Dynamic Optimization in Air Quality Modeling

Transcription

1 Dynamic Optimization in Air Quality Modeling A. Caboussat Department of Mathematics, University of Houston Houston, Texas Project supported by the U. S. Environmental Protection Agency Grant X Rice-UH Optimization Seminar, Rice University, March 30, 200 Dynamic Optimization in Air Quality Modeling p.1/42

2 Acknowledgments N. R. Amundson, J. W. He, A. V. Martynenko, University of Houston, Department of Mathematics, Houston, Texas. J. H. Seinfeld, Caltech, Department of Chemical Engineering, Pasadena, California. S. Clegg, University of East Anglia, UK. J. Rappaz, C. Landry, Ecole Polytechnique Fédérale de Lausanne. Rice-UH Optimization Seminar, Rice University, March 30, 200 Acknowledgments p.2/42

3 Aerosols Life Cycle Aerosols particles have effects on human health, visibility reduction in urban and regional areas, acid rain, alteration of the earth s radiation balance, oxidation due to aqueous droplets, cloud and ozone formation, etc. Rice-UH Optimization Seminar, Rice University, March 30, 200 Aerosol Life Cycle p.3/42

4 Motivations Modeling and computation of the physical state and chemical composition of atmospheric aerosol particles. "The chemical and physical properties of aerosols are needed to estimate and predict direct and indirect climate forcing", (IPCC, Third Assessment Report, 2001). At present the knowledge of aerosol composition and transformation is limited and high uncertainty remains on their environmental effects. Current aerosol models do not always predict accurately the phase state and the growth phenomena of atmospheric aerosols, since they rely on many a priori assumptions. Rice-UH Optimization Seminar, Rice University, March 30, 200 Motivations p.4/42

5 Aerosol Particles Single aerosol particle: c(t) aerosol R(t) b(t) p equil gas Thermodynamics : Find the equilibrium state of the particle and the repartition between solid, liquid and gas phases. Dynamics : Find the gas-particle partitioning of chemical species. Rice-UH Optimization Seminar, Rice University, March 30, 200 Aerosol Particles p.5/42

6 Outline Modeling of the thermodynamic equilibrium and mass transfer for organic aerosol particles. Static Optimization. Determination of the convex hull of the energy function. The notion of phase simplex. A primal-dual interior-point method for the computation of the minimum of energy. Dynamic Optimization. Differential equations and algebraic constraints. An extended optimization problem and sequential quadratic programming techniques. Rice-UH Optimization Seminar, Rice University, March 30, 200 Outline p./42

7 Global Air Quality Problem System of coupled PDEs for the concentrations b = ( b i ) of chemical components in the aerosol and c = ( c i ) in the bulk gas: (1) (2) i (r,x, t) + (u(x, t)) x bi (r,x, t) t b I i(r,x, t) bs (r,x, t) r x (K(x, t) x bi (r,x, t) ) + ( Is (r,x, t) b i (r,x, t) ) r = r 0 β(r r, r s (r r ) b,x, t) r r bi (r,x, t)dr b i (r,x, t) β(r, r s (r ) b,x, t) 0 r dr + S i (r,x, t), t c i(x, t) + (u(x, t)) x c i (x, t) x (K(x, t) x c i (x, t)) b s (r,x, t) + I i (r,x, t)dr = f i ( c(x, t)) + E i (x, t). r 0 Rice-UH Optimization Seminar, Rice University, March 30, 200 Global Air Quality Problem p./42

8 Global Air Quality Problem System of coupled PDEs for the concentrations b = ( b i ) of chemical components in the aerosol and c = ( c i ) in the bulk gas: (1) (2) i (r,x, t) + (u(x, t)) x bi (r,x, t) t b I i(r,x, t) bs (r,x, t) r x (K(x, t) x bi (r,x, t) ) + ( Is (r,x, t) b i (r,x, t) ) r = r 0 β(r r, r s (r r ) b,x, t) r r bi (r,x, t)dr b i (r,x, t) β(r, r s (r ) b,x, t) 0 r dr +S i (r,x, t), t c i(x, t) + (u(x, t)) x c i (x, t) x (K(x, t) x c i (x, t)) b s (r,x, t) + I i (r,x, t)dr = f i ( c(x, t)) + E i (x, t). r 0 Rice-UH Optimization Seminar, Rice University, March 30, 200 Global Air Quality Problem p./42

9 Global Air Quality Problem System of coupled PDEs for the concentrations b = ( b i ) of chemical components in the aerosol and c = ( c i ) in the bulk gas: (1) (2) i (r,x, t)+ (u(x, t)) x bi (r,x, t) t b I i(r,x, t) bs (r,x, t) r x (K(x, t) x bi (r,x, t) ) + ( Is (r,x, t) b i (r,x, t) ) r = r 0 β(r r, r s (r r ) b,x, t) r r bi (r,x, t)dr b i (r,x, t) β(r, r s (r ) b,x, t) 0 r dr + S i (r,x, t), t c i(x, t) + (u(x, t)) x c i (x, t) x (K(x, t) x c i (x, t)) b s (r,x, t) + I i (r,x, t)dr = f i ( c(x, t)) + E i (x, t). r 0 Rice-UH Optimization Seminar, Rice University, March 30, 200 Global Air Quality Problem p./42

10 Global Air Quality Problem System of coupled PDEs for the concentrations b = ( b i ) of chemical components in the aerosol and c = ( c i ) in the bulk gas: (1) (2) i (r,x, t) + (u(x, t)) x bi (r,x, t) t b I i(r,x, t) bs (r,x, t) r x (K(x, t) x bi (r,x, t) ) + ( Is (r,x, t) b i (r,x, t) ) r = r 0 β(r r, r s (r r ) b,x, t) r r bi (r,x, t)dr b i (r,x, t) β(r, r s (r ) b,x, t) 0 r dr + S i (r,x, t), t c i(x, t) + (u(x, t)) x c i (x, t) x (K(x, t) x c i (x, t)) b s (r,x, t) + I i (r,x, t)dr = f i ( c(x, t)) + E i (x, t). r 0 Rice-UH Optimization Seminar, Rice University, March 30, 200 Global Air Quality Problem p./42

11 Global Air Quality Problem System of coupled PDEs for the concentrations b = ( b i ) of chemical components in the aerosol and c = ( c i ) in the bulk gas: (1) (2) i (r,x, t) + (u(x, t)) x bi (r,x, t) t b I i(r,x, t) bs (r,x, t) r x (K(x, t) x bi (r,x, t) ) + ( Is (r,x, t) b i (r,x, t) ) r = r 0 β(r r, r s (r r ) b,x, t) r r bi (r,x, t)dr b i (r,x, t) β(r, r s (r ) b,x, t) 0 r dr + S i (r,x, t), t c i(x, t) + (u(x, t)) x c i (x, t) x (K(x, t) x c i (x, t)) b s (r,x, t) + I i (r,x, t)dr = f i ( c(x, t)) + E i (x, t). r 0 Splitting algorithm and discretization in time. Rice-UH Optimization Seminar, Rice University, March 30, 200 Global Air Quality Problem p./42

12 Evaporation Term The coupling is a result of condensation and evaporation processes: I i (r,x, t) = h i (r) ( c i (x, t) η(r) 1 RT(x, t) pequil i ) (b((r,x, t)), (η(r) Kelvin constant of particle mass r, h i (r) molecular transfer coefficient, T temperature, p equil i (r, x, t) fugacity of gas species i.) The physical phenomena in the system of partial differential equations are decoupled with a time splitting scheme and discretized in space. The resulting subsystem is governed by two main processes, namely the thermodynamic equilibrium inside the particle and the mass transfer between the bulk gas and the particle. Rice-UH Optimization Seminar, Rice University, March 30, 200 Evaporation Term p.8/42

13 Dynamics and Mass Transfer d dt c(t) d dt b(t) = h(r) = h(r) ( c(t) η(r) ( c(t) η(r) ) 1 RT(t) pequil (b(t)), c(0) = c 0 ) 1 RT(t) pequil (b(t)), b(0) = b 0 We need to solve a phase equilibrium problem for the internal variables (y α,x α ) for the calculation of p equil i (b) with min y α,x α s. t. p equil (b(t)) = p vapor exp( g(x α )), P y α g(x α ) P y α x α = b(t), y α 0, e T x α = 1, x α > 0, α = 1,...,P. Rice-UH Optimization Seminar, Rice University, March 30, 200 Dynamics and Mass Transfer p.9/42

14 Model Problem Since the system is conservative, c(t) + b(t) = b(0) + c(0) = b tot and one can eliminate b: d c(t) = h(r) dt P min y α,x α s. t. ( c(t) η(r) p ) vapor RT(t) e g(x α) y α g(x α ) P y α x α = b tot c(t), y α 0, e T x α = 1, x α > 0, α = 1,...,P. (1) Static Minimization - determination of the convex hull of the energy; (2) Dynamic Optimization - extended optimization problem. Rice-UH Optimization Seminar, Rice University, March 30, 200 Model Problem p.10/42

15 Static Optimization min y α,x α s. t. P y α g(x α ) P y α x α = b, y α 0, e T x α = 1, x α > 0, α = 1,...,P. x α virtual mole-fraction compositions (e T = (1, 1,...,1)). y α total amount in each phase and allows to track of the existence of the phase α. Although the number of phase classes is specified a priori, the number of phases existing at the equilibrium is not known a priori, but is a result of the equilibrium computation. Rice-UH Optimization Seminar, Rice University, March 30, 200 Static Optimization p.11/42

16 Reduced Problem min y α,z α s. t. P y α f(z α ) P y α z α = b, P y α = 1, y α 0, α = 1,...,P, z α int( n ), α = 1,...,P. z α = (x α ) N 1 R N, for all x α R N+1 such that e T x α = 1. N = conv{0,e 1,,e n } is a N-simplex in R N. Rice-UH Optimization Seminar, Rice University, March 30, 200 Reduced Problem p.12/42

17 Convex Hull Convex hull of a function f : R N R: convf is the greatest convex function majored by f. (convf)(b) = inf { P y α f(z α ) P y α z α = b, y α 0, } P y α = 1. Carathéodory s Theorem: For a set C in R N, every point of conv(c) belongs to some simplex with vertices in C and thus can be expressed as a convex combination of N + 1 points of C (not necessarily different). When C is connected, N points suffice. (convf)(b) = inf { N+1 y α f(z α ) N+1 y α z α = b, y α 0, N+1 y α = 1 }. Rice-UH Optimization Seminar, Rice University, March 30, 200 Reduced Problem p.13/42

18 Properties of the Energy Function Regularity of the energy function: f C (int N ) C 0 ( N ). x 0 N, w proper lim x x 0 f w =. Each vertex of the N-simplex N (pure components) belongs to a connected convex region of f. Determination of a tangent plane to the energy function. Rice-UH Optimization Seminar, Rice University, March 30, 200 Properties of the Energy Function p.14/42

19 Properties of the Energy Function No isolated connected convex regions (for global optimum). The function f is strictly convex in a neighborhood of a solution (for uniqueness). Rice-UH Optimization Seminar, Rice University, March 30, 200 Properties of the Energy Function p.14/42

20 Supporting Tangent Plane Gibbs Free Energy Supporting tangent plane If C is convex and z C, there exists a hyperplane Θ with z Θ and C is included in one of its closed half-spaces. Rice-UH Optimization Seminar, Rice University, March 30, 200 Supporting Tangent Plane p.15/42

21 Some Results on the Convex Hull The feed b can be decomposed into a stable phase splitting: b = P y α z α, z k int( N ), k = 1,...,P. The convex hull is continuously differentiable on int( N ). The following relations hold for all (k, m) = 1,...,P : f(z k ) = f(z m ) f(z k ) f(z k ) z k = f(z m ) f(z m ) z m Moreover the tangent plane Θ(z) = f(z 1 ) + f(z 1 ) (z z 1 ) is tangent to f at all active vertices and always below the graph of f. Existence and uniqueness [Rabier, Griewank (1992)]. Acknowledgments: A. Anantharanam, Ecole Polytechnique de Paris Rice-UH Optimization Seminar, Rice University, March 30, 200 Some Results on the Convex Hull p.1/42

22 Global vs. Local Minima We need a criterion to distinguish the local minima from the global minimum. Let (y 1,...,y N+1,z 1,...,z N+1 ) be a local minimum of the reduced optimization problem for b int( N ). Then if y k > 0, then z k int( N ). Consider Y = (y 1,...,y n+1,z 1,...,z n+1 ) a feasible point for the optimization problem with P = n + 1 fixed phases. Y is a global minimum if and only if Y is a local minimum and f(z) Θ(z) on n, where Θ is the hyperplane associated with Y. Rice-UH Optimization Seminar, Rice University, March 30, 200 Global vs. Local Minima p.1/42

23 Global vs. Local Methods Global Methods for Phase equilibrium calculations: No global optimization methods for the problem of convexification of the Gibbs energy; Complexity of global methods grows exponentially with the size of the problem; Local Methods: Local optimization methods can handle large-scale problems; They can miss the global solution; They require an good initial guess for the variables; They are sensitive to algorithm parameter values; Local optimization techniques, together with good initial guess Rice-UH Optimization Seminar, Rice University, March 30, 200 Global vs. Local Methods p.18/42

24 An Interior-Point Method Introduction of a log-barrier penalty function to ensure the non-negativeness of the total number of elements in each phase. min y α,x α s. t. N+1 N+1 y α g(x α ) y α x α = b, e T x α = 1, x α > 0, y α 0, α = 1,...,N + 1. ν is a penalty parameter, which tends to zero. The phase α disappears when y α 0. x α indicates only the location of a virtual phase. All phases exist at the beginning and then are selected by the algorithm. Rice-UH Optimization Seminar, Rice University, March 30, 200 An Interior-Point Method p.19/42

25 An Interior-Point Method Introduction of a log-barrier penalty function to ensure the non-negativeness of the total number of elements in each phase. min y α,x α s. t. N+1 N+1 y α g(x α ) ν y α x α = b, N+1 ln(y α ) e T x α = 1, x α > 0, α = 1,...,N + 1. ν is a penalty parameter, which tends to zero. The phase α disappears when y α 0. x α indicates only the location of a virtual phase. All phases exist at the beginning and then are selected by the algorithm. Rice-UH Optimization Seminar, Rice University, March 30, 200 An Interior-Point Method p.19/42

26 Karush-Kuhn-Tucker Conditions KKT conditions (stationary points of the Lagrangian): y α ( g(x α ) + λ) + ζ α e = 0, α = 1,...,N + 1, g(x α ) + λ T x α ν y α = 0, α = 1,...,N + 1, e T x α 1 = 0, α = 1,...,N + 1, P y α x α b = 0, KKT System (Newton method), (with 2 g(x α ) singular): y α 2 g(x α ) g(x α ) + λ y α e ( g(x α ) + λ) T ν (y α ) 2 (x α ) T 0 (y α ) T x α 0 0 e T p x p y p λ p ζα = b x b y b λ b ζα Rice-UH Optimization Seminar, Rice University, March 30, 200 Karush-Kuhn-Tucker Conditions p.20/42

27 Projected KKT System Projection on the constraints e T x α = 1: y α 2 f(z α ) f(z α ) + η y α 0 ( f(z α ) + η) T ν (y α ) 2 (z α ) T e (y α ) T z α 0 0 e T p z p y p η p θ = b z b y b η b θ KKT systems for chemical systems are usually ill-conditioned. Design of numerical linear algebra techniques. Direct decomposition techniques of the block-structured system (range-space + null-space). Control of the inertia of the matrices arising in the resolution. Rice-UH Optimization Seminar, Rice University, March 30, 200 Projected KKT System p.21/42

28 Projected KKT System Projection on the constraints e T x α = 1: y α 2 f(z α ) f(z α ) + η y α 0 ( f(z α ) + η) T ν (y α ) 2 (z α ) T e (y α ) T z α 0 0 e T p z p y p η p θ = b z b y b η b θ Assumptions: The Hessian 2 f(z α ) is positive definite (second order conditions) The iterates z 1,z 2,...,z N+1 are linearly independent (linear independent constraint qualification). Rice-UH Optimization Seminar, Rice University, March 30, 200 Projected KKT System p.21/42

29 Projected KKT System Projection on the constraints e T x α = 1: y α 2 f(z α ) f(z α ) + η y α 0 ( f(z α ) + η) T ν (y α ) 2 (z α ) T e (y α ) T z α 0 0 e T p z p y p η p θ = b z b y b η b θ Key Issue: Initialization of z α in order to guarantee the second order conditions during the whole algorithm. Rice-UH Optimization Seminar, Rice University, March 30, 200 Projected KKT System p.21/42

30 Numerical Results The existence of the phases depends on the feed vector b. z 1 z 2 Gibbs Free Energy b Tangent plane conv f(b) < f(b), conv f(z i ) = f(z i ), i = 1, 2 Rice-UH Optimization Seminar, Rice University, March 30, 200 Numerical Results p.22/42

31 Numerical Results The existence of the phases depends on the feed vector b. z 1 Gibbs Free Energy b Tangent plane conv f(b) = f(b) = f(z 1 ) = conv f(z 1 ). Rice-UH Optimization Seminar, Rice University, March 30, 200 Numerical Results p.22/42

32 Results in One Dimension Energy is defined on the (0, 1) segment. Both extremities of the segment are in a convex region by assumption. Convex Hull: normalized GFE water mole fraction 1 hexacosanol Rice-UH Optimization Seminar, Rice University, March 30, 200 Results in One Dimension p.23/42

33 Results in Two Dimensions Non-convex Energy Function: Determination of a tangent plane. Rice-UH Optimization Seminar, Rice University, March 30, 200 Results in Two Dimensions p.24/42

34 Convergence of Phase Simplexes Phase simplex for b fixed: Three vertices Two vertices Convergence of the phase simplexes in approximately 20 iterations (ν 0 = 10 3, tolerance = 10 ). Rice-UH Optimization Seminar, Rice University, March 30, 200 Convergence of Phase Simplexes p.25/42

35 Convergence of Phase Simplexes (2) Convergence towards three active vertices: 1 Vertices x α Barycentric Coordinates y α Particular choice of initial guess (near the vertices) and numerical parameters to avoid local minima. Rice-UH Optimization Seminar, Rice University, March 30, 200 Convergence of Phase Simplexes (2) p.2/42

36 Convex Hull C 2 H 54 O H2 O C 9 H 14 O 4 Computational cost: 20 s. for grid points, 25 iterations in average (tolerance = 10 ). Rice-UH Optimization Seminar, Rice University, March 30, 200 Convex Hull p.2/42

37 Results in Higher Dimensions Optimization in R 4 : For b = (0.002, 0.002, 0.002, 13.0). Solution with 2 active vertices (y 1 = y 3 = 0, y 2 = , y 4 = ). Convergence in 40 iterations (CPU time = s. / ν 0 = 10 3, tolerance = 10 ) Optimization in R 18 : For b int 18. Solution with 2 active vertices / 1 active constraints. Convergence in 41 iterations (CPU time = s. / ν 0 = 10 3, tolerance = 10 ) Rice-UH Optimization Seminar, Rice University, March 30, 200 Results in Higher Dimensions p.28/42

38 Dynamic Optimization Coupling of the evolution of the gas concentrations with the KKT conditions for the modeling of mass transfer. d c(t) = h(r) dt ( c(t) η(r) p ) vapor RT(t) exp( g(x α)). Under the optimum constraints: min y α,x α s. t. N+1 N+1 y α g(x α ) ν P ln(y α ) y α x α = b tot c, e T x α = 1, x α > 0, α = 1,...,N + 1. Rice-UH Optimization Seminar, Rice University, March 30, 200 Dynamic Optimization p.29/42

39 Dynamic Optimization Coupling of the evolution of the gas concentrations with the KKT conditions for the modeling of mass transfer. d dt c(t) = h(r) ( c(t) η(r) p ) vapor RT(t) exp( λ). Replaced by the KKT conditions: y α ( g(x α ) + λ) + ζ α e = 0, α = 1,...,N + 1, g(x α ) + λ T x α ν = 0, y α α = 1,...,N + 1, e T x α 1 = 0, α = 1,...,N + 1, N+1 y α x α + c(t) b tot = 0. Rice-UH Optimization Seminar, Rice University, March 30, 200 Dynamic Optimization p.29/42

40 DAE Discontinuities d dt c = f(t, c, x), 0 = g ν (t, c, x). Without inequalities (i.e. for given ν) : stiff system of DAE (difference of scales between concentrations and of speed between the reactions). Existence of a solution relies on the implicit function theorem. Convergence of implicit time-discretization schemes. With inequalities (i.e. for ν 0) : bifurcation problem. Discontinuities of the derivatives of the solution. Bifurcation between local and global optimum. Rice-UH Optimization Seminar, Rice University, March 30, 200 DAE Discontinuities p.30/42

41 An Implicit Discretization c n+1 c n ( = h(r n ) c n+1 η(r n ) p vapor τ RT n+1 exp( λ n+1)), yα n+1 ( g(x n+1 α ) + λ n+1 ) + ζα n+1 e = 0, α = 1,...,N + 1, g(x n+1 α ) + (λ n+1 ) T x n+1 α ν yα n+1 = 0, α = 1,...,N + 1, e T x n+1 α 1 = 0, α = 1,...,N + 1, P yα n+1 x n+1 α + c n+1 b tot = 0 Nonlinear system solved by a Newton method. The particle mass r is discretized explicitly, thanks to different time scales. Rice-UH Optimization Seminar, Rice University, March 30, 200 An Implicit Discretization p.31/42

42 Extended KKT System Newton system H c 0 0 B 0 0 y α 2 g(x α ) g(x α ) + λ y α e 0 ( g(x α ) + λ) T ν (y α ) 2 (x α ) T 0 I (y α ) T x α 0 0 p c p x p y p λ = b c b x b y b λ 0 e T p ζα b ζα H c is positive definite. Design of numerical algebra techniques with sequential quadratic programming and/or Schur complement methods. Control of the inertia of the matrices. Rice-UH Optimization Seminar, Rice University, March 30, 200 Extended KKT System p.32/42

43 Sequential Quadratic Programming Resolution of the extended linear system by resolution of a sequence of convex quadratic optimization problems (sequential quadratic programming). General Formulation in terms of primal and dual variables. ( Hk A T k A k 0 ) ( px p λ ) = ( fk + A T k λ k c k ) is equivalent to min p 1 2 pt H k p + f T k p = 0 s. t. A k p k + c k = 0 Control of the inertia of the matrices. Rice-UH Optimization Seminar, Rice University, March 30, 200 Sequential Quadratic Programming p.33/42

44 Sequential Quadratic Programming (2) In our case, the sequential quadratic programming problem can be expressed as the convex problem min p c { } 1 2 pt c H c p c + b T c p c + G(p c ), where G(p c ) is the optimum value of min p x α,p yα 1 2 ( P (p xα p yα ) T pxα Θ α p yα ) + P ( b xα byα ) T ( pxα p yα ) s. t. e T p xα = b ζα P P y α p xα + x α p yα = b λ p c Rice-UH Optimization Seminar, Rice University, March 30, 200 Sequential Quadratic Programming (2) p.34/42

45 Gas-Particle Partitioning Trajectory of the feed vector b (mixing inside the particle): 1-hexacosanol water pinic acid Convergence towards a stationary solution inside the particle. Detections of the phase separations. Rice-UH Optimization Seminar, Rice University, March 30, 200 Gas-Particle Partitioning p.35/42

46 Mass Conservation 25 Gas-Particle Aerosol Total 12 Particle Phases Mass 10 Moles Gas Time Time Mass conservation in the gas-particle system and convergence to a stationary solution for the particle phases. Rice-UH Optimization Seminar, Rice University, March 30, 200 Mass Conservation p.3/42

47 Computation of Particle Radius The radius of the particle r(t) is computed by conservation of mass in the (spherical) particle: 4 3 πr(t)3 }{{} Volume = n s i=1 b i (t)m c,i ρ i, }{{} Approximated ratio Mass/density where m c the molecular weight vector of the components set and ρ i is the density of the component i. Rice-UH Optimization Seminar, Rice University, March 30, 200 Computation of Particle Radius p.3/42

48 Aerosol Growth Evolution of r Radius Time The characteristic times for gas-particle equilibrium compare well with Meng, Seinfeld (199). Rice-UH Optimization Seminar, Rice University, March 30, 200 Aerosol Growth p.38/42

49 Extension to a Population of Particles Population of aerosol particles: c(t) aerosol R 2 (t) p equil 2 R 1 (t) p equil 1 b 2 (t) b 1 (t) gas b 3 (t) p equil 3 b 4 (t) p equil 4 Differences of sizes/reaction speeds/modeling of internal energy increase the stiffness of the problem. Rice-UH Optimization Seminar, Rice University, March 30, 200 Extension to a Population of Particles p.39/42

50 Extension to a Population of Particles Population of aerosol particles i = 1,...,N. d dt c(t) N = h(r i ) i=1 d dt b i(t) = h(r i ) ( c(t) η(r i ) ( c(t) η(r i ) ) 1 RT(t) pequil (b i (t)), c(0) = c 0 ) 1 RT(t) pequil (b i (t)) p equil (b i (t)) = p vapor exp ( g i (x i α) ),, b i (0) = b 0,i min y i α,xi α s. t. P i y i αg i (x i α) P yαx i i α = b i (t), y i α 0, e T x i α = 1, x i α > 0, α = 1,...,P i. Rice-UH Optimization Seminar, Rice University, March 30, 200 Extension to a Population of Particles p.39/42

51 Rice-UH Optimization Seminar, Rice University, March 30, 200 Extension to a Population of Particles Numerical linear algebra techniques: 2 4 H 1 0 B C 1 0 O 1 A I A T H 2 0 B 2 C O 2 A I A T D D H p c1 p x1 p λ1 p c2 p x2 p λ p c0 3 5 = 2 4 r c1 r x1 r λ1 r c2 r x2 r λ r c0 3 5 Properties of the Schur complement? Current work with A. Leonard, Undergraduate Student, UH Extension to a Population of Particles p.39/42

52 Tracking of Discontinuities Phase separations when activation/deactivation of an inequality constraint. Event location techniques for the tracking of the discontinuities: Phase becomes inactive when y α = 0. Phase becomes active when y α > 0. Trade-off with warm-start techniques Rice-UH Optimization Seminar, Rice University, March 30, 200 Tracking of Discontinuities p.40/42

53 Tracking of Discontinuities First order Euler scheme for the differential-algebraic system. Detection of vanishing phases y α = 0: Taylor expansion of y α = y α (b n ) + h dy α dt (bn ) + O((h ) 2 ). Estimation of the derivatives with a sensitivity approach, to obtain the appropriate time step. Predictor-corrector two-step Adams method for the computation of next time step. Convergence result: b(t ) b n+1 R h N Ch, where t is the time of impact and h (0, h) is the time step of impact. This order is due to the Euler sc heme! Current work with C. Landry, Graduate Student, EPFL Rice-UH Optimization Seminar, Rice University, March 30, 200 Tracking of Discontinuities p.40/42

54 Current and Future Work Accurate detection of the discontinuities (event location theory). Extension to a population of particles and numerical linear algebra. Higher order numerical schemes. Bifurcation theory for differential-algebraic equations. Convex hull of a finite family of energy functions. Coupling with active sets methods for mixtures of aerosols. Determination of the convex hull of a finite family of energy functions (mixtures of aerosols) and design of interior-point active-sets methods. Rice-UH Optimization Seminar, Rice University, March 30, 200 Current and Future Work p.41/42

55 Rice-UH Optimization Seminar, Rice University, March 30, 200 p.42/42