Title: A Homotopic Approach to Domain Determination and Solution Refinement for the Stationary Fokker- Planck Equation

Transcription

1 Elsevier Editorial System(tm) for Probabilistic Engineering Mechanics Manuscript Draft Manuscript Number: Title: A Homotopic Approach to Domain Determination and Solution Refinement for the Stationary Fokker- Planck Equation Article Type: Full Length Article Keywords: Stationary Fokker-Planck equation; Galerkin projection; Hilbert space approximation; Homotopic recursion Corresponding Author: Mr. Mrinal Kumar, Corresponding Author's Institution: Texas A&M University First Author: Mrinal Kumar Order of Authors: Mrinal Kumar; Suman Chakravorty, Ph.D.; John L Junkins, Ph.D. Abstract: An iterative approach for the solution refinement of the stationary Fokker-Planck equation is presented. The recursive use of a modified norm induced on the solution domain by the most recent estimate of the stationary probability density function is shown to significantly improve the accuracy of the approximation over the standard L_2-norm based Galerkin error projection. The modified norm is argued to be naturally suited to the problem and hence preferable over the standard L_2 norm because the former requires substantially fewer degrees of freedom for the same order of approximation accuracy, making it immediately attractive for the Fokker-Planck equation in higher dimensions. Additionally, it is shown that the modified norm can be utilized to progress through a homotopy of dynamical systems, D_p, in order to determine the domain of the stationary distribution of a nonlinear system of interest (corresponding to p = 1), by starting with a known dynamical system (corresponding to p = 0 and not necessarily linear) and working upwards. The partition of unity finite element method (PUFEM) is used for numerical implementation. Novel weight functions are employed for easy extension to higher dimensional problems.

2 Manuscript A Homotopic Approach to Domain Determination and Solution Refinement for the Stationary Fokker-Planck Equation Mrinal Kumar, Suman Chakravorty 1, John L. Junkins 2 Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA. Abstract An iterative approach for the solution refinement of the stationary Fokker-Planck equation is presented. The recursive use of a modified norm induced on the solution domain by the most recent estimate of the stationary probability density function is shown to significantly improve the accuracy of the approximation over the standard L 2 -norm based Galerkin error projection. The modified norm is argued to be naturally suited to the problem and hence preferable over the standard L 2 norm because the former requires substantially fewer degrees of freedom for the same order of approximation accuracy, making it immediately attractive for the Fokker-Planck equation in higher dimensions. Additionally, it is shown that the modified norm can be utilized to progress through a homotopy of dynamical systems, D p, in order to determine the domain of the stationary distribution of a nonlinear system of interest (corresponding to p = 1), by starting with a known dynamical system (corresponding to p = 0 and not necessarily linear) and working upwards. The partition of unity finite element method (PUFEM) is used for numerical implementation. Novel weight functions are employed for easy extension to higher dimensional problems. Key words: Stationary Fokker-Planck equation, Galerkin projection, Hilbert space approximation, Homotopic recursion Corresponding author. Graduate Research Assistant addresses: mrinal@neo.tamu.edu (Mrinal Kumar ), schakrav@aeromail.tamu.edu (Suman Chakravorty ), junkins@tamu.edu (John L. Junkins ). 1 Assistant Professor 2 Distinguished Professor, Holder of the Royce Wisenbaker Chair Preprint submitted to Elsevier 1 March 2007

3 1 INTRODUCTION The subject of uncertainty propagation was introduced to the scientific community in the 1860s through the investigations of Maxwell and Boltzmann into the random nature of gaseous motion. Since then, the field has benefitted from the works of several giants, like Lord Rayleigh, Albert Einstein, Max Planck, Andrian Fokker, Andrey Kolmogorov and several others. Consequently, following a somewhat heuristic start, the field has grown with firm foundations in the principles of stochastic differential equations and probability theory. Today, the study of uncertainty propagation through stochastic systems continues to permeate a multitude of fields in science and engineering. In essence, it involves the study of the time evolution (and when it exists, the steady state) of the probability density function (pdf) that characterizes the underlying stochastic process. The well known Fokker-Planck-Kolmogorov equation [1], or simply the Fokker-Planck equation (FPE) is of central importance in this context because it captures the exact time evolution of the state-pdf of dynamical systems driven by white-noise excitation. Thus, the FPE provides the exact description of the uncertainty propagation problem for stochastic systems under white-noise forcing [2]. Unfortunately, analytical solutions of the FPE exist only for linear dynamical systems and a handful of nonlinear systems possessing special structure [3]. Such systems represent a very small fraction of the enormous variety of stochastic systems encountered in science and engineering. Therefore, several approximate methods have been used for the purpose of uncertainty propagation through nonlinear systems. Some popular methods include Monte Carlo simulations [4, 5], Gaussian closure [6 8] (or higher order closures) [9], equivalent linearization and stochastic averaging [6, 10]. Despite the widespread use of these approximate methods, attempts to numerically solve the FPE have never been abandoned because of the above mentioned fact that it provides the exact description of the uncertainty propagation problem under white noise excitation. In other words, the solution of the FPE is valid in situations where all other approximate methods prove inadequate; namely, long term uncertainty propagation and/or high degree of nonlinearity in the underlying dynamical system. In the recent literature, several numerical methods have been proposed for solving the FPE. These include global approximation techniques [11,12], finitedifference (FD) [13] and finite-element methods (FEM) [14, 16], multi-scale FEM [15], and the meshless FEM [2, 17]. Most of these techniques are based on the variational formulation (weak form) of the FPE, and solve the equation in an integral/average sense over the solution domain. However, all these methods encounter multiple difficulties that are inherently involved in attempting the FPE solution numerically [2]. The primary difficulties include computational costs associated with the discretization of high dimensional state-spaces, positivity of the obtained solution, and the non-existence of a unique solution 2

4 domain of finite size and orientation, especially when no prior knowledge about the underlying dynamical system is available [2, 18]. In addition, the obtained solution must satisfy the normality constraint in order to be a valid pdf. The issue of high dimensionality has been addressed in Kumar et al. [2], in which an efficient numerical scheme based on the local, meshless, partition of unity finite element method (PUFEM) [19] has been utilized to solve the stationary FPE. In this paper, we address the issue of domain determination and solution refinement of the stationary FPE using an iterative, homotopic, weighted Galerkin approximation scheme. The inner product involved in the residual error projection is modified and weighted by the most recent approximation of the true pdf. This is in contrast with the traditional Galerkin approach in which the standard L 2 space equipped with the Lebesgue measure is used as the projection space. We argue that the best weight to modify the standard L 2 inner product is the true solution of the stationary FPE. The resulting modified projection space is denoted by L(dΨ ), where dψ is the probability measure induced on R N by the true solution of the FPE, Ψ. Unfortunately the true pdf, Ψ, is the object of study in the current case, and is unknown. Therefore, an iterative scheme is developed in which the most recently obtained pdf is used as a weight to converge to the true solution. While the accuracy of the final iteration is limited by the approximation-ability of the utilized basis functions, it is shown through numerical examples that for the same order of accuracy, the weighted norm approach requires substantially fewer degrees of freedom (e.g. number of nodes in a finite-element discretization) than the traditional L 2 approach. This augurs well for extension to the FPE in higher dimensions, wherein the L 2 approach would require an enormous number of degrees of freedom. A similar approach has also been used by other researchers to obtain improvement in the approximation to the FPE [12]. In this paper, we prove the stability of this approach by showing its closeness to the Hilbert projection theorem. A similar result for the transient FPE has been presented in a different paper [18], for which both the stability and convergence proofs have been provided. The current paper also discusses a recursive homotopic approach for the determination of the solution domain of a nonlinear stochastic system of interest. It is shown that it is possible to recursively track the domain of distribution for the desired system, starting with a dynamical system whose response is known. Hence, the proposed homotopic approach provides a method of tackling the issue of existence of no prior knowledge about the solution. A numerical example is presented to illustrate how this approach can be used to track the domain for the Duffing oscillator by varying its nonlinearity parameter. 3

5 2 The Fokker-Planck Equation and its Weak Form The Fokker-Planck equation provides the exact description of the uncertainty propagation problem for dynamical systems driven by white-noise excitation. Consider a general N-dimensional white-noise driven nonlinear dynamical system with uncertain initial conditions, given by the following Itô stochastic differential equation: dx = f(t, x)dt + g(t, x)dw, E[x(t 0 )] = x 0. (2.1) where, W represents a Wiener process with the correlation function Qδ(t 1 t 2 ), and x 0 represents the nominal initial state. The initial probability distribution of the state is given by the pdf W(t 0, x), which captures the state uncertainty at time t 0. Then, for the system given by (2.1), the time evolution of W(t 0, x) is described by the following FPE, which is a second order, linear PDE in W(t, x): t W(t, x) = L FPW(t, x). (2.2) where, N L FP = D (1) i (.) + i=1 x i N i=1 j=1 D (1) (t, x) = f(t, x) N 2 D (2) ij (.), (2.3) x i x j g(t, x) Qg(t, x), (2.4) x D (2) (t, x) = 1 2 g(t, x)qgt (t, x), (2.5) where, L FP is called the Fokker-Planck operator, D (1) is known as the drift coefficient vector and D (2) is the diffusion coefficient matrix. 2.1 FPE: Difficulties in Numerical Implementation Solving the FPE numerically for the pdf, W, is a formidable problem because of the following issues: (1) Positivity of the pdf: W(t, x) 0, t & x. (2) Normalization constraint of the pdf: W(t, x)dv = 1. (3) Dimensionality: The Fokker-Planck operator in Eq.2.3 involves partial derivatives with respect to all the states of the underlying system. For example, for a dynamical system that describes 2 dimensional motion, 4

6 like planar motion of a point mass under a central two-body gravitational force field, the corresponding FPE would require the discretization of not just the position coordinates, but also the velocity coordinates, i.e. a total of four (state) spatial dimensions, in addition to time. For most numerical methods which involve a mesh-based discretization procedure, this represents a major stumbling block due to the enormous pre-processing required in grid generation and the subsequent book-keeping of interelement boundaries for ensuring solution continuity during evaluation of the integrals. (4) No unique solution domain for numerical implementation: The true domain of solution of the FPE is (, ) N. However, the discretization procedure of any numerical method requires a finite domain. Heuristic methods are typically used to define a conservatively sized domain in order to include the significant portion of the pdf to be approximated, say, W > If no prior knowledge is available about the dynamical system under consideration (especially highly nonlinear systems), it is generally difficult to obtain the location, orientation, and size of such a finite domain that achieves the mentioned tolerance. Issues 1 and 2 represent additional constraints that the solution of Eq. 2.2 must satisfy in order to be a valid pdf. Since these constraints are not built into the structure of the FPE, they must be accommodated in the numerical solution. While (2) can be enforced by a simple renormalization of the obtained solution, (1) is a tough proposition. Several researchers have used a log-transformation of the FPE to ensure positivity (the inverse transform (exponential) applied to the solution obtained in the transformed coordinates ensures positive values) [11, 12]. However, this approach converts the linear PDE (Eq. 2.2) into a nonlinear PDE, which is generally not desirable. Dimensionality (issue 3) continues to be the primary deterring factor preventing numerical methods from ready application to the FPE for general dynamical systems. Recently, promising success in computational efficiency and time of execution has been shown with the use of meshless methods like MLPG (Meshless Local Petrov Galerkin method [20]) and PUFEM [2,17]. In this paper, we address the issue of domain determination and solution refinement for the stationary FPE with an iterative homotopic approach that employs a modified inner product for the residual error projection. We utilize the meshless PUFEM approach for developing the weak form approximation. We mention that any approximation method is equally compatible with this iterative approach. PUFEM has been chosen for its several computational advantages [2] and easy extension to higher dimensional problems. 5

7 2.2 The Variational Formulation (Weak Form) of the stationary FPE The variational formulation of the stationary FPE involves the determination of a solution W U, such that the following system of projection equations is satisfied: L FP ( W)υdΩ = 0, υ V. (2.6) Ω Generally, the approximation space U (also known as the trial space) and the projection space V (also called the test space) are infinite dimensional. Therefore, we have: W(x) = c k φ k (x). (2.7) k=1 For the numerical implementation of the variational form, the approximation and projection spaces are truncated to finite dimensional subspaces, i.e. the trial and test functions are chosen from finite dimensional subspaces U n U and V n V respectively. We have: n Ŵ(x) = c k φ k (x). (2.8) k=1 In the Galerkin approach, the residual error resulting from the truncated approximation is projected onto the space of the trial functions, i.e. V n = U n, or υ k = φ k. We note here that in the case of the transient FPE, the fourier coefficients c k would be time varying, and (2.8) would be equivalent to the separation of variables. Writing (2.6) using the inner product notation, we get: ( n ) L FP c k φ k, φ j = 0, j = 1, 2,... n. (2.9) k=1 In the above equation, <.,. > represents the standard inner product on L 2 (Ω). In the next section, we discuss the modification of the above inner product for solution refinement of the stationary FPE. 3 Modification of the L 2 Inner Product and Space Homotopy The variational formulation of the FPE described in the above section employs the traditional L 2 inner product. We claim that there exists a natural measure for the above problem which defines a modified inner product, and which can be used to our advantage to obtain accurate approximations with a small number of degrees of freedom. This natural measure is characterized by the true solution of the FPE. Besides redefining the inner product, it also implicitly defines the domain of solution by providing weightage to only the significant 6

8 regions of the pdf. The problem with using the actual solution as the weight is that it is unknown. We therefore develop an iterative scheme in which the most recent approximation of the true solution is used to weight the L 2 inner product. In the following development, we show the closeness of this approach to the optimal approximation of the pdf obtained from the normal equations derived from the Hilbert projection theorem. We begin with the following assumptions on the approximation space U n and the initial estimate Ŵ0, of the true solution Ψ : Assumption 3.1 The true solution is exactly approximable by the trial space U n using the Hilbert projection theorem, i.e. there exist {c k }, k = 1, 2,..., n, such that: n Ψ (x) = c k φ k (x). k=1 Assumption 3.2 A sufficiently close approximation Ŵ of the true solution is available to start the iterative process: Ŵ Ψ < ɛ. Assumption 3.3 The basis functions φ k form an orthonormal set with respect to the standard Euclidean inner product, i.e. L 2 (Ω). Assumption 3.1 has been made primarily for convenience and is equivalent to saying that the approximation space U n equals U. It can be relaxed to read sufficiently well approximable (to within ɛ ) instead of exactly approximable, and the results still hold but the mathematical development becomes tedious without adding significant insight. The stability proof that follows depends on the closeness of the starting approximation, i.e. Assumption 3.2. Thus, if the L 2 error norm of the starting approximation is bound above by ɛ, it is shown below that the error norm resulting from the next step of iteration is at most scaled by a constant factor. If the scaling factor (which depends on the particular system under consideration) is less than 1, a contraction mapping is obtained and convergence follows, but in general this might not be the case. Finally, Assumption 3.3 is made also purely for the sake of convenience of evaluating integrals, and the actual approximation space chosen need not satisfy this condition. In the following, we set up the equations for the Hilbert projection approach to find the coefficients c k in Assumption (3.1). We redefine the inner product 7

9 .,. as the following: φ i, φ j = Ω R N φ i (x)φ j (x)ψ (x)dx. (3.1) Then, the Hilbert coefficients, c k for the true solution ψ are given by the following equation: n c k φ k, φ j = Ψ, φ j j = 1, 2,..., n. (3.2) k=1 Following Assumption 3.3, c k = Ψ, φ k. Next, we define a new inner product,.,., which is induced on the solution domain Ω by the current approximation Ŵ, of the true solution ψ : φ i, φ j = Ω R N φ i (x)φ j (x)ŵ(x)dx. (3.3) Using the above inner product, the projection equations 2.9 for the variational formulation can be rewritten as the following: n n L FP ( c kφ k ), φ j + α{ c kφ k, φ j Γ k=1 k=1 Ψ, φ j Γ } = 0 j = 1, 2,... n, (3.4) where, c k denote the fourier coefficients of the approximation obtained from the weighted Galerkin approximation shown above. Also, α is a penalty parameter which has been introduced to enforce the boundary conditions, and.,. Γ denotes the evaluation of the integral over the boundary. 3.1 Closeness of the Hilbert and Galerkin Approximations In the above section, two sets of coefficients were discussed, c k and c k, corresponding to the Hilbert projection method and the Galerkin method weighted with the most recent pdf approximation respectively. Eq. 3.4 represents a system of linear equations in c k, which can be expressed as follows: B C + B ΓC = F Γ, (3.5) Also, following Assumption 3.1, the Hilbert approximation of ψ (x) satisfies the Galerkin variational form of the stationary FPE exactly, i.e.: 8

10 n n L FP ( c k φ k ), φ j + α{ c k φ k, φ j Γ k=1 k=1 Ψ, φ j Γ } = 0 j = 1, 2,... n. (3.6) Note that the inner product in the Hilbert projection equation is weighted by the true solution, ψ, and hence the notation.,. is used. Then, Eq. 3.6 reduces to the following linear system: BC + B Γ C = F Γ. (3.7) In Eq. (3.5), vector C represents the Galerkin coefficient vector while in Eq. (3.6), C represents the Hilbert coefficient vector. The various other matrices and vectors are defined as follows: B = [ L FP (φ k ), φ j ], (3.8) B Γ = α [ φ k, φ j Γ ], (3.9) F Γ = α [ Ψ, φ j Γ ], (3.10) B = [ L FP (φ k ), φ j ], (3.11) B Γ = α [ φ k, φ j Γ ], (3.12) F Γ = α [ Ψ, φ j Γ ]. (3.13) As the first step towards showing the closeness of C proximity of Eq. 3.5 to Eq. 3.7 and write Eq. 3.5 as: to C, we prove the BC + (B Γ + 3 )C = F Γ (3.14) Comparing Eq. 3.5 and Eq. 3.14, we have: 1 = F Γ F Γ, (3.15) 2 = BC B C, (3.16) 3 = B Γ B Γ. (3.17) Then, we have the following lemma for the upper bounds of various i : Lemma 3.4 Given the validity of Assumptions 3.1 and 3.2, the following inequalities hold: 1 K 1 ɛ, 2 K 2 L FP ɛ, 3 K 3 ɛ, where, K 1 -K 3 are finite constants,. represents the Euclidean norm for the vectors 1 and 2, and the matrix norm induced by the Euclidean norm for 9

11 3, and L FP represents the operator norm of the Fokker-Planck operator. Proof 1 Consider 1 = [δ 1 j ]: δj 1 = α{ Ψ, φ j Γ Ψ, φ j Γ } = α Ψ φ j (Ŵ Ψ )dγ (3.18) Γ δ 1 j 2 α 2 Γ α 2 Γ α 2 Γ α 2 Γ Ψ 2 φ j 2 Ŵ Ψ 2 dγ (3.19) Ψ 2 Ŵ Ψ 2 dγ Γ φ j 2 dγ (3.20) Ψ 2 Ŵ Ψ 2 dγ.1 (3.21) Ψ 2 dγ Γ Ŵ Ψ 2 dγ (3.22) α 2.1.ɛ 2 (3.23) δj 1 α ɛ (3.24) In the above, the Cauchy-Schwarz inequality has been applied in going from Eq to Eq and from Eq to Eq Additionally, weaker forms of Assumptions 3.2 and 3.3 (since only boundary integrals are involved) have been used in Eq and Eq Thus, from Eq. 3.24, we conclude that there exists K 1 < such that: 1 K 1 ɛ. (3.25) Next, looking at 2 = [δ 2 j ], and following similar arguments as above, we obtain: 10

12 n δj 2 = c kl FP (φ k ), φ j k=1 n c kl FP (φ k ), φ j (3.26) k=1 n = c kl FP (φ k )(Ψ Ŵ)dΩ Ω k=1 n δj 2 2 c kl FP (φ k ) 2 Ŵ Ψ 2 dω Ω k=1 n L FP ( c kφ k ) 2 ɛ 2 k=1 n L FP 2 c kφ k 2 ɛ 2 k=1 L FP 2 C 2 ɛ 2 (3.27) In Eq. 3.27, the norm of the Galerkin coefficient vector, C is a finite quantity because it contains the coefficients of the various basis functions used to approximate pdf s that have well behaved functional forms (i.e. without δ-function like singularities). Hence, bounding it above by a finite quantity, we can show that there exists a K 2 < such that: 2 K 2 L FP ɛ. (3.28) Finally, considering 3 = [δ 3 kj]: δkj 3 = α{ φ k, φ j Γ φ k, φ j Γ } = α φ k φ j (Ŵ Ψ )dγ Γ δ 3 kj 2 α 2 Γ φ k 2 φ j 2 Ŵ Ψ 2 dγ α ɛ 2 (3.29) A weak form of Assumption 3.3 (over the boundary) has been used in Eq Thus, K 3 <, such that: 3 K 3 ɛ (3.30) This completes the proof of the lemma. We now proceed to show the stability of the iterative approach by establishing an upper bound for the error of the approximation resulting from the weighted Galerkin approach. We make the following additional assumptions: 11

13 Assumption 3.5 The quantity ɛ is small enough such that (B+B Γ ) Assumption 3.6 The operator norm of Fokker-Planck operator is bounded above as L FP = M <. This leads us to the following result: Lemma 3.7 Given the validity of Assumptions 3.5 and 3.6, the following upper bound exists on the L 2 error norm between the weighted Galerkin and Hilbert approximations of the FPE: C C Kɛ (3.31) Proof 2 Let us adopt the following notation: B + B Γ = B G, and = Σ. Then, Eqs. 3.7 and 3.14 become: (B G 3 )C = F Γ Σ. (3.32) B G C = F Γ. (3.33) Thus, we have: C C = B 1 G F Γ (B G 3 ) 1 (F Γ Σ ) = {B 1 G (B G 3 ) 1 }F Γ + (B G 3 ) 1 Σ Taking the norm (standard L 2 ) on both sides and applying the triangle inequality, C C B 1 G (B G 3 ) 1 F Γ + (B G + 3 ) 1 Σ (3.34) Furthermore, following Assumption 3.5, we obtain the following expansion: (B G 3 ) 1 = B 1 G + B 2 G 3... Thus, using the result for the upper bound of 3 from Lemma 3.4 (Eq. 3.30), we obtain: (B G 3 ) 1 B 1 G + B 1 B 1 G G 2 K 3 ɛ, (B G 3 ) 1 B 1 G 2 K 3 ɛ. Also, combining Eqs and 3.28: Σ K Σ (1 + L FP )ɛ, (3.35) where K Σ = max(k 1, K 2 ). Denoting B 1 G as P, F Γ as Q, and L FP as 12

14 M, Eq becomes: C C QP 2 K 3 ɛ + K Σ (P + P 2 K 3 ɛ)(1 + M)ɛ. Dropping out terms of order higher than O(ɛ), we get: C C (QP 2 K 3 + P K Σ (1 + M))ɛ, C C Kɛ. This completes the proof of the lemma. Therefore, we see that the error in the next iteration of the process is scaled by the constant K, which comprises of several norms associated with the underlying system. If this quantity is less than 1, we obtain a contraction mapping and the error reduces to zero in the limit. However, this is not true in general. In either case, the method is stable for a finite number of iterations and will not lead to divergence (except in certain pathological cases discussed below). Superior convergence characteristics has been shown for two dynamical systems through numerical simulations in the results section. Looking closely at the constant K, we observe that the norm of the inverse of the Hilbert stiffness matrix, B G appears in its expression. If this matrix is illconditioned or singular, the method loses its stability. This situation may arise in certain conditions (e.g. local approximation schemes which involve degrees of freedom with local domain of influence) and is discussed in detail below. On the other hand, the norm of the vector F does not cause problems as it involves the integral of the true solution along the domain boundary, which is a very small quantity ( 10 9 or lower). In the numerical examples shown below, we show that convergence is achievable using the (local) PUFEM algorithm for the variational formulation, in conjunction with suitable patching of solutions from successive iterations. 3.2 Space Homotopy for Domain Determination In the above section we assumed that the solution domain on which the iterations are carried out is known a-priori. In general, this might not be the case, especially for nonlinear systems. In this section, we demonstrate the implementation of a space homotopy via a family of single parameter dynamical systems to track the domain of the stationary distribution for the system of interest. The underlying assumption is the existence of a family of dynamical 13

15 systems, D p indexed by the homotopy parameter p: D p : dx = f(x, p)dt + g(x, p)dw, p [0, 1], (3.36) where D 1 corresponds to the dynamical system of interest and D 0 corresponds to a stochastic dynamical system whose response is known, i.e., the stationary FPE associated with it can be solved. Let ψ p(x) denote the true solution of the FPE associated with dynamical system D p. We make the following assumption about the family of dynamical systems D p and the solution of the associated FPE s, ψ p: Assumption 3.8 Given any p [0, 1], and any ɛ > 0, there exists δ > 0 such that for all p B δ (p), (open ball of radius δ centered at p) ψ p(x) ψ p (x) ɛ. In essence, the above assumption assumes the existence of a one parameter family of dynamical systems such that the solutions to the associated FPE s change smoothly over this parameter space - in other words, a homotopy. We next consider only those dynamical systems for which the constant K appearing in Lemma 3.7 is less than unity (hence leading to a contraction mapping and ensuring convergence). We then state the following obvious result as a proposition: Proposition 3.9 Consider dynamical systems D p with K < 1 in Lemma 3.7. Then, given ψ 1, such that ψ 1 Ψ ɛ and that ɛ is sufficiently small, a sequence of functions {ψ n } n=1 can be constructed recursively, starting with ψ 1 such that ψ n Ψ 0 as n, p [0, 1]. The proof is trivial because of Lemma 3.7 and the contraction mapping argument for K < 1. A note about the notation: when we write ψ i, we refer to the i th function of a sequence {ψ i } N i=1. However, when we write ψ i, we refer to the true solution of the FPE for the dynamical system D p=i. Then, with Assumption 3.8 in mind, we have the following result pertaining to how the solution of the FPE associated with our system of interest (D 1 ), i.e., ψ 1(x) = Ψ (x) can be obtained recursively given the knowledge of the solution of the FPE associated with the system D 0. The result uses Proposition 3.9 in conjunction with successive approximation. Proposition 3.10 Let ɛ p be sufficiently small such that proposition 3.9 is satisfied for any ψ satisfying ψ ψp ɛ p. Let inf p [0,1] ɛ p = ɛ > 0. Then, under assumptions , 3.5, 3.6 and 3.8, given ψ0, the exact solution of the FPE corresponding to D 0, there exists a finite sequence of functions {ψ n } M n=1 14

16 s.t. ψ M = Ψ. Moreover, this sequence can be obtained in a recursive fashion starting with ψ 0. (i.e. ψ 1 = ψ 0) Proof 3 Let δ p be such that if p B δp (p) then ψp ψp ɛ. Note that this 2 is possible due to Assumption 3.8. Consider the open covering p [0,1] B δp (p) of the set [0, 1]. Since [0, 1] is compact, there exists a finite subcover of [0, 1] given by M i=1 B δpi (p i ). Let δ i δ pi (p i ) and ψp i ψi. Let us assume that ψi is known and we need to obtain ψi+1. By definition, there exists a p s.t. p p i < δ i and p p i+1 < δ i+1. Then, it follows from construction that ψ i ψ i+1 ψ i ψ p + ψ p ψ i+1 ɛ. (3.37) Then, due to Proposition 3.9, starting with ψ i, it is possible to obtain ψ i+1 in a recursive fashion. Note that the above holds for all i = 0, 1,, M 1. Thus, in this fashion we can obtain the sequence {ψ 0, ψ 1,, ψ M = Ψ } recursively starting with ψ 0. This completes the proof of the proposition. In summary, the development above (space homotopy in conjunction with solution refinement) can be presented as the following algorithm: (1) Find a homotopy of dynamical systems D p, p [0, 1], such that D p (p = 1) corresponds to the system of interest and D p (p = 0) corresponds to a known system, in the sense that its associated stationary FPE can be solved. (2) Select a finite number of points p i [0, 1], i = 1,, M; that are sufficiently close. From the rest of the algorithm, rename the dynamical system corresponding to p i, i.e. D pi, as D i, and the true solution of the associated stationary FPE, ψp i, as ψi. Notice that the selection of points p i can be done online - i.e. if p i+1 is found to be not close enough to p i, it is possible to go back and redo the previous iteration. (3) Following the new index based notation, notice that the exact solution for system D 1 is known (ψ1). Also, the solution we are after (for p = 1) is, in the new notation, ψm = Ψ. Set i = 2. (4) Determine the solution ψi in the following manner: (a) Set j = 1 and the current weight for L 2 norm modification, W = ψi 1. (b) Using W as the weight in the modified norm approach, obtain ψ j i, i.e. the j th approximation for ψi. (c) If ψ j i = ψ i, goto step 5. Else, set j = j + 1 and W = ψ j 1 i step 4b. (5) If i = M, stop. Else, set i = i + 1 and goto step 4a. and goto 15

17 The above algorithm hence involves two loops. The outer loop runs over the homotopical sequence of dynamical systems, from the known to the desired. The inner loop performs successive refinements upon the solution obtained for each particular dynamical system by utilizing the modified norm approach. The process is started with the known system, whose solution for the FPE is available, serving as the first weight for norm-modification in the outlined approach. We mention that measuring error in the inner loop (as to how close we are to the true solution for any particular dynamical system) is not a trivial job, since the true solutions are not known. In practice, it is possible to look at the equation error in order to measure the closeness of approximation to the truth. 4 Numerical Implementation In this section, we discuss the use of the PUFEM algorithm for the numerical implementation of the above methodology and discuss the associated difficulties and suggest possible fixes. PUFEM is a powerful local, meshless, nodebased finite element approximation method which has been used successfully to solve several difficult partial differential equations. Its application to the stationary FPE has been discussed in detail in [2]. Here, we only provide the basic equations of the variational formulation of the stationary FPE with this method. The PUFEM approximation of the pdf in the n th iteration, Ŵ n, can be written as: Ŵ n (x) = P Q s s=1 k=1 c skϕ s (x)ζ sk (x), (4.1) where, ϕ s (x) are node-centered overlapping compactly supported positive weight functions which bring about an implicit discretization [2] of the solution domain Ω and satisfy the property of partition of unity, i.e. s ϕ s (x) = 1, x Ω. The functions ζ sk (x), k = 1,..., Q s are basis functions used in the approximation space defined locally on the compact support of the corresponding partition of unity function ϕ s. The product {ϕ s ζ sk } is called the shape function, and the set forms a conforming approximation space over the global solution domain Ω. It is important to note here that the coefficients c sk represent the amplitudes of the local shape functions inside compactly supported domains. Additional details can be obtained from [2]. Using this approximation space, the weighted Galerkin variational equations for the stationary FPE are given by: Ω s Q sp i=1 L FP (c iφ i )φ j Ŵ n 1 dω + α Γ s Γ Q sp i=1 c iφ i φ j Ŵ n 1 dγ (4.2) 16

18 = α Γ s Γ Ψ φ j Ŵ n 1 dγ, j = 1,..., Q s P where, φ i = ϕ s (x)ψ sk (x), Ω s is the local domain of influence of the shape functions ϕ s ψ sk, i.e. the compact support of ϕ s. Similarly, Γ s Γ represents the intersection of the local element boundary with the global domain boundary. The resulting elements of the matrices involved in the linear system of equations (3.5) are: B ij = L FP (ϕ k ψ kl )ϕ p ψ pq Ŵ n 1 dω, (4.3) Ω s B Γij = α ϕ k ψ kl ϕ p ψ pq Ŵ n 1 dγ, (4.4) F Γj = α Γ s Γ Γ s Γ Ψ ϕ p ψ pq Ŵ n 1 dγ, (4.5) where, i = ( k 1 s=1 Q s + l ) and j = ( p 1 s=1 Q s + q ). 4.1 Conditioning of the Stiffness Matrix As seen in Sec.3.1, the condition number of the stiffness matrix B H is an important issue in the stability of the above approach. Unfortunately, if a local approximation scheme such as PUFEM is used for either the weighted Galerkin or Hilbert approaches, the stiffness matrix invariably turns out to be ill-conditioned. This is because of the following reason - the pdf used as the weight gives relative weightage to different regions of the domain, thus distinguishing the regions of greater significance (close to the mean) from the regions of low significance (e.g. regions beyond 3σ for a Gaussian distribution). Also, in a local scheme, the shape functions and their coefficients (c i or c i) have local influence. In other words, the integrals associated with the shape functions close to the boundary are evaluated on local domains only near the boundary region. By virtue of the exponentially low weight given to these regions by the weighting pdf, these integrals get nearly washed out in comparison with the integrals evaluated on local domains close to the mean. Consequently, the entries in the stiffness matrix B corresponding to the coefficients of local shape functions near the boundary domains diminish severely in comparison with the entries for the coefficients of the shape functions in the interior. This effect makes the boundary coefficients unobservable, and the resulting stiffness matrix numerically ill-conditioned. 17

19 4.1.1 A Numerical Fix We thus conclude that using a pdf as the weight for the modification of the inner product causes ill-conditioning of the stiffness matrix in local approximation techniques because it renders the local coefficients near the boundary regions unobservable. A natural solution to this problem is to extract the portion of the stiffness matrix which has acceptable conditioning for inversion, and to retain the solution for the remaining coefficients from the previous iteration. In this manner, not all coefficients are modified in going from one iteration to another as the coefficients close to the boundary do not change. This method also gives a simple way of trimming the domain of solution from one iteration to the next - by identifying and pruning regions which receive weightage below a specified tolerance from the weighting pdf. However, we mention that selective modification of the coefficients usually leads to discontinuities and/or ripple formation in and around the concerned local domains. To counter this, the two sets of coefficients (from the current and the previous iterations) are patched together to produce a smooth surface. This patching procedure can be done using the PUFEM algorithm with the help of blending functions, such as those mentioned in [2]. This approach provided highly acceptable results as illustrated in the next section. 5 Results In this section, we present numerical examples to illustrate the theoretical ideas presented above. We show that with the modified norm approach, it is possible to obtain high accuracy while using a small number of degrees of freedom for highly nonlinear systems. The dynamical systems considered in this section reside in 2-D state space. 5.1 Solution Refinement of the Stationary FPE: Results We consider the two nonlinear systems described below: System 1 Consider the following 2-D damped Duffing oscillator: ẍ + ηẋ + αx + βx 3 = gg(t) (5.1) 18

20 We assign the parameters appearing above the following values: α = 15, β = 30, η = 10, g = 1 (soft-spring case). The analytical solution of the stationary FPE for the above system is given by the following expression: { W s (x, ẋ) = C exp 2 η [ ]} αx 2 g 2 Q 2 + βx4 4 + ẋ2, (5.2) 2 where, C is a normalization constant. Fig.1(a) shows the true stationary distribution for this system, which is a bimodal pdf System 2 Consider the following 2-D nonlinear oscillator [11]: ẍ + βẋ + x + α(x 2 + ẋ 2 )ẋ = gg(t) (5.3) We set the following values: α = 0.125, β = 0.5, g = The analytical solution of the stationary FPE for the above system is known, and given by the following expression: W s (x, ẋ) = C exp { 1 2g 2 [ ] } α 2 (x2 + ẋ 2 ) 2 + β(x 2 + ẋ 2 ), (5.4) From Fig. 2(a), we see that from the top-view, the true stationary distribution for this system looks like a ring. Notice that the stationary distributions for both systems are exponentials of a polynomial function. As the first exercise, we evaluate the various norms involved to ensure that the systems described above conform with the theory presented in section 3. In particular, we will demonstrate that the numerical fix suggested in section to tackle the issue of unobservability of the boundary nodes indeed causes the constant K appearing in Lemma 3.7 to be less than unity, hence leading to a contraction mapping, which in turn implies convergence. Table 1 contains the various quantities that appear in lemmas 3.4 and 3.7. These are ballpark numbers and give order of magnitude estimates. The constants K 1 K 3 appearing in Lemma 3.4 have been computed by evaluating the various domain and boundary integrals. The operator norm, L FP has been computed via discretization. Notice that using a pdf as the weight to modify the L 2 norm causes the stiffness matrix to be nearly singular. However, the numerical fix suggested in section brings down the ill-conditioning significantly, enough to make the constant K appearing in Lemma 3.7 less than unity. Notice however,that the value of K suggests that convergence is expected to be faster for system 1 than system 2. This was indeed observed and is illustrated in the convergence plots presented below. 19

21 Table 1 Approximate estimates of various norms and constants appearing in the theory, for systems 1 and 2. The numerical fix proposed in section ensures convergence by improving the conditioning of the stiffness matrix. Quantity/Norm System 1 System 2 K K K L FP = M B 1 G = P, before fix B 1 G fixed = P fixed F Γ = Q K (Lemma 3.7) We now proceed to the actual results of solution refinement for the described systems. Figs. 1(a)-1(f) show results for system 1 (soft-spring Duffing oscillator). Fig. 1(b) shows the error surface using the PUFEM algorithm with the standard L 2 inner product approach on a grid equipped with local quadratic basis functions. This error surface serves as a reference for the standard L 2 approach. We next perform the iterative refinement process, starting with the L 2 solution computed on a much coarser grid (12 12). This solution is also the weight for inner-product modification in the first iteration. Upon using the modified inner-product approach in conjunction with patching of neighboring iteration approximations, the accuracy improves significantly, which is evident in the error surface shown in Fig. 1(c). The true power of this approach is illustrated in Fig.1(d), in which the convergence characteristics for three comparable methods have been shown. The graph corresponding to the iterative PUFEM shows that the process is commenced on a coarse grid, and the use of the pdf obtained after every iteration to improve the inner-product space for the subsequent iteration leads to significant drops in error. Once no further accuracy is possible with the mesh, a switch is made to a finer grid (14 14), beginning with the last pdf obtained from the previous (12 12) grid as the weight for the first iteration on the new grid. The spacing between circles on the iterative PUFEM graph illustrates the drop in error after every iteration. Thus, huddling of the circles signifies saturation on a particular grid, and a switch to a finer grid is made following such behavior. In the graph shown, iterations have been terminated after saturation of the (16 16) grid, and the final error surface is shown in Fig. 1(c). The most significant contribution of this result is that it shows that it is possible to achieve extremely accurate approximations with a small number 20

22 of degrees of freedom. For example, compare on Fig. 1(d) the error after the final iteration on the grid ( 1536 PUFEM DOFs using quadratic bases) with the error of the L 2 approach on a grid ( 5400 PUFEM DOFs with quadratic bases). Fig. 1(e) illustrates (for iteration #3) the phenomenon of ripple formation when selective update of coefficients is carried out by pruning out the unobservable coefficients which are weighted out by the exponentially low weightage provided by the pdf. As expected, ripples form on either side of the two weighty modes, where the pdf drops off suddenly to extremely small values on either side. However, the process of patching the current solution with the previous iteration smoothes out these ripples and a relatively better solution is obtained (Fig. 1(f)). Similar results are obtained for system 2 (Fig. 2), and it is again possible to obtain high accuracy with a much smaller number of approximation nodes, as compared with the standard L 2 approach. However, the results are not as drastic here because for this system, it is possible to obtain fairly accurate results with even the standard L 2 approach. Also, the convergence rate is slower as visible in Fig. 2(c), which is also evident from the numerical value of constant K in Table 1. We mention that we have obtained similar encouraging results for several other 2-D oscillators. 5.2 Space Homotopy: Results In order to illustrate the use of the homotopic approach for domain determination of the stationary FPE, we consider the following Duffing oscillator which is a modified version of the one used in the previous section: ẍ = αx βẋ + ɛ(x 3 + σ) + w (5.5) The homotopy parameter p in the above system is ɛ, variation in which generates a family of dynamical systems of varying nonlinearity. The role of the parameter σ is to shift the domain of the significant portion of the pdf as the homotopy parameter is varied. Its presence allows us to validate the fact that the proposed method can successfully track changes in the domain as p (= ɛ) changes from 0 to 1. Also, α is assumed to be positive (corresponding to a hard spring with a solitary stable equilibrium point). Fig. 3 shows the results for the above system. In Fig. 3(a), the variation of the x-coordinate of the stable equilibrium point of the system is shown with ɛ. The marked values (stars) on this curve depict the values of the parameter used enroute to the desired dynamical system, corresponding to ɛ = 1. It was found necessary to take small steps initially (see figure) in order to satisfy the assumptions 21

23 stated in Lemma 3.7. In general, the nature of this variation will depend on the particular manner by which the homotopic parameter influences the system under consideration. We mention that the execution of space homotopy also involves the solution refinement process described above. Before we can proceed from a particular value of p to the next, it must be ensured that the approximation obtained for the current p has converged to within the acceptable tolerance, which requires the refinement iterations illustrated above. This is also apparent from the algorithm detailed in section 3.2. Fig. 3(b) shows the smooth variation of the converged solutions obtained for each ɛ (= p). In Fig. 3(c), we see that the domain inside which the dynamical system D 1 is solved is completely disconnected from the domain for the known system, D 0. However, through the iterative process of homotopic approximations, (of which only 4 are shown in this figure) the desired result is achievable. Finally, Fig. 3(d) shows the closeness of the final iteration on the dynamical system corresponding to p = ɛ = 1 (drawn surface) with the analytical result (shown with crosses), which is known in this case for p = ɛ = 1. 6 Conclusion A homotopic, iterative approach to solution refinement and domain determination for the stationary Fokker-Planck equation has been presented. A modification of the standard L 2 inner product by using the most recent approximation of the actual pdf has been shown to improve solution accuracy beyond that achievable by the standard inner product using the same number of degrees of freedom. This approach also provides a natural way to determine the solution domain for nonlinear systems by working through a one parameter family of dynamical systems. It has been shown that the use of the modified inner product leads to a stable iterative process (barring pathological cases like pdf s with δ-function like singularities) for a finite number of iterations, and convergence can be guaranteed for certain dynamical systems (although not in general). However, with the use of patching of adjacent approximations, superior convergence has been illustrated for significantly nonlinear systems. Therefore, it is possible with the current approach to obtain high accuracy with a small number of degrees of freedom - which is a significant advantage when the problem of interest resides in higher dimensions. References [1] H. Risken, The Fokker Planck Equation: Methods of Solution and Applications, Springer Series in Synergetics, Springer-Verlag,

24 (a) Analytical Result: Damped (b) Error Surface: Standard L 2 Approach Duffing Oscillator (c) Error Surface at the End of the Iterative Process (d) Comparative Convergence Characteristics: Duffing Oscillator (e) Ripple Formation in the Duffing Oscillator at the Boundary of Patching Solutions from Adjacent (f) Smoothing of the Ripples by the Transition Region between Low Iterations. and High Weightage Regions of the Weighting pdf. Fig. 1. Simulation Results for the Damped Duffing Oscillator [2] M. Kumar, P. Singla, S. Chakravorty and J. L. Junkins, The Partition of Unity Finite Element Approach to the Staitonary Fokker-Planck Equation, 2006 AIAA/AAS Astrodynamics Specialist Conf., Aug 21-24, 2006, Keystone, CO, USA. [3] A. T. Fuller, Analysis of Nonlinear Stochastic Systems by Means of the Fokker-Planck Equation, Int. J. Control, Vol. 9, No. 6, 1969, pp

25 (a) Analytical Result: System 2 (b) Error Surface at the end of the Iterative Process (c) Comparative Convergence Characteristics: System 2 Fig. 2. Simulation Results for System 2 [4] E. A. Johnson, S. F. Wojtkiewicz, L. A. Bergman and B. F. Spencer, Observations with Regard to Massively Parallel Computation for Monte Carlo Simulation of Stochastic Dynamical Systems, Int. J. Nonlin. Mech., Vol. 32, No. 4, 1997, pp [5] N. Harnpornchai, H.J. Pradlwarter and G.I. Schnëller, Stochastic Analysis of Dynamical Systems by Phase-Space-Controlled Monte Carlo Simulation, Comp. Meth. Appl. Mech. Engr., Vol. 168, 1999, pp [6] J. B. Roberts and P. D. Spanos, Random Vibration and Statistical Linearization, Dover Publications, [7] W. F. Wu and Y. K. Lin, Cumulant-Neglect Closure for Nonlinear Oscillators Under Parametric and External Excitations, Int. J. Nonlin. Mech., Vol.19, No. 4, 1984, pp [8] H. J. Pradlwarter and M. Vasta, Numerical Solution of the Fokker-Planck Equation via Gaussian Superposition Representation, Structural Safety and Reliability: Proc. ICOSSAR 97-7th Int. Conf. on Structural Safety and Reliability, Eds. N. Shairaishi, M. Shinozuka and Y. K. Wen, Vol. 2, 1997, Kyoto, Japan, pp [9] R. S. Park and D. Scheers, Nonlinear Mapping of Gaussian State Uncertainties: Theory and Applications to Spacecraft Control and 24

26 (a) Variation of the x-coordinate of the Stable Equilibrium with the Homotopy Parameter (p = ɛ). p = 1 p p = 0 (b) The Progression of Iterations from a Known Dynamical System, D 0 to the Unknown, D 1. Navigation, AAS/AIAA Astrodynamics Specialist Conference, 7-11 Aug., 2005, Lake Tahoe, CA, USA, AAS Paper No [10] H. J. Pradlwarter, Nonlinear Stochastic Response Distributions by Local Statistical Linearization, Int. J. Nonlin. Mech., Vol. 36, No. 7, 2001, pp [11] G. Muscolino, G. Ricciardi and M. Vasta, Stationary and Non Stationary Probability Density Function for Non-Linear Oscillators, Int. J. Nonlin. Mech., Vol. 32, No. 6, 1997, pp [12] M. Di. Paola and A. Sofi, Approximate Solution of the Fokker-Planck- Kolmogorov Equation, Prob. Engr. Mech., Vol. 17, 2002, pp [13] S. F. Wojtkiewicz, and L. A. Bergman and B. F. Spencer Jr., High Fidelity Numerical Solutions of the Fokker-Planck Equation, Proc. of ICOSSAR 97: 25

27 (c) Movement of the Domain of Solution Along the Iterative Procedure. (d) A Comparison of the Final Iteration with the True Solution. Fig. 3. Illustration of Space Homotopy by Variation of Dynamical Systems, D 0 D 1. the 7 th International Conference on Structural Safety and Reliability, Eds. A. Bazzani, J. Ellison, H. Mais and G. Turchetti, Nov 24-28, 1997, Kyoto, Japan. [14] E. A. Johnson, S. F. Wojtkiewicz, L. A. Bergman and B. F. SpencerJr., Finite Element and Finite Difference Solutions to the Transient Fokker-Planck Equation, Proc. of a Workshop: Nonlinear and Stochastic Beam Dynamics in Accelerators - A Challenge to Theoretical and Numerical Physics, Eds. A. Bazzani, J. Ellison, H. Mais and G. Turchetti, Lüneburg, Germany, 1997, [15] A. Masud and L. A. Bergman, Application of Multi-Scale Finite Element Methods to the Solution of the Fokker-Planck Equation, Comp. Meth. App. Mech. Engr., Accepted. 26