction Peter

On the KalmanYakubovichPopov Lemma and its Application in Model Order Reduction Peter

Elgersburg Workshop 2013 February 1114, 2013 On the KalmanYakubovichPopov Lemma and its Application in Model Order Reduction Peter Benner Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex

1.25k views • 98 slides
Time Domain Models Box & Jenkins popularized an approach to time series analysis based on

Time Domain Models Box & Jenkins popularized an approach to time series analysis based on

Time Domain Models Box & Jenkins popularized an approach to time series analysis based on Auto-Regressive Integrated Moving Average (ARIMA) models. 1 Autoregressive Models Autoregressive model of order p (AR( p )): x t =

212 views • 18 slides
Mathematical Properties of PML in time domain Oliver Pfeifer by P. Joly Overview Some

Mathematical Properties of PML in time domain Oliver Pfeifer by P. Joly Overview Some

Mathematical Properties of PML in time domain Oliver Pfeifer by P. Joly Overview Some properties of hyperbolic systems of first order PDE Well-posedness of the PML model Stability of the PML-method Cauchy problem in R 2 Lets have a

798 views • 33 slides
Order Reduction of Large Scale DAE Models J.D. Hedengren and T. F. Edgar Department of Chemical

Order Reduction of Large Scale DAE Models J.D. Hedengren and T. F. Edgar Department of Chemical

Order Reduction of Large Scale DAE Models J.D. Hedengren and T. F. Edgar Department of Chemical Engineering The University of Texas at Austin 1 Outline Motivation Two Step Process for DAE Model Reduction 1. Reduction of

539 views • 29 slides
Guaranteed Control of Switched Control Systems Using Model Order Reduction and Bisection ent 1 ,

Guaranteed Control of Switched Control Systems Using Model Order Reduction and Bisection ent 1 ,

Guaranteed Control of Switched Control Systems Using Model Order Reduction and Bisection ent 1 , Florian De Vuyst 1 , Christian Rey 2 , Adrien Le Co Ludovic Chamoin 2 , Laurent Fribourg 3 April 11, 2015 1 CMLA Centre de Math ematiques et de

1.69k views • 135 slides
Efficient, Parametrically-Robust Nonlinear Model Reduction using Local Reduced-Order Bases

Efficient, Parametrically-Robust Nonlinear Model Reduction using Local Reduced-Order Bases

Introduction Local Reduced-Order Models Application Conclusion Efficient, Parametrically-Robust Nonlinear Model Reduction using Local Reduced-Order Bases Matthew J. Zahr and Charbel Farhat Farhat Research Group Stanford University SIAM

545 views • 39 slides
F S k F k n p.j.cameron@qmw.ac.uk 1 k n are the Stirling numbers k

F S k F k n p.j.cameron@qmw.ac.uk 1 k n are the Stirling numbers k

Connection with model theory A countable first-order structure is countably categorical if it is the unique countable model of its Orbits on n -tuples first-order theory. Theorem M is

334 views • 4 slides
Dynamic Partial-Order Reduction for Model Checking Software Cormac Flanagan Patrice Godefroid

Dynamic Partial-Order Reduction for Model Checking Software Cormac Flanagan Patrice Godefroid

Dynamic Partial-Order Reduction for Model Checking Software Cormac Flanagan Patrice Godefroid UC Santa Cruz Bell Labs Presented by: Ulrich Mller Model Checking Given a multithreaded program Wed like to check for deadlocks and

462 views • 17 slides
an and d Im Impl plementa ementation tion (01 0120 20442 4423) ) Naming aming an and

an and d Im Impl plementa ementation tion (01 0120 20442 4423) ) Naming aming an and

Net etwork work Ke Kerne nel l Ar Archi chitect tectures ures an and d Im Impl plementa ementation tion (01 0120 20442 4423) ) Naming aming an and Ad Addressing ressing Chaiporn Jaikaeo Chaiporn.j@ku.ac.th Department of

375 views • 19 slides
Low temperature transport in correlated systems and its evolution under pressure V. Zlatic, R.

Low temperature transport in correlated systems and its evolution under pressure V. Zlatic, R.

Low temperature transport in correlated systems and its evolution under pressure V. Zlatic, R. M., J. K. Freericks (Phys. Rev. B 78 , 045113 (2008)) Thermoelectric device efficiency in the temperature range of interest given by the Figure of

881 views • 30 slides
Doing fieldwork: Doing fieldwork: Some personal experiences Some personal experiences Rik De

Doing fieldwork: Doing fieldwork: Some personal experiences Some personal experiences Rik De

Doing fieldwork: Doing fieldwork: Some personal experiences Some personal experiences Rik De Busser Rik De Busser Research Centre for Linguistic Typology Research Centre for Linguistic Typology Institute of Linguistics, Academia Sinica

545 views • 41 slides
On the Lindell-Pinkas Secure Computation of Logarithms: From Theory to Practice Raphael S. Ryger

On the Lindell-Pinkas Secure Computation of Logarithms: From Theory to Practice Raphael S. Ryger

On the Lindell-Pinkas Secure Computation of Logarithms: From Theory to Practice Raphael S. Ryger Onur Kardes Rebecca N. Wright Yale University Stevens Institute of Technology Rutgers University New Haven, CT USA Hoboken, NJ USA Piscataway,

794 views • 60 slides
Slide 1 ___________________________________ 8.3 Re le va nt Costs o Re le va nt Costs De finition

Slide 1 ___________________________________ 8.3 Re le va nt Costs o Re le va nt Costs De finition

Slide 1 ___________________________________ 8.3 Re le va nt Costs o Re le va nt Costs De finition ___________________________________ o c osts that diffe r fr om one alte r na tive busine ss opportunity to a nothe r o AKA avoida ble c osts

664 views • 5 slides
Chapter 4: Relevant Costs and Benefits for Decision- Making Agenda Sunk/Opportunity Costs

Chapter 4: Relevant Costs and Benefits for Decision- Making Agenda Sunk/Opportunity Costs

Chapter 4: Relevant Costs and Benefits for Decision- Making Agenda Sunk/Opportunity Costs Decision Relevance Differential Analysis 2 1 Its all relevant Sunk Costs outlays of resources or effort from past periods.

398 views • 15 slides
APT TECHNICAL CPD - MAF SHORT TERM DECISION MAKING Short term decision making and pricing

APT TECHNICAL CPD - MAF SHORT TERM DECISION MAKING Short term decision making and pricing

APT TECHNICAL CPD - MAF SHORT TERM DECISION MAKING Short term decision making and pricing strategy Nicholas Riemer Nicholas.Riemer@firstrand.co.za Agenda Workflow to understanding short term decision making and pricing strategy What

272 views • 16 slides
CbPC Business Plans: Principles to Guide Development of a Sustainable Program Lynn Hill

CbPC Business Plans: Principles to Guide Development of a Sustainable Program Lynn Hill

CbPC Business Plans: Principles to Guide Development of a Sustainable Program Lynn Hill Spragens, MBA Spragens & Associates, LLC Lynn@Lspragens.com Tuesday, April 12, 2016 Join us for upcoming CAPC webinars and virtual office hours

641 views • 33 slides

More recommend