Suboptimal feedback control of PDEs by solving Hamilton-Jacobi Bellman equations on sparse grids
|
|
- Daniela Newton
- 5 years ago
- Views:
Transcription
1 Suboptimal feedback control of PDEs by solving Hamilton-Jacobi Bellman equations on sparse grids Jochen Garcke joint work with Axel Kröner, INRIA Saclay and CMAP, Ecole Polytechnique Ilja Kalmykov, Universität Bonn (who is looking for a PhD position) Department Numerical Data-Driven Prediction Institute for Numerical Simulation
2 Outline 1 Optimal control 2 Sparse Grids 3 Numerical Results 4 Higher Order Methods
3 Optimal Control of Low Dim. Approx. of PDEs continuous problem described by PDE semi-discretization in space model reduction semi-discrete control problem HJB-equation sparse grids feedback operator
4 Optimal Control of Low Dim. Approx. of PDEs continuous problem described by PDE semi-discretization in space model reduction semi-discrete control problem HJB-equation sparse grids feedback operator
5 Optimal Feedback Control optimal feedback control of a dynamical system min J(u) = u U ad T 0 l(y(t), u(t)) dt, ẏ(t) = f (y(t), u(t)), t > 0 y(0) = y 0 s.t. state y(t) R d, initial state y 0 R d control u(t) U m R m (often called action) Lipschitz continuous dynamics f : R d R m R d running cost with polynomial growth l : R d R m R set of admissible controls U ad = {u L 2 ([0, T ]; U m ) U m R m compact} aim: feedback law u = K(t, y (t))
6 Semi-Lagrangian scheme from DPP one derives semi-lagrangian scheme { } v k (x) = min v k+1 (y x ( t)) + t l(x, u) u U (SL) for k = N 1,..., 0 with v N (x) = 0, time step t > 0, x R d y x ( t) state obtained by time discretization scheme from x evaluation of the right hand side in (SL), either: (potentially expensive) comparison over a finite set U finite U in linear quadratic case, i.e. f (x, u) = Ax + Bu, l(x, u) = 1 2 A R d d, B R d m ( x T Mx + u T Ru ), M R d d, R R m m the optional feedback control is given by (needing approx. for v) u (t) = P U ( R 1 B T v(y (t), t) )
7 Optimal Control of Low Dim. Approx. of PDEs continuous problem described by PDE semi-discretization in space model reduction semi-discrete control problem HJB-equation sparse grids feedback operator
8 Application to a PDE Control Problem wave equation (following Kröner, Kunisch, Zidani (2015)) ŷ tt c ŷ = Bu in (0, T ) Ω, ŷ(0) = ŷ 0, ŷ t (0) = ŷ 1 in Ω, ŷ = 0 on (0, T ) Ω initial state and velocity ŷ 0 H 1 (Ω) and ŷ 1 L 2 (Ω) control operator B := (sin(πx),..., sin(mπy)) (can be generalized) formulate as first order system in time with y 1 = y, y 2 = ẏ which we can write as y 1 t = y 2, y 2 t y 1 = Bu, y 1 (0) = ŷ 0, y 2 (0) = ŷ 1, y t + Ay = (0, Bu) T, y(0) = y 0 ( ) 0 Id with y = (y 1, y 2 ), y 0 = (ŷ 0, ŷ 1 ) T Y 1, and A = c 0
9 Semi-Discrete Formulation of Control Problem semi-discrete formulation of control problem by method of lines for a given basis we define b := (ϕ 1,..., ϕ d ), d N, with ϕ i : Ω R A := (( ϕ i (x), ϕ j (x)) i,j=1,...,d ) M := ((ϕ i (x), ϕ j (x)) i,j=1,...,d ) (stiffness matrix) (mass matrix) in our numerical examples we later choose ϕ i (x) := sin(iπx), i = 1,..., d, and obtain A = diag((1/2(iπ) 2 ) i=1,...,d ), M = diag((1/2) i=1,...,d )
10 Resulting Semi-Discrete PDE Control Problem this now fits into control formulation from beginning min J(u) = u U ad T 0 l(y(t), u(t)) dt, ẏ(t) = f (y(t), u(t)), t > 0 y(0) = y 0 for initial value y 0 R 2d and t [0, T ). dynamics f (x, u) = Ax + Bu with running cost l(x, u) := β x x1 T Mx 1 + β u u T u if value function is differentiable in x, t optimal feedback control u (y (t), t) = P U ( 1 ) B T x v(y (t), t) β u where P U projection on set of admissible controls Hamilton-Jacobi Bellman equation with dimension 2d s.t.
11 Optimal Control of Low Dim. Approx. of PDEs continuous problem described by PDE semi-discretization in space model reduction semi-discrete problem HJB-equation sparse grids feedback operator
12 Interpolation with Hierarchical Basis φ φ φ 3,1 3,2 φ 3,4 3,3 φ φ3,6 3,5 φ3,7 nodal basis V 1 V 2 V 3 φ φ φ 3,1 2,1 φ 1,1 3,3 φ 3,5 φ 2,3 φ 3,7 hierarchical basis V 3 = W 3 W2 V1
13 Hier. Basis Functions in Higher Dimensions d-dimensional piecewise d-linear functions d φ l,j (x) := φ lt,jt (x t ) t=1 hierarchical difference space W l (e t is t-th unit vector) W l := V l \ d V l et, hier. diff. space represented by W l = span{φ l,j j B l } with { B l := j N d j t = 1,..., 2 lt 1, j t odd, t = 1,..., d, if l t > 1, j t = 0, 1, 2, t = 1,..., d, if l t = 1 t=1 full grid space in hierarchical basis V s n := l n W l }.
14 Hierarchical Subspaces W l W 1,1 W 1,2 W 1,3 W 1,4 W 2,1 W 2,2 W 2,3 W 2,4 W 3,1 W 3,2 W 3,3 W 3,4 W 4,1 W 4,2 W 4,3 W 4,4
15 Sparse Grids we define the sparse grid function space Vn s V n as Vn s := l 1 n+d 1 W l every f Vn s can now be represented as fn s (x) = α l,j φ l,j (x) j B l l 1 n+d 1 approximation property in H 2 mix sparse grid needs O(h 1 n f f s n 2 = O(h 2 n log(h 1 n ) d 1 ) (log(h 1 )) d 1 ) points n
16 Sparse Grids in two and three dimensions
17 Problems with Sparse Grids: Monotonicity interpolation of peaked Gaussian fct. with sparse grid n = 2 f (x 1, x 2 ) := exp ( 100(x 1 0.5) 2) exp ( 100(x 2 0.5) 2) 0-level set of interpolant is pink in the right picture sparse grid interpolation does not preserve positivity
18 Spatially Adaptive Sparse Grids to approximate functions which either do not fulfil smoothness condition at all or strongly vary due to finite but locally large derivatives adaptive refinement may be used start with a regular grid of level 2 (left) to populate index set I refine one grid point by creating all children (middle) to keep grid consistent, missing parents are created (right) usually hierarchical surplus α l,j is used as refinement indicator
19 Basic Sparse Grid SL Scheme evaluate for x Q I, t > 0, K = T / t, k = K 1,..., 0, ( ) v k (x) = min tl(x, u) + v k+1 (y x ( t)), u U v k (x) = 0 y x ( t) state obtained by time discretization scheme from x Algorithm 1: Adaptive SL-SG scheme Data: refinement constant ε, coarsening constant η Result: sequence of adaptive sparse grid solutions v k V I(k) initialize I(K) for k = K 1,..., 0 do iterate in time with t = T /K initialize I(k 1) with I(k) adaptively interpolate min u U ( vk (y x ( t)) + tl(x, u) ) coarsen v k 1 V I(k 1) see Bokanowski, G., Griebel, and Klompmaker (2013) compute v k 1
20 Further Discretization Aspects to determine minimizing control within the SL-scheme we use minimization by comparison over finite subset U σ U or gradient of the value function u n (x, s) = P U ( 1 ) B T h v n+1 (x, s) β u per finite differences ( h v n+1 (x, s) ) i := v n+1 (x + h e i, s) v n+1 (x h e i, s) 2h for computation of y x ( t) we use second order Heun scheme in our experiments we focus on discretization error in space, while using time resolution which is good enough reference solution v r in 2D computed with a higher order finite difference code on a uniform mesh by an ENO scheme compute reference trajectories y r in state space and u r in control space using a Riccati approach
21 Example 2D: Simplified Semi-Discrete Wave Equation dynamics f (x, u) = Ax + Bu with running cost l(x, u) := β x x T 1 Mx 1 + β u u T u example based on harmonic oscillator β x = 2, β u = 0.1, T = 1, t = 0.01, ( ) ( ) A =, B =, initial data x R 2, domain Q = [ 1, 1] 2, U = [ 3.5, 3.5].
22 Example 2D: Convergence of value function 0.47 normal hat (gradient) fold out hat (gradient) normal hat (compare) fold out hat (compare) v v L ε 10 4
23 Example 2D: Convergence of value function normal hat (adaptive) fold out hat (adaptive) normal hat (regular) fold out hat (regular) 0.96 v v L nodes (end)
24 Example 2D: Value function (a) ε = (b) ε = Figure: adaptive sparse grid with normal and fold out hat functions
25 Example 2D: Convergence in the Trajectory normal hat (gradient) fold out hat (gradient) normal hat (compare) fold out hat (compare) normal hat (level) fold out hat (level) normal hat (eps) fold out hat (eps) y y L y y L ε (a) L error in the trajectory vs. ε nodes (b) L error in the trajectory vs. nodes Figure: Comparing the scheme on adaptive sparse grids using normal hat and fold out basis functions
26 Example 2D: Convergence in the Control normal hat (gradient) fold out hat (gradient) normal hat (compare) fold out hat (compare) normal hat (level) fold out hat (level) normal hat (eps) fold out hat (eps) u u L 0.56 u u L ε 10 4 (a) L error in the control vs. ε nodes (b) L error in the control vs. nodes Figure: Comparing the scheme using the gradient approach on adaptive and regular sparse grids using normal hat and fold out basis functions
27 Example: Semi-Discrete Wave Equation (4D) reminder: wave equation as first order system in time ẏ(t) = f w (y(t), u(t)), t > 0, y(0) = y 0 with dynamics f w : R 2d R m R, f w (x, u) := Ax + Bu where ( A := 0 I d cm 1 A 0 ), B := ( ) 0, b R m d, y b 0 R 2d cost consider setup l w (x, u) := β x x T 1 Mx 1 + β u u T u β x = 2, β u = 0.1, T = 4, t = 0.01, c = 0.05.
28 Example: Convergence in the Control normal hat fold out hat normal hat (eps) fold out hat (eps) fold out hat (level) u u L 0.63 u u L ε (a) L error in the control vs. ε nodes (b) L error in the control vs. nodes Figure: Error in the control comparing the scheme based on sparse grids using normal hat and fold out basis functions, regular and adaptive sparse grids (4D).
29 Example: semi-discrete wave equation (4D) SL-SG Riccati SL-SG Riccati SL-SG Riccati time (a) y time (b) y time (c) u SL-SG Riccati SL-SG Riccati SL-SG Riccati time (d) y time (e) y time (f) u 2
30 Example: semi-discrete wave equation (6D) 10 1 fold out hat (gradient) fold out hat (gradient) u u L 10 0 u u L ε (a) error in the control vs. ε nodes (b) error in the control vs. nodes Figure: The adaptive sparse grids scheme using fold out basis functions (6D). Need to decrease time step to t = (larger entries in stiffness matrix).
31 Example: semi-discrete wave equation (8D) 10 1 fold out hat (gradient) fold out hat (gradient) u u L 10 0 u u L ε (a) error in the control vs. ε nodes (b) error in the control vs. nodes Figure: The adaptive sparse grids scheme using fold out basis functions (8D). Need to decrease time step to t = (larger entries in stiffness matrix).
32 Bilinear System of Schrödinger type we write a Schrödinger type equation as a real-valued system f s : R 2d R R, f s (x, u) := Ax + uby with ( 0 cm A = 1 ) A cm 1, B = M 1 ˆB, M = A 0, as before discrete cost functional is given by J(u, y) := T 0 l(y(t), u(t)) dt running cost l s (x, u) := β x x T Mx + β u u T u, where x = (x 1, x 2 ) R 2d, u U ( ) M 0, ˆB R 2d 2d 0 M
33 Example 2D setup A = ( ) 0 0.5, B = ( ) 0 0.5, M = with T = 1, t = 1/500, β x = 2, β u = 0.1, U = [ 4, 4] in this case the control bounds are active ( )
34 2D-Bilinear Schrödinger Type Problem 10 0 normal hat fold out hat error nodes (end) (a) error in the value function vs. nodes (b) fold-out basis functions and ε = Figure: Bilinear equation: Sparse grid (2D).
35 Higher Order Methods we aimed to investigate (adaptive) time stepping in particular in combination with adaptive sparse grids but got side-tracked into investigating higher order time stepping which resulted into also looking at higher order (B-Splines) sparse grids we can give so far just preliminary results
36 Higher Order Runge-Kutta Methods look at our control formulation from beginning T min J(u) = l(y(t), u(t)) dt, s.t. u U ad 0 ẏ(t) = f (y(t), u(t)), t > 0, y(0) = y 0 we need time discretisation for y(t) and quadrature rule to evaluate min u Uad J(u) with higher order RK-scheme this could look like (misusing notation) ( ) v k (x) = min t c i l(x τ i, u i ) + v k+1 (ŷ x ( t)) u 1,u 2,u 3,... i where ŷ x takes into account actions u i and RK-scheme with intermediate steps τ i therefore RK4 would have O(dc 4 ) complexity if complexity for computing one control is d c see also Falcone, Ferretti (1994).
37 Simplified Semi-Discrete Wave - Using Heun v v L l = 4 and 2 control points l = 6 and 2 control points l = 8 and 2 control points l = 4 and 1 control point l = 6 and 1 control point l = 8 and 1 control point v v L l = 4 and 2 control points l = 6 and 2 control points l = 8 and 2 control points l = 4 and 1 control point l = 6 and 1 control point l = 8 and 1 control point t Function evaluations (a) error in value function vs. time steps (b) error in value function vs. evaluations Figure: Error to the FD reference solution for Heun time integrator and different control strategies. Evaluation of control using compare approach with 40 points.
38 Structure Preserving Runge-Kutta Methods observe that for explicit or diagonally implicit RK schemes, the last element u n s of discrete control vector affects only the last RK-stage k s in fact, condition on RK-coefficients can be formulated, so that control optimization can be applied separately to each stage class of RK methods, which fulfils this are diagonally implicit symplectic Runge-Kutta (DISRK) schemes re-use implicit collocation RK-scheme for quadrature originally developed for long time integration of Hamiltonian systems are equivalent to composition of implicit midpoint schemes Ψ h = Φ bsh Φ b2 h Φ b1 h we can construct a SL scheme, which has O(d c s) complexity for the minimization problem problem for s > 1 DISRK goes backwards in time for some steps!?
39 Simplified Semi-Discrete Wave - Revisited v v L SG level 2 SG level 4 SG level 6 SG level 8 SG level 10 SG level 12 v v L SG level 2 SG level 4 SG level 6 SG level 8 SG level 10 SG level t t (a) Heun, d 2 c cost (b) implicit midpoint, 2 d c cost Figure: Error against FD reference, T = 1.0
40 Simplified Semi-Discrete Wave - Revisited l = 6 l = 8 l = 10 l = 12 l = l = 10 eval. on [ 1.0, 1.0] 2 l = 10 eval. on [ 0.5, 0.5] 2 l = 12 eval. on [ 1.0, 1.0] 2 l = 12 eval. on [ 0.5, 0.5] 2 v v L v v L t t (a) implicit midpoint / DISRK1 (b) implicit midpoint / DISRK1 Error against Riccati-Solution, T = 0.1 measured on full domain [ 1, 1] 2 and smaller [ 0.5, 0.5] 2 similarly we observed for DISRK3 an order 3 with error to 10 6
41 Simplified Semi-Discrete Wave Higher Order Discretization v v L l = 10 modified linear l = 4 mod. BSpline deg. 3 l = 5 mod. BSpline deg. 3 l = 6 mod. BSpline deg. 3 l = 7 mod. BSpline deg t (c) linear SG vs. B-Splines -SG results using Sparse Grids with B-Splines using DISRK1, measure error on smaller domain
42 Conclusion optimal feedback control PDE problems lead to HJB equations use model reduction to reduce complexity SL-scheme on sparse grids for HJB equations value function in coefficients of spectral basis can be smooth higher order time stepping schemes and higher order discretisation can be effective for smooth problems sparse grids with B-splines converge nicer, but have higher computing costs due to need to solve a linear equation system are there are other composite methods, which are better suitable?
43 Key References O. Bokanowski, J. G., M. Griebel, and I. Klompmaker An adaptive sparse grid semi-lagrangian scheme for first order Hamilton-Jacobi Bellman equations, J. of Sci. Comp., (2013) M. Falcone, R. Ferretti, Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations, Numerische Mathematik. (1994). J. G., A. Kröner, Suboptimal feedback control of PDEs by solving HJB equations on adaptive sparse grids, J. of Sci. Comp., (2016). A. Kröner, K. Kunisch, H. Zidani, Optimal feedback control for the undamped wave equation equation, ESAIM: Control Optim. Calc. Var. (2015).
Suboptimal feedback control of PDEs by solving HJB equations on adaptive sparse grids
Wegelerstraße 6 53115 Bonn Germany phone +49 228 73-3427 fax +49 228 73-7527 www.ins.uni-bonn.de Jochen Garcke and Axel Kröner Suboptimal feedback control of PDEs by solving HJB equations on adaptive sparse
More informationSuboptimal feedback control of PDEs by solving HJB equations on adaptive sparse grids
Suboptimal feedback control of PDEs by solving HJB equations on adaptive sparse grids Jochen Garcke, Axel Kröner To cite this version: Jochen Garcke, Axel Kröner. Suboptimal feedback control of PDEs by
More informationThe HJB-POD approach for infinite dimensional control problems
The HJB-POD approach for infinite dimensional control problems M. Falcone works in collaboration with A. Alla, D. Kalise and S. Volkwein Università di Roma La Sapienza OCERTO Workshop Cortona, June 22,
More informationNumerical approximation for optimal control problems via MPC and HJB. Giulia Fabrini
Numerical approximation for optimal control problems via MPC and HJB Giulia Fabrini Konstanz Women In Mathematics 15 May, 2018 G. Fabrini (University of Konstanz) Numerical approximation for OCP 1 / 33
More informationAn adaptive RBF-based Semi-Lagrangian scheme for HJB equations
An adaptive RBF-based Semi-Lagrangian scheme for HJB equations Roberto Ferretti Università degli Studi Roma Tre Dipartimento di Matematica e Fisica Numerical methods for Hamilton Jacobi equations in optimal
More informationFiltered scheme and error estimate for first order Hamilton-Jacobi equations
and error estimate for first order Hamilton-Jacobi equations Olivier Bokanowski 1 Maurizio Falcone 2 2 1 Laboratoire Jacques-Louis Lions, Université Paris-Diderot (Paris 7) 2 SAPIENZA - Università di Roma
More informationThe Direct Transcription Method For Optimal Control. Part 2: Optimal Control
The Direct Transcription Method For Optimal Control Part 2: Optimal Control John T Betts Partner, Applied Mathematical Analysis, LLC 1 Fundamental Principle of Transcription Methods Transcription Method
More informationNumerical Methods for Optimal Control Problems. Part I: Hamilton-Jacobi-Bellman Equations and Pontryagin Minimum Principle
Numerical Methods for Optimal Control Problems. Part I: Hamilton-Jacobi-Bellman Equations and Pontryagin Minimum Principle Ph.D. course in OPTIMAL CONTROL Emiliano Cristiani (IAC CNR) e.cristiani@iac.cnr.it
More informationpacman A classical arcade videogame powered by Hamilton-Jacobi equations Simone Cacace Universita degli Studi Roma Tre
HJ pacman A classical arcade videogame powered by Hamilton-Jacobi equations Simone Cacace Universita degli Studi Roma Tre Control of State Constrained Dynamical Systems 25-29 September 2017, Padova Main
More informationAlgorithms for Scientific Computing
Algorithms for Scientific Computing Hierarchical Methods and Sparse Grids d-dimensional Hierarchical Basis Michael Bader Technical University of Munich Summer 208 Intermezzo/ Big Picture : Archimedes Quadrature
More informationImplicit-explicit exponential integrators
Implicit-explicit exponential integrators Bawfeh Kingsley Kometa joint work with Elena Celledoni MaGIC 2011 Finse, March 1-4 1 Introduction Motivation 2 semi-lagrangian Runge-Kutta exponential integrators
More informationOrdinary Differential Equations II
Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 29 Almost Done! No
More informationModule 4: Numerical Methods for ODE. Michael Bader. Winter 2007/2008
Outlines Module 4: for ODE Part I: Basic Part II: Advanced Lehrstuhl Informatik V Winter 2007/2008 Part I: Basic 1 Direction Fields 2 Euler s Method Outlines Part I: Basic Part II: Advanced 3 Discretized
More informationNumerical schemes of resolution of stochastic optimal control HJB equation
Numerical schemes of resolution of stochastic optimal control HJB equation Elisabeth Ottenwaelter Journée des doctorants 7 mars 2007 Équipe COMMANDS Elisabeth Ottenwaelter (INRIA et CMAP) Numerical schemes
More informationOrdinary Differential Equations II
Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 33 Almost Done! Last
More informationHamilton-Jacobi-Bellman Equation Feb 25, 2008
Hamilton-Jacobi-Bellman Equation Feb 25, 2008 What is it? The Hamilton-Jacobi-Bellman (HJB) equation is the continuous-time analog to the discrete deterministic dynamic programming algorithm Discrete VS
More informationOPTIMAL CONTROL. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 28
OPTIMAL CONTROL Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 28 (Example from Optimal Control Theory, Kirk) Objective: To get from
More informationMultilevel stochastic collocations with dimensionality reduction
Multilevel stochastic collocations with dimensionality reduction Ionut Farcas TUM, Chair of Scientific Computing in Computer Science (I5) 27.01.2017 Outline 1 Motivation 2 Theoretical background Uncertainty
More informationDeterministic Dynamic Programming
Deterministic Dynamic Programming 1 Value Function Consider the following optimal control problem in Mayer s form: V (t 0, x 0 ) = inf u U J(t 1, x(t 1 )) (1) subject to ẋ(t) = f(t, x(t), u(t)), x(t 0
More informationControlled Diffusions and Hamilton-Jacobi Bellman Equations
Controlled Diffusions and Hamilton-Jacobi Bellman Equations Emo Todorov Applied Mathematics and Computer Science & Engineering University of Washington Winter 2014 Emo Todorov (UW) AMATH/CSE 579, Winter
More informationNumerical Optimal Control Overview. Moritz Diehl
Numerical Optimal Control Overview Moritz Diehl Simplified Optimal Control Problem in ODE path constraints h(x, u) 0 initial value x0 states x(t) terminal constraint r(x(t )) 0 controls u(t) 0 t T minimize
More informationInterpolation and function representation with grids in stochastic control
with in classical Sparse for Sparse for with in FIME EDF R&D & FiME, Laboratoire de Finance des Marchés de l Energie (www.fime-lab.org) September 2015 Schedule with in classical Sparse for Sparse for 1
More informationENO and WENO schemes. Further topics and time Integration
ENO and WENO schemes. Further topics and time Integration Tefa Kaisara CASA Seminar 29 November, 2006 Outline 1 Short review ENO/WENO 2 Further topics Subcell resolution Other building blocks 3 Time Integration
More informationChapter Two: Numerical Methods for Elliptic PDEs. 1 Finite Difference Methods for Elliptic PDEs
Chapter Two: Numerical Methods for Elliptic PDEs Finite Difference Methods for Elliptic PDEs.. Finite difference scheme. We consider a simple example u := subject to Dirichlet boundary conditions ( ) u
More informationPart IB Numerical Analysis
Part IB Numerical Analysis Definitions Based on lectures by G. Moore Notes taken by Dexter Chua Lent 206 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
More informationLecture 3: Hamilton-Jacobi-Bellman Equations. Distributional Macroeconomics. Benjamin Moll. Part II of ECON Harvard University, Spring
Lecture 3: Hamilton-Jacobi-Bellman Equations Distributional Macroeconomics Part II of ECON 2149 Benjamin Moll Harvard University, Spring 2018 1 Outline 1. Hamilton-Jacobi-Bellman equations in deterministic
More informationThe Newton-ADI Method for Large-Scale Algebraic Riccati Equations. Peter Benner.
The Newton-ADI Method for Large-Scale Algebraic Riccati Equations Mathematik in Industrie und Technik Fakultät für Mathematik Peter Benner benner@mathematik.tu-chemnitz.de Sonderforschungsbereich 393 S
More informationScientific Computing I
Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Neckel Winter 2013/2014 Module 8: An Introduction to Finite Element Methods, Winter 2013/2014 1 Part I: Introduction to
More informationTaylor expansions for the HJB equation associated with a bilinear control problem
Taylor expansions for the HJB equation associated with a bilinear control problem Tobias Breiten, Karl Kunisch and Laurent Pfeiffer University of Graz, Austria Rome, June 217 Motivation dragged Brownian
More informationAlgorithms for Scientific Computing
Algorithms for Scientific Computing Finite Element Methods Michael Bader Technical University of Munich Summer 2016 Part I Looking Back: Discrete Models for Heat Transfer and the Poisson Equation Modelling
More informationSemi-Lagrangian Formulations for Linear Advection Equations and Applications to Kinetic Equations
Semi-Lagrangian Formulations for Linear Advection and Applications to Kinetic Department of Mathematical and Computer Science Colorado School of Mines joint work w/ Chi-Wang Shu Supported by NSF and AFOSR.
More informationFitting multidimensional data using gradient penalties and combination techniques
1 Fitting multidimensional data using gradient penalties and combination techniques Jochen Garcke and Markus Hegland Mathematical Sciences Institute Australian National University Canberra ACT 0200 jochen.garcke,markus.hegland@anu.edu.au
More informationA High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations
Motivation Numerical methods Numerical tests Conclusions A High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations Xiaofeng Cai Department of Mathematics
More informationAdaptive spline interpolation for Hamilton Jacobi Bellman equations
Adaptive spline interpolation for Hamilton Jacobi Bellman equations Florian Bauer Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany Lars Grüne Mathematisches Institut, Universität
More informationLagrangian Analysis of 2D and 3D Ocean Flows from Eulerian Velocity Data
Flows from Second-year Ph.D. student, Applied Math and Scientific Computing Project Advisor: Kayo Ide Department of Atmospheric and Oceanic Science Center for Scientific Computation and Mathematical Modeling
More informationOrdinary Differential Equations
Chapter 13 Ordinary Differential Equations We motivated the problem of interpolation in Chapter 11 by transitioning from analzying to finding functions. That is, in problems like interpolation and regression,
More informationOptimal Control Theory
Optimal Control Theory The theory Optimal control theory is a mature mathematical discipline which provides algorithms to solve various control problems The elaborate mathematical machinery behind optimal
More informationIdentification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016
Prof. Dr. Eleni Chatzi Lecture 4-09. March, 2016 Fundamentals Overview Multiple DOF Systems State-space Formulation Eigenvalue Analysis The Mode Superposition Method The effect of Damping on Structural
More informationA Monotone Scheme for Hamilton-Jacobi Equations via the Nonstandard Finite Difference Method
A Monotone Scheme for Hamilton-Jacobi Equations via the Nonstandard Finite Difference Method Roumen Anguelov, Jean M-S Lubuma and Froduald Minani Department of Mathematics and Applied Mathematics University
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
More informationNumerical Methods - Initial Value Problems for ODEs
Numerical Methods - Initial Value Problems for ODEs Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Initial Value Problems for ODEs 2013 1 / 43 Outline 1 Initial Value
More informationBlock-Structured Adaptive Mesh Refinement
Block-Structured Adaptive Mesh Refinement Lecture 2 Incompressible Navier-Stokes Equations Fractional Step Scheme 1-D AMR for classical PDE s hyperbolic elliptic parabolic Accuracy considerations Bell
More informationValue Function and Optimal Trajectories for some State Constrained Control Problems
Value Function and Optimal Trajectories for some State Constrained Control Problems Hasnaa Zidani ENSTA ParisTech, Univ. of Paris-saclay "Control of State Constrained Dynamical Systems" Università di Padova,
More informationFast Marching Methods for Front Propagation Lecture 1/3
Outline for Front Propagation Lecture 1/3 M. Falcone Dipartimento di Matematica SAPIENZA, Università di Roma Autumn School Introduction to Numerical Methods for Moving Boundaries ENSTA, Paris, November,
More informationA note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations
A note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations S. Hussain, F. Schieweck, S. Turek Abstract In this note, we extend our recent work for
More informationRunge-Kutta and Collocation Methods Florian Landis
Runge-Kutta and Collocation Methods Florian Landis Geometrical Numetric Integration p.1 Overview Define Runge-Kutta methods. Introduce collocation methods. Identify collocation methods as Runge-Kutta methods.
More informationProper Orthogonal Decomposition in PDE-Constrained Optimization
Proper Orthogonal Decomposition in PDE-Constrained Optimization K. Kunisch Department of Mathematics and Computational Science University of Graz, Austria jointly with S. Volkwein Dynamic Programming Principle
More informationNumerical Mathematics
Alfio Quarteroni Riccardo Sacco Fausto Saleri Numerical Mathematics Second Edition With 135 Figures and 45 Tables 421 Springer Contents Part I Getting Started 1 Foundations of Matrix Analysis 3 1.1 Vector
More informationGreedy control. Martin Lazar University of Dubrovnik. Opatija, th Najman Conference. Joint work with E: Zuazua, UAM, Madrid
Greedy control Martin Lazar University of Dubrovnik Opatija, 2015 4th Najman Conference Joint work with E: Zuazua, UAM, Madrid Outline Parametric dependent systems Reduced basis methods Greedy control
More informationLecture V: The game-engine loop & Time Integration
Lecture V: The game-engine loop & Time Integration The Basic Game-Engine Loop Previous state: " #, %(#) ( #, )(#) Forces -(#) Integrate velocities and positions Resolve Interpenetrations Per-body change
More informationExponential integrators for semilinear parabolic problems
Exponential integrators for semilinear parabolic problems Marlis Hochbruck Heinrich-Heine University Düsseldorf Germany Innsbruck, October 2004 p. Outline Exponential integrators general class of methods
More informationLecture IV: Time Discretization
Lecture IV: Time Discretization Motivation Kinematics: continuous motion in continuous time Computer simulation: Discrete time steps t Discrete Space (mesh particles) Updating Position Force induces acceleration.
More informationNUMERICAL ANALYSIS 2 - FINAL EXAM Summer Term 2006 Matrikelnummer:
Prof. Dr. O. Junge, T. März Scientific Computing - M3 Center for Mathematical Sciences Munich University of Technology Name: NUMERICAL ANALYSIS 2 - FINAL EXAM Summer Term 2006 Matrikelnummer: I agree to
More informationEN Applied Optimal Control Lecture 8: Dynamic Programming October 10, 2018
EN530.603 Applied Optimal Control Lecture 8: Dynamic Programming October 0, 08 Lecturer: Marin Kobilarov Dynamic Programming (DP) is conerned with the computation of an optimal policy, i.e. an optimal
More informationFast Structured Spectral Methods
Spectral methods HSS structures Fast algorithms Conclusion Fast Structured Spectral Methods Yingwei Wang Department of Mathematics, Purdue University Joint work with Prof Jie Shen and Prof Jianlin Xia
More informationStrong Stability-Preserving (SSP) High-Order Time Discretization Methods
Strong Stability-Preserving (SSP) High-Order Time Discretization Methods Xinghui Zhong 12/09/ 2009 Outline 1 Introduction Why SSP methods Idea History/main reference 2 Explicit SSP Runge-Kutta Methods
More informationLehrstuhl Informatik V. Lehrstuhl Informatik V. 1. solve weak form of PDE to reduce regularity properties. Lehrstuhl Informatik V
Part I: Introduction to Finite Element Methods Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Necel Winter 4/5 The Model Problem FEM Main Ingredients Wea Forms and Wea
More informationThe Kuratowski Ryll-Nardzewski Theorem and semismooth Newton methods for Hamilton Jacobi Bellman equations
The Kuratowski Ryll-Nardzewski Theorem and semismooth Newton methods for Hamilton Jacobi Bellman equations Iain Smears INRIA Paris Linz, November 2016 joint work with Endre Süli, University of Oxford Overview
More informationNumerical Solution I
Numerical Solution I Stationary Flow R. Kornhuber (FU Berlin) Summerschool Modelling of mass and energy transport in porous media with practical applications October 8-12, 2018 Schedule Classical Solutions
More informationOptimal Control. Quadratic Functions. Single variable quadratic function: Multi-variable quadratic function:
Optimal Control Control design based on pole-placement has non unique solutions Best locations for eigenvalues are sometimes difficult to determine Linear Quadratic LQ) Optimal control minimizes a quadratic
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Linear Multistep methods Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the
More informationDefect-based a-posteriori error estimation for implicit ODEs and DAEs
1 / 24 Defect-based a-posteriori error estimation for implicit ODEs and DAEs W. Auzinger Institute for Analysis and Scientific Computing Vienna University of Technology Workshop on Innovative Integrators
More informationSemi-implicit Krylov Deferred Correction Methods for Ordinary Differential Equations
Semi-implicit Krylov Deferred Correction Methods for Ordinary Differential Equations Sunyoung Bu University of North Carolina Department of Mathematics CB # 325, Chapel Hill USA agatha@email.unc.edu Jingfang
More informationOptimal Control. Lecture 18. Hamilton-Jacobi-Bellman Equation, Cont. John T. Wen. March 29, Ref: Bryson & Ho Chapter 4.
Optimal Control Lecture 18 Hamilton-Jacobi-Bellman Equation, Cont. John T. Wen Ref: Bryson & Ho Chapter 4. March 29, 2004 Outline Hamilton-Jacobi-Bellman (HJB) Equation Iterative solution of HJB Equation
More informationMath 7824 Spring 2010 Numerical solution of partial differential equations Classroom notes and homework
Math 7824 Spring 2010 Numerical solution of partial differential equations Classroom notes and homework Jan Mandel University of Colorado Denver May 12, 2010 1/20/09: Sec. 1.1, 1.2. Hw 1 due 1/27: problems
More informationConstrained Minimization and Multigrid
Constrained Minimization and Multigrid C. Gräser (FU Berlin), R. Kornhuber (FU Berlin), and O. Sander (FU Berlin) Workshop on PDE Constrained Optimization Hamburg, March 27-29, 2008 Matheon Outline Successive
More informationHigh-order filtered schemes for time-dependent second order HJB equations
High-order filtered schemes for time-dependent second order HJB equations Olivier Bokanowski, Athena Picarelli, and Christoph Reisinger November 15, 2016 Abstract In this paper, we present and analyse
More informationCOSC 3361 Numerical Analysis I Ordinary Differential Equations (II) - Multistep methods
COSC 336 Numerical Analysis I Ordinary Differential Equations (II) - Multistep methods Fall 2005 Repetition from the last lecture (I) Initial value problems: dy = f ( t, y) dt y ( a) = y 0 a t b Goal:
More informationOrdinary Differential Equations
Ordinary Differential Equations We call Ordinary Differential Equation (ODE) of nth order in the variable x, a relation of the kind: where L is an operator. If it is a linear operator, we call the equation
More informationn 1 f n 1 c 1 n+1 = c 1 n $ c 1 n 1. After taking logs, this becomes
Root finding: 1 a The points {x n+1, }, {x n, f n }, {x n 1, f n 1 } should be co-linear Say they lie on the line x + y = This gives the relations x n+1 + = x n +f n = x n 1 +f n 1 = Eliminating α and
More informationCS 257: Numerical Methods
CS 57: Numerical Methods Final Exam Study Guide Version 1.00 Created by Charles Feng http://www.fenguin.net CS 57: Numerical Methods Final Exam Study Guide 1 Contents 1 Introductory Matter 3 1.1 Calculus
More informationFast Numerical Methods for Stochastic Computations
Fast AreviewbyDongbinXiu May 16 th,2013 Outline Motivation 1 Motivation 2 3 4 5 Example: Burgers Equation Let us consider the Burger s equation: u t + uu x = νu xx, x [ 1, 1] u( 1) =1 u(1) = 1 Example:
More informationReview Higher Order methods Multistep methods Summary HIGHER ORDER METHODS. P.V. Johnson. School of Mathematics. Semester
HIGHER ORDER METHODS School of Mathematics Semester 1 2008 OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE
More informationThe family of Runge Kutta methods with two intermediate evaluations is defined by
AM 205: lecture 13 Last time: Numerical solution of ordinary differential equations Today: Additional ODE methods, boundary value problems Thursday s lecture will be given by Thomas Fai Assignment 3 will
More informationEnergy-Preserving Runge-Kutta methods
Energy-Preserving Runge-Kutta methods Fasma Diele, Brigida Pace Istituto per le Applicazioni del Calcolo M. Picone, CNR, Via Amendola 122, 70126 Bari, Italy f.diele@ba.iac.cnr.it b.pace@ba.iac.cnr.it SDS2010,
More informationFinite Differences for Differential Equations 28 PART II. Finite Difference Methods for Differential Equations
Finite Differences for Differential Equations 28 PART II Finite Difference Methods for Differential Equations Finite Differences for Differential Equations 29 BOUNDARY VALUE PROBLEMS (I) Solving a TWO
More information5. Ordinary Differential Equations. Indispensable for many technical applications!
Indispensable for many technical applications! Numerisches Programmieren, Hans-Joachim Bungartz page 1 of 30 5.1. Introduction Differential Equations One of the most important fields of application of
More informationNumerical Analysis Preliminary Exam 10 am to 1 pm, August 20, 2018
Numerical Analysis Preliminary Exam 1 am to 1 pm, August 2, 218 Instructions. You have three hours to complete this exam. Submit solutions to four (and no more) of the following six problems. Please start
More information2.29 Numerical Fluid Mechanics Fall 2011 Lecture 20
2.29 Numerical Fluid Mechanics Fall 2011 Lecture 20 REVIEW Lecture 19: Finite Volume Methods Review: Basic elements of a FV scheme and steps to step-up a FV scheme One Dimensional examples d x j x j 1/2
More informationIndex. C 2 ( ), 447 C k [a,b], 37 C0 ( ), 618 ( ), 447 CD 2 CN 2
Index advection equation, 29 in three dimensions, 446 advection-diffusion equation, 31 aluminum, 200 angle between two vectors, 58 area integral, 439 automatic step control, 119 back substitution, 604
More informationAn Introduction to Algebraic Multigrid (AMG) Algorithms Derrick Cerwinsky and Craig C. Douglas 1/84
An Introduction to Algebraic Multigrid (AMG) Algorithms Derrick Cerwinsky and Craig C. Douglas 1/84 Introduction Almost all numerical methods for solving PDEs will at some point be reduced to solving A
More informationA Survey of Computational High Frequency Wave Propagation II. Olof Runborg NADA, KTH
A Survey of Computational High Frequency Wave Propagation II Olof Runborg NADA, KTH High Frequency Wave Propagation CSCAMM, September 19-22, 2005 Numerical methods Direct methods Wave equation (time domain)
More informationRegularity and approximations of generalized equations; applications in optimal control
SWM ORCOS Operations Research and Control Systems Regularity and approximations of generalized equations; applications in optimal control Vladimir M. Veliov (Based on joint works with A. Dontchev, M. Krastanov,
More informationModified Equations for Variational Integrators
Modified Equations for Variational Integrators Mats Vermeeren Technische Universität Berlin Groningen December 18, 2018 Mats Vermeeren (TU Berlin) Modified equations for variational integrators December
More informationThe Finite Element Method for the Analysis of Non-Linear and Dynamic Systems: Non-Linear Dynamics Part I
The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems: Non-Linear Dynamics Part I Prof. Dr. Eleni Chatzi Dr. Giuseppe Abbiati, Dr. Konstantinos Agathos Lecture 5/Part A - 23 November,
More informationOverview of the Numerics of the ECMWF. Atmospheric Forecast Model
Overview of the Numerics of the Atmospheric Forecast Model M. Hortal Seminar 6 Sept 2004 Slide 1 Characteristics of the model Hydrostatic shallow-atmosphere approimation Pressure-based hybrid vertical
More informationLena Leitenmaier & Parikshit Upadhyaya KTH - Royal Institute of Technology
Data mining with sparse grids Lena Leitenmaier & Parikshit Upadhyaya KTH - Royal Institute of Technology Classification problem Given: training data d-dimensional data set (feature space); typically relatively
More informationIndex. higher order methods, 52 nonlinear, 36 with variable coefficients, 34 Burgers equation, 234 BVP, see boundary value problems
Index A-conjugate directions, 83 A-stability, 171 A( )-stability, 171 absolute error, 243 absolute stability, 149 for systems of equations, 154 absorbing boundary conditions, 228 Adams Bashforth methods,
More informationPartitioned Methods for Multifield Problems
C Partitioned Methods for Multifield Problems Joachim Rang, 6.7.2016 6.7.2016 Joachim Rang Partitioned Methods for Multifield Problems Seite 1 C One-dimensional piston problem fixed wall Fluid flexible
More informationA Gauss Lobatto quadrature method for solving optimal control problems
ANZIAM J. 47 (EMAC2005) pp.c101 C115, 2006 C101 A Gauss Lobatto quadrature method for solving optimal control problems P. Williams (Received 29 August 2005; revised 13 July 2006) Abstract This paper proposes
More informationSuboptimal Open-loop Control Using POD. Stefan Volkwein
Institute for Mathematics and Scientific Computing University of Graz, Austria PhD program in Mathematics for Technology Catania, May 22, 2007 Motivation Optimal control of evolution problems: min J(y,
More informationarxiv: v1 [math.na] 29 Jul 2018
AN EFFICIENT DP ALGORITHM ON A TREE-STRUCTURE FOR FINITE HORIZON OPTIMAL CONTROL PROBLEMS ALESSANDRO ALLA, MAURIZIO FALCONE, LUCA SALUZZI arxiv:87.8v [math.na] 29 Jul 28 Abstract. The classical Dynamic
More informationMultistage Methods I: Runge-Kutta Methods
Multistage Methods I: Runge-Kutta Methods Varun Shankar January, 0 Introduction Previously, we saw that explicit multistep methods (AB methods) have shrinking stability regions as their orders are increased.
More informationECE257 Numerical Methods and Scientific Computing. Ordinary Differential Equations
ECE257 Numerical Methods and Scientific Computing Ordinary Differential Equations Today s s class: Stiffness Multistep Methods Stiff Equations Stiffness occurs in a problem where two or more independent
More informationPOD for Parametric PDEs and for Optimality Systems
POD for Parametric PDEs and for Optimality Systems M. Kahlbacher, K. Kunisch, H. Müller and S. Volkwein Institute for Mathematics and Scientific Computing University of Graz, Austria DMV-Jahrestagung 26,
More informationA Semi-Lagrangian Scheme for the Open Table Problem in Granular Matter Theory
A Semi-Lagrangian Scheme for the Open Table Problem in Granular Matter Theory M. Falcone and S. Finzi Vita Abstract We introduce and analyze a new scheme for the approximation of the two-layer model proposed
More informationConsistency and Convergence
Jim Lambers MAT 77 Fall Semester 010-11 Lecture 0 Notes These notes correspond to Sections 1.3, 1.4 and 1.5 in the text. Consistency and Convergence We have learned that the numerical solution obtained
More informationContinuous adjoint based error estimation and r-refinement for the active-flux method
Continuous adjoint based error estimation and r-refinement for the active-flux method Kaihua Ding, Krzysztof J. Fidkowski and Philip L. Roe Department of Aerospace Engineering, University of Michigan,
More informationImplicitly Defined High-Order Operator Splittings for Parabolic and Hyperbolic Variable-Coefficient PDE Using Modified Moments
Implicitly Defined High-Order Operator Splittings for Parabolic and Hyperbolic Variable-Coefficient PDE Using Modified Moments James V. Lambers September 24, 2008 Abstract This paper presents a reformulation
More informationSuppose that we have a specific single stage dynamic system governed by the following equation:
Dynamic Optimisation Discrete Dynamic Systems A single stage example Suppose that we have a specific single stage dynamic system governed by the following equation: x 1 = ax 0 + bu 0, x 0 = x i (1) where
More information