Probabilistic Engineering Mechanics 24 (2009) 265–277

Contents lists available at ScienceDirect

Probabilistic Engineering Mechanics

journal homepage: www.elsevier.com/locate/probengmech

A homotopic approach to domain determination and solution refinement for thestationary Fokker–Planck equationMrinal Kumar ∗,1, Suman Chakravorty 2, John L. Junkins 3Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA

a r t i c l e i n f o

Article history:Received 1 March 2007Received in revised form30 June 2008Accepted 24 July 2008Available online 3 August 2008

Keywords:Stationary Fokker–Planck equationGalerkin projectionHilbert space approximationHomotopic recursion

a b s t r a c t

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 estimateof the stationary probability density function, is shown to significantly improve the accuracy of theapproximation over the standard L2-norm based Galerkin error projection. The modified norm is arguedto be naturally suited to the problem, and hence preferable over the standard L2-norm, because the formerrequires substantially fewer degrees of freedom for the same order of approximation accuracy, making itimmediately attractive for the Fokker–Planck equation in higher dimensions. Additionally, it is shown thatthe modified norm can be utilized to progress through a homotopy of dynamical systems,Dp, in order todetermine the domain of the stationary distribution of a nonlinear system of interest, (corresponding top = 1) by starting with a known dynamical system (corresponding to p = 0) and working upwards. Thepartition of unity finite element method is used for numerical implementation. The meshless nature ofthis technique facilitates the application of themodified-norm approach to higher dimensional problems.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The subject of uncertainty propagation was introduced to thescientific community in the 1860s through the investigations ofMaxwell and Boltzmann into the random nature of gaseous mo-tion. Since then, the field has benefitted from the work of sev-eral giants, like Lord Rayleigh, Albert Einstein, Max Planck, AndrianFokker, Andrey Kolmogorov and several others. Consequently, fol-lowing a somewhat heuristic start, the field has grown with firmfoundations in the principles of stochastic differential equationsand probability theory. Today, the study of uncertainty propaga-tion through stochastic systems continues to permeate amultitudeof fields in science and engineering. In essence, it involves the studyof time evolution (andwhen it exists, the steady state) of the prob-ability density function (pdf ) that characterizes the underlyingstochastic process. The well known Fokker–Planck-Kolmogorovequation [1], or simply the Fokker–Planck equation (FPE) is of cen-tral importance in this context, because it captures the exact timeevolution of the state-pdf of dynamical systems driven by white-noise excitation. Thus, the FPE provides an exact description ofthe uncertainty propagation problem for stochastic systems un-der white-noise forcing [2]. Unfortunately, analytical solutions of

∗ Corresponding author.E-mail addresses:mrinal@neo.tamu.edu (M. Kumar),

schakrav@aeromail.tamu.edu (S. Chakravorty), junkins@tamu.edu (J.L. Junkins).1 Graduate Research Assistant.2 Assistant Professor.3 Distinguished Professor, Holder of the Royce Wisenbaker Chair.

0266-8920/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.probengmech.2008.07.006

the FPE exist only for linear dynamical systems, and a handfulof nonlinear systems possessing special structure [3]. Such sys-tems represent a very small fraction of the enormous variety ofstochastic systems encountered in science and engineering. There-fore, several approximate methods have been used for the pur-pose of uncertainty propagation through nonlinear systems. Somepopular methods include Monte Carlo simulations [4,5], Gaussianclosure [6–8] (or higher order closures) [9], equivalent lineariza-tion and stochastic averaging [6,10]. Despite the widespread useof these approximate methods, attempts to numerically solve theFPE have never been abandoned, because of the above mentionedfact that it provides the exact description of the uncertainty prop-agation problem under white noise excitation. In other words, thesolution of the FPE is valid in situations where all other approx-imate methods prove inadequate; namely, long term uncertaintypropagation and/or high degree of nonlinearity in the underlyingdynamical system.In the recent literature, several numerical methods have been

proposed for solving the FPE. These include global approximationtechniques [11,12], finite-difference (FD) [13] and finite-elementmethods (FEM) [14,16], multi-scale FEM [15], and the meshlessFEM [2,17]. Most of these techniques are based on the variationalformulation (weak form) of the FPE, and solve the equation inan integral/average sense over the solution domain. However, allthese methods encounter multiple difficulties that are inherentlyinvolved in attempting the FPE solution numerically [2]. Theprimary difficulties include computational costs associated withthe discretization of high dimensional state-spaces, positivity of

266 M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277

the obtained solution, and the non-existence of a unique solutiondomain of finite size and orientation, especially when no priorknowledge about the underlying dynamical system is available [2,18]. In addition, the solution thus obtained must satisfy thenormality constraint, in order to be a valid pdf . The issue of highdimensionality has been addressed in Kumar et al. [2], in which anefficient numerical scheme based on the local, meshless, partitionof unity finite element method (PUFEM) [20] has been utilized tosolve 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 innerproduct involved in the residual error projection is modified, andweighted by the most recent approximation of the true pdf . Thisis in contrast with the traditional Galerkin approach, in whichthe standard L2 space equipped with the Lebesgue measure isused as the projection space. We argue that the best weight tomodify the standard L2 inner product is the true solution of thestationary FPE. The resulting modified projection space is denotedby L(dΨ ?), where dΨ ? is the probability measure induced on RN

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 mostrecently obtained pdf is used as a weight to converge to thetrue solution. While the accuracy of the final iteration is limitedby the approximation-ability of the utilized basis functions, itis shown through numerical examples that for the same orderof accuracy, the weighted-norm approach requires substantiallyfewer degrees of freedom (i.e. the size of the discretized problem)than the traditional L2 approach. In other words, if the samenumber of degrees of freedom (DOFs) were used for a givenproblem, the method of weighted norms would provide muchbetter accuracy than the standard L2-norm approach. The FPE iswell known to suffer from the curse of dimensionality, i.e. the DOFsincrease exponentiallywith the dimensionality of the problem. Theweighted norm approach offers a relief from this curse by cuttingdown the problem size, leading to a smaller number of DOFs fora given level of accuracy. This augurs well for extension to theFPE in higher dimensions, because a modification of the norm cancurb the exponential increase in problem size. A similar approachhas also been used by other researchers to obtain an improvementin the approximation to the FPE [12]. In this paper, we prove thestability of this approach, by showing its closeness to the Hilbertprojection theorem. A similar result for the transient FPE has beenpresented in a different paper [18], for which both the stabilityand convergence proofs have been provided. Theoretically, theidea of weighted norms could be incorporated into any of thenumerical methods discussed above, by modifying the definitionof the inner product. In the current work, we utilize the partitionof unity finite elementmethod (PUFEM), which is a meshless finiteelement approach that offers several benefits over the traditionalFEM. Its meshless nature requires minimal bookkeeping in thedomain discretization process, leading to an easy extension tohigher dimensional problems. Furthermore, it offers scope for localrefinement, by permitting the use of higher order polynomials inselected regions of the solution domain. (local p-refinement) Thevarious local approximations are blended together using partitionof unity weight functions, also known as PU pasting functions. Thecurrent paper utilizes novel weight functions of polynomial form,that allow blending with any desired order of continuity acrosslocal domains. [19] Further details of this method can be found inKumar et al. [2]. In the current work, we shall restrict our attentionto the idea of weighted norms, while keeping in mind that PUFEMoffers the best numerical framework for the extension of this ideato higher dimensional problems.The current paper also discusses a recursive homotopic

approach for determination of the solution domain of a nonlinear

stochastic system of interest. It is shown that it is possible torecursively track the domain of distribution for the desired system,startingwith a dynamical systemwhose response is known.Hence,the proposed homotopic approach provides a method of tacklingthe issue of existence of no prior knowledge about the solution. Anumerical example is presented to illustrate how this approach canbe used to track the domain for the Duffing oscillator, by varyingits nonlinearity parameter.

2. The Fokker–Planck equation and its weak form

The Fokker–Planck equation provides the exact descriptionof the uncertainty propagation problem for dynamical sys-tems driven by white-noise excitation. Consider a general N-dimensional white-noise driven nonlinear dynamical system withuncertain initial conditions, given by the following stochastic dif-ferential equation:dx = f(t, x)dt + g(t, x)dW , E[x(t0)] = x0 (2.1)where, W represents a Wiener process with the correlationfunction Qδ(t1 − t2), and x0 represents the nominal initial state.The initial probability distribution of the state is given by thepdfW(t0, x), which captures the state uncertainty at time t0. Then,for the system given by (2.1), the time evolution of W(t0, x) isdescribed by the following FPE, (in Stratonovich form) which is asecond order, linear PDE inW(t, x):∂

∂tW(t, x) = LF P W(t, x) (2.2)

where,

LF P =

[−

N∑i=1

∂

∂xiD(1)i (.)+

N∑i=1

N∑j=1

∂2

∂xi∂xjD(2)ij (.)

], (2.3)

D(1)(t, x) = f(t, x)+12∂g(t, x)∂x

Qg(t, x), (2.4)

D(2)(t, x) =12g(t, x)QgT(t, x), (2.5)

where, LF P is called the Fokker–Planck operator, D(1) is knownas the drift coefficient vector and D(2) is the diffusion coefficientmatrix. Note that Eq. (2.4) represents the Stratonovich form of thedrift vector. There exists another formknown as the Itô form, whichis generally different from the Stratonovich form, and is consideredbymathematicians to be the rigorously correct expression for D(1).In engineering fields however, the Stratonovich form is preferredto avoid Itô calculus, which is required to deal with the Itô form.The two forms are identical in the case of state additive noise,i.e. when g(t, x) = g(t). This is typically the case with most reallife stochastic systems, and we consider only such systems in thispaper.Note that Eq. (2.2) represents the transient FPE. In the current

paper, we are interested in finding the time-invariant (stationary)solution to Eq. (2.2), i.e. LF P W(x) = 0. The conditions for theexistence of such a solution are well known, and can be foundin [3]. A necessary condition is that the system dynamics be timeinvariant, i.e. f(t, x) = f(x) and g(t, x) = g(x). Since we arecurrently interested in state-additive noise, we have g(t, x) =constant. The existence of finite intensity noise and at least oneattractor is also necessary for the existence of a nontrivial andunique stationary solution of the FPE [3].

2.1. FPE: Difficulties in numerical implementation

Solving the FPE numerically for the pdf , W , is a formidableproblem 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.

M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277 267

(3) Dimensionality: The Fokker–Planck operator in Eq. (2.3)involves partial derivatives with respect to all the states of theunderlying system. For example, for a dynamical system thatdescribes 2 dimensional motion, like planar motion of a pointmass under a central two-body gravitational force field, thecorresponding FPE would require the discretization of not justthe position coordinates, but also the velocity coordinates, i.e. atotal of four (state) ‘‘spatial’’ dimensions, in addition to time.For most numerical methods which involve a mesh-baseddiscretization procedure, this represents a major stumblingblock due to the enormous pre-processing required in gridgeneration, and the subsequent book-keeping of inter-elementboundaries for ensuring solution continuity during evaluationof the integrals.

(4) No unique solution domain for numerical implementation: Thetrue domain of solution of the FPE is RN . However, thediscretization procedure of any numerical method requiresa finite domain. Heuristic methods are typically used todefine a conservatively sized domain in order to includethe ‘‘significant’’ portion of the pdf to be approximated,say, W > 10−9. If no prior knowledge is available aboutthe dynamical system under consideration (especially highlynonlinear systems), it is generally difficult to obtain thelocation, orientation, and size of such a finite domain thatachieves the mentioned tolerance.

Issues 1 and 2 represent additional constraints that the solutionof Eq. (2.2) must satisfy in order to be a valid pdf . Since theseconstraints are not built into the structure of the FPE, theymust be accommodated in the numerical solution. While (2)can be enforced by a simple renormalization of the obtainedsolution, (1) is a tough proposition. Several researchers haveused a log-transformation of the FPE to ensure positivity (theinverse transform (exponential) applied to the solution obtainedin the transformed coordinates ensures positive values) [11,12].However, this approach converts the linear PDE (Eq. (2.2)) into anonlinear PDE, which is generally not desirable. Dimensionality(issue 3) continues to be the primary deterring factor preventingnumerical methods from ready application to the FPE for generalnonlinear systems. Recently, promising success in computationalefficiency and time of execution has been shown with the useof meshless methods, like MLPG (Meshless Local Petrov Galerkinmethod [21]) and PUFEM [2,17]. In this paper,we primarily addressthe issue of domain determination and solution refinement for thestationary FPE with an iterative homotopic approach that employsa modified inner product for the residual error projection. Weutilize the meshless PUFEM approach for developing the weakform approximation. Wemention that any approximationmethodis equally compatible with this iterative approach. PUFEM hasbeen chosen for its several computational advantages [2] and easyextension to higher dimensional problems.

2.2. The variational formulation (weak form) of the stationary FPE

The variational formulation of the stationary FPE involves thedetermination of a solution W ∈ U, such that the following systemof projection equations is satisfied:∫Ω

LF P (W)υdΩ = 0, ∀υ ∈ V. (2.6)

Generally, the approximation space U (also known as the trialspace) and the projection space V (also called the test space) areinfinite dimensional. Therefore, we have:

W(x) =∞∑k=1

ckφk(x). (2.7)

For the numerical implementation of the variational form, theapproximation and projection spaces are truncated to finitedimensional subspaces, i.e. the trial and test functions are chosen

from finite dimensional subspaces Un ⊂ U and Vn ⊂ Vrespectively. We have:

W(x) =n∑k=1

ckφk(x). (2.8)

In the Galerkin approach, the residual error resulting from thetruncated approximation, is projected onto the space of the trialfunctions, i.e. Vn = Un, or υk = φk. We note here that inthe case of the transient FPE, the fourier coefficients ck would betime varying, and (2.8) would be equivalent to the separation ofvariables. Writing (2.6) using the inner product notation, we get:⟨

LF P

(n∑k=1

ckφk

), φj

⟩= 0, j = 1, 2, . . . n. (2.9)

In the above equation, 〈., .〉 represents the standard inner producton L2(Ω). In the next section, we discuss the modification of theabove inner product for solution refinement of the stationary FPE.

3. Modification of the L2 inner product and space homotopy

The variational formulation of the FPE described in the abovesection, employs the traditional L2 inner product. We claim thatthere exists a natural measure for the above problem, whichdefines a modified inner product, and which can be used to ouradvantage to obtain accurate approximations with a small numberof degrees of freedom. This natural measure is characterized by thetrue solution of the FPE. Besides redefining the inner product, it alsoimplicitly defines the domain of solution byprovidingweightage toonly the significant regions of the pdf . The obvious problem withusing the actual solution as the weight is that it is unknown. Wetherefore develop an iterative scheme in which the most recentapproximation of the true solution is used to weight the L2 innerproduct. In the following development, we show the closeness ofthis approach to the optimal approximation of the pdf obtainedfrom the normal equations derived from the Hilbert projectiontheorem.Webeginwith the following assumptions on the approximation

spaceUn and the initial estimate W0, of the true solution Ψ ?:

Assumption 3.1. The true solution is exactly approximable by thetrial spaceUn using the Hilbert projection theorem, i.e. there existck, k = 1, 2, . . . , n, such that:

Ψ ?(x) =n∑k=1

ckφk(x).

Assumption 3.2. A ‘‘sufficiently’’ close approximation W of thetrue solution is available to start the iterative process:

‖W − Ψ ?‖ < ε.

Assumption 3.3. The basis functions φk form an orthonormal setwith respect to the standard Euclidean inner product, i.e. L2(Ω).Assumption 3.1 has been made primarily for convenience, and

is equivalent to saying that the approximation space Un equalsU∞. It can be relaxed to read ‘‘sufficiently well approximable’’ (towithin ε?) instead of ‘‘exactly approximable’’, and the results stillhold but the mathematical development becomes tedious withoutadding significant insight. The stability proof that follows dependson the closeness of the starting approximation, i.e. Assumption 3.2.Thus, if the L2 error norm of the initial approximation is boundabove by ε, it is shown below that the error norm resulting fromthe next step of iteration is at most scaled by a constant factor. Ifthe scaling factor (which depends on the particular system underconsideration) is less than 1, a contraction mapping is obtainedand convergence follows, but in general this might not be thecase. Finally, Assumption 3.3 is made also purely for the sake of

268 M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277

convenience of evaluating integrals, and the actual approximationspace chosen need not satisfy this condition.In the following, we set up the equations for the Hilbert

projection approach to find the coefficients ck in Assumption 3.1.We redefine the inner product 〈., .〉 as the following:

〈φi, φj〉 ,

∫Ω∈RN

φi(x)φj(x)Ψ ?(x)dx. (3.1)

Then, the Hilbert coefficients, ck for the true solution ψ? are givenby the following equation:

n∑k=1

ck⟨φk, φj

⟩=⟨Ψ ?, φj

⟩j = 1, 2, . . . , n. (3.2)

Following Assumption 3.3, ck = 〈Ψ ?, φk〉.Next, we define a new inner product, 〈〈., .〉〉, which is induced

on the solution domainΩ by the current approximation W , of thetrue solution ψ?:

〈〈φi, φj〉〉 ,

∫Ω∈RN

φi(x)φj(x)W(x)dx. (3.3)

Using the above inner product, the projection equation (2.9) for thevariational formulation can be rewritten as the following:⟨⟨

LF P

(n∑k=1

c ′kφk

), φj

⟩⟩+ α

⟨⟨n∑k=1

c ′kφk, φj

⟩⟩Γ

−⟨⟨Ψ ?, φj

⟩⟩Γ

= 0 j = 1, 2, . . . n, (3.4)

where, c ′k denote the fourier coefficients of the approximationobtained from theweighted Galerkin approximation shown above.Also, α is a penalty parameter which has been introducedto enforce the boundary conditions, and 〈〈., .〉〉Γ denotes theevaluation 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,ck and c ′k, corresponding to the Hilbert projection methodand the Galerkin method weighted with the most recent pdfapproximation respectively. Eq. (3.4) represents a system of linearequations in c ′k, which can be expressed as follows:

B′C′ + B′Γ C′= F′Γ . (3.5)

Next, following Assumption 3.1, the Hilbert approximation ofψ?(x) satisfies the Galerkin variational form of the stationary FPEexactly, i.e.:⟨

LF P

(n∑k=1

ckφk

), φj

⟩+ α

⟨n∑k=1

ckφk, φj

⟩Γ

−⟨Ψ ?, φj

⟩Γ

= 0 j = 1, 2, . . . n. (3.6)

Note that the inner product in the Hilbert projection equation isweighted by the true solution, ψ?, and hence the notation 〈., .〉 isused. Then, Eq. (3.6) reduces to the following linear system:

BC+ BΓ C = FΓ . (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 =[〈LF P (φk), φj〉

], (3.8)

BΓ = α[〈φk, φj〉Γ

], (3.9)

FΓ = α[〈Ψ ?, φj〉Γ

], (3.10)

B′ =[〈〈LF P (φk), φj〉〉

], (3.11)

B′Γ = α[〈〈φk, φj〉〉Γ

], (3.12)

F′Γ = α[〈〈Ψ ?, φj〉〉Γ

]. (3.13)

As the first step towards showing the closeness of C′ to C, we provethe proximity of Eqs. (3.5)–(3.7) and write Eq. (3.5) as:

BC′ + (BΓ +∆3)C′ = FΓ +∆1 +∆2. (3.14)

Comparing Eqs. (3.5) and (3.14), we have:

∆1 = F′Γ − FΓ , (3.15)

∆2 = BC′ − B′C′, (3.16)

∆3 = B′Γ − BΓ . (3.17)

Then,wehave the following lemma for the upper bounds of various∆i:

Lemma 3.4. Given the validity of Assumptions 3.1 and 3.2, thefollowing inequalities hold:

‖∆1‖ ≤ K1ε,‖∆2‖ ≤ K2‖LF P ‖ε,

‖∆3‖ ≤ K3ε,

where, K1–K3 are finite constants, ‖.‖ represents the Euclidean normfor the vectors ∆1 and ∆2, and the matrix norm induced by theEuclidean norm for ∆3, and ‖LF P ‖ represents the operator norm ofthe Fokker–Planck operator.

Proof 1. Consider∆1 = [δ1j ]:

δ1j = α〈〈Ψ ?, φj〉〉Γ − 〈Ψ?, φj〉Γ

= α

∫Γ

Ψ ?φj(W − Ψ?)dΓ (3.18)

⇒ |δ1j |2≤ |α|2

∫Γ

∣∣Ψ ?∣∣2 ∣∣φj∣∣2 |W − Ψ ?

|2dΓ (3.19)

≤ |α|2∫Γ

∣∣Ψ ?∣∣2 |W − Ψ ?

|2dΓ

∫Γ

|φj|2dΓ (3.20)

≤ |α|2∫Γ

∣∣Ψ ?∣∣2 |W − Ψ ?

|2dΓ .1 (3.21)

≤ |α|2∫Γ

∣∣Ψ ?∣∣2 dΓ ∫

Γ

|W − Ψ ?|2dΓ (3.22)

≤ |α|2.1.ε2 (3.23)

⇒ |δ1j | ≤ |α|ε. (3.24)

In the above, the Cauchy–Schwarz inequality has been appliedin going fromEqs. (3.19) and (3.20), and fromEqs. (3.21) and (3.22).Additionally, weaker forms of Assumptions 3.2 and 3.3 (since onlyboundary integrals are involved) have been used in Eqs. (3.21) and(3.23). Thus, from Eq. (3.24), we conclude that there exists K1 <∞such that:

‖∆1‖ ≤ K1ε. (3.25)

Next, looking at ∆2 = [δ2j ], and following similar arguments asabove, we obtain:

M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277 269

δ2j =

⟨n∑k=1

c ′kLF P (φk), φj

⟩−

⟨⟨n∑k=1

c ′kLF P (φk), φj

⟩⟩

=

∫Ω

n∑k=1

c ′kLF P (φk)(Ψ?− W)dΩ (3.26)

⇒ |δ2j |2≤

∫Ω

∣∣∣∣∣ n∑k=1

c ′kLF P (φk)

∣∣∣∣∣2

|W − Ψ ?|2dΩ

≤

∥∥∥∥∥LF P

(n∑k=1

c ′kφk

)∥∥∥∥∥2

ε2

≤ ‖LF P ‖2

∥∥∥∥∥ n∑k=1

c ′kφk

∥∥∥∥∥2

ε2

≤ ‖LF P ‖2‖C′‖2ε2. (3.27)

In Eq. (3.27), the norm of the Galerkin coefficient vector, ‖C′‖ is afinite quantity, because it contains the coefficients of the variousbasis functions used to approximate pdf s that have well behavedfunctional forms (i.e. without δ-function like singularities). Hence,bounding it above by a finite quantity, we can show that thereexists a K2 <∞ such that:

‖∆2‖ ≤ K2‖LF P ‖ε. (3.28)

Finally, considering∆3 = [δ3kj]:

δ3kj = α〈〈φk, φj〉〉Γ − 〈φk, φj〉Γ

= α

∫Γ

φkφj(W − Ψ?)dΓ

⇒ |δ3kj|2≤ |α|2

∫Γ

|φk|2|φj|

2|W − Ψ ?

|2dΓ

≤ |α|21.1.ε2. (3.29)

A weak form of Assumption 3.3 (over the boundary) has been usedin Eq. (3.29). Thus, ∃K3 <∞, such that:

‖∆3‖ ≤ K3ε. (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 approximationresulting from the weighted Galerkin approach. We make thefollowing additional assumptions:

Assumption 3.5. The quantity ε is small enough such that ‖(B +BΓ )−1‖‖∆3‖ ≤ 1.

Assumption 3.6. The operator norm of Fokker–Planck operator isbounded above as ‖LF P ‖ = M <∞.

This leads us to the following result:

Lemma 3.7. Given the validity of Assumptions 3.5 and 3.6, thefollowing upper bound exists on the L2 error norm between theweighted Galerkin and Hilbert approximations of the FPE:

‖C′ − C‖ ≤ Kε. (3.31)

Proof 2. Let us adopt the following notation: B′ + B′Γ = BG, and∆1 +∆2 = ∆Σ . Then, Eqs. (3.7) and (3.14) become:

(BG −∆3)C = F′Γ −∆Σ . (3.32)

BGC′ = F′Γ . (3.33)

Thus, we have:

C′ − C = B−1G F′

Γ − (BG −∆3)−1(F′Γ −∆Σ )

= B−1G − (BG −∆3)−1F′Γ + (BG −∆3)

−1∆Σ .

Taking the norm (standard L2) on both sides, and applying thetriangle inequality,

‖C′ − C‖ ≤ ‖B−1G − (BG −∆3)−1‖‖F′Γ ‖

+‖(BG +∆3)−1‖‖∆Σ‖. (3.34)

Furthermore, following Assumption 3.5, we obtain the followingexpansion:

(BG −∆3)−1 = B−1G + B−2G ∆3 − · · · .

Thus, using the result for the upper bound of ∆3 from Lemma 3.4Eq. (3.30), we obtain:

‖(BG −∆3)−1‖ ≤ ‖B−1G ‖ + ‖B−1G ‖

2K3ε,

‖B−1G − (BG −∆3)−1‖ ≤ ‖B−1G ‖

2K3ε.

Also, combining Eqs. (3.25) and (3.28):

‖∆Σ‖ ≤ KΣ (1+ ‖LF P ‖)ε, (3.35)

where KΣ = max(K1, K2). Denoting ‖B−1G ‖ as P , ‖F′

Γ ‖ as Q , and‖LF P ‖ asM , Eq. (3.34) becomes:

‖C′ − C‖ ≤ QP2K3ε + KΣ (P + P2K3ε)(1+M)ε.

Dropping out terms of order higher than O(ε), we get:

‖C′ − C‖ ≤ (QP2K3 + PKΣ (1+M))ε,⇒ ‖C′ − C‖ ≤ Kε.

This completes the proof of the lemma.

Therefore, we see that the error in the next iteration of theprocess is scaled by the constant K , which is comprised of severalnorms associated with the underlying system. If this quantity isless than 1, we obtain a contractionmapping and the error reducesto zero in the limit. However, this is not true in general. In eithercase, the method is stable for a finite number of iterations, andwill not lead to divergence (except in certain pathological casesdiscussed below). Superior convergence characteristics has beenshown for dynamical systems in twoand three dimensions throughnumerical simulations in the results section.Looking closer at the constant K , we observe that the norm

of the inverse of the Hilbert stiffness matrix, BG appears in itsexpression. If this matrix is ill-conditioned or singular, the methodloses its stability. This situation may arise in certain conditions(e.g. local approximation schemes which involve degrees offreedom with local domain of influence) and is discussed in detailbelow. On the other hand, the norm of the vector F does not causeproblems as it involves the integral of the true solution alongthe domain boundary, which is a very small quantity (10−9 orlower). In the numerical examples shown below, we show thatconvergence is achievable using the (local) PUFEM algorithm forthe variational formulation, in conjunction with suitable patchingof solutions from successive iterations.

3.2. Space homotopy for domain determination

In the above section, we assumed that the solution domain onwhich the iterations are carried out is known a-priori. In general,this might not be the case, especially for nonlinear systems. In thissection, we demonstrate the implementation of a space homotopyvia a family of single parameter dynamical systems to track thedomain of the stationary distribution for the systemof interest. The

270 M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277

underlying assumption is the existence of a family of dynamicalsystems,Dp indexed by the homotopy parameter p:

Dp : dx = f (x, p)dt + g(x, p)dW , p ∈ [0, 1], (3.36)where D1 corresponds to the dynamical system of interest, andD0 corresponds to a stochastic dynamical system whose responseis known, i.e., the stationary FPE associated with it can be solved.Let ψ?

p (x) denote the true solution of the FPE associated withdynamical system Dp. We make the following assumption aboutthe family of dynamical systems Dp and the solution of theassociated FPE’s, ψ?

p :

Assumption 3.8. Given any p ∈ [0, 1], and any ε > 0, there existsδ > 0, such that for all p′ ∈ Bδ(p), (open ball of radius δ centeredat p) ‖ψ?

p (x)− ψ?p′(x)‖ ≤ ε.

In essence, the above assumption assumes the existence of aoneparameter family of dynamical systems, such that the solutionsto 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 (henceleading to a contraction mapping and ensuring convergence). Wethen state the following obvious result as a proposition:

Proposition 3.9. Consider dynamical systems Dp with K < 1 inLemma 3.7. Then, given ψ1, such that ‖ψ1 − Ψ ?

‖ ≤ ε and that ε issufficiently small, a sequence of functions ψn∞n=1 can be constructedrecursively, starting with ψ1 such that ‖ψn − Ψ ?

‖ → 0 as n →∞,∀p ∈ [0, 1].

The proof is trivial, because of Lemma 3.7 and the contractionmapping 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 iNi=1. However, when we write ψ?i ,

we refer to the true solution of the FPE for the dynamical systemDp=i.Then, with Assumption 3.8 in mind, we have the following

result pertaining to how the solution of the FPE associatedwith our system of interest (D1), i.e., ψ?

1 (x) = Ψ ?(x) can beobtained recursively given the knowledge of the solution of the FPEassociated with the system D0. The result uses Proposition 3.9 inconjunction with successive approximation.

Proposition 3.10. Let εp be sufficiently small, such that Proposi-tion 3.9 is satisfied for any ψ satisfying ‖ψ − ψ?

p‖ ≤ εp. Letinfp∈[0,1] εp = ε > 0. Then, under Assumptions 3.1–3.3, 3.5, 3.6 and3.8, givenψ?

0 , the exact solution of the FPE corresponding toD0, thereexists a finite sequence of functions ψnMn=1 s.t.ψ

M= Ψ ∗. Moreover,

this sequence can be obtained in a recursive fashion starting withψ?0 .

(i.e. ψ1 = ψ?0 ).

Proof 3. Let δp be such that if p′ ∈ Bδp(p) then ‖ψ?p − ψ

?p′‖ ≤

ε2 .

Note that this is possible due to Assumption 3.8.Consider the open covering

⋃p∈[0,1] Bδp(p) of the set [0, 1].

Since [0, 1] is compact, there exists a finite subcover of [0, 1] givenby⋃Mi=1 Bδpi (pi).

Let δi ≡ δpi(pi) and ψ?pi ≡ ψ

?i .

Let us assume thatψ?i is known and we need to obtainψ

?i+1. By

definition, there exists a p s.t. |p − pi| < δi and |p − pi+1| < δi+1.Then, it follows from construction that

‖ψ?i − ψ

?i+1‖ ≤ ‖ψ

?i − ψ

?p‖ + ‖ψ

?p − ψ

?i+1‖ ≤ ε. (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 thesequence 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 inconjunction with solution refinement) can be presented as thefollowing algorithm:(1) Find a homotopy of dynamical systems Dp, p ∈ [0, 1], suchthat Dp(p = 1) corresponds to the system of interest andDp(p = 0) corresponds to a known system, in the sense thatits associated stationary FPE can be solved.

(2) Select a finite number of points pi ∈ [0, 1], i = 1, . . . ,M;that are ‘‘sufficiently’’ close. For the rest of the algorithm, referto the dynamical system corresponding to pi, namely Dpi , byits index, i.e. as Di, and the true solution of the associatedstationary FPE, ψ?

pi as ψ?i . Notice that the selection of points pi

can be done online—i.e. if pi+1 is found to be ‘‘not close enough’’to pi, it is possible to go back and redo the previous iteration.

(3) Following the new ‘‘index based’’ notation, notice that theexact solution for systemD1 is known (ψ?

1 ). Also, the solutionwe are after (for p = 1) is, in the new notation, ψ?

M = Ψ?. Set

i = 2.(4) Determine the solution ψ?

i in the following manner:(a) Set j = 1 and the current weight for norm modification,W = ψ?

i−1.(b) Using W as the weight in the modified norm approach,obtain ψ ji , i.e. the jth approximation for ψ

?i .

(c) If ψ ji = ψ?i , goto step 5. Else, set j = j + 1 andW = ψ

j−1i

and goto step 4(b).(5) If i = M , stop. Else, set i = i+ 1 and goto step 4(a).The above algorithm involves two loops. The outer loop runs

over the homotopic sequence of dynamical systems, from the‘‘known’’ to the ‘‘desired’’. The inner loop performs successiverefinements upon the solution obtained for each dynamicalsystem, by utilizing the modified-norm approach. The algorithmstarts with the known system (p = 0) whose solution for thestationary FPE is available, serving as the first weight for norm-modification in the outlined approach. A schematic of the abovealgorithm is presented in Section 5, (see Fig. 6) which depictspictorially its salient features and the two loops described above.We mention here that measuring error in the inner loop (as tohow close we are to the true solution for any particular dynamicalsystem) 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 tomeasure the closeness of approximation to the truth.

4. Numerical implementation

In this section, we discuss the use of the PUFEM algorithmfor the numerical implementation of the above methodology,and discuss the associated difficulties and suggest possible fixes.PUFEM is a powerful local, meshless, node-based finite elementapproximation method, which has been used successfully to solveseveral difficult partial differential equations. Its application tothe stationary FPE has been discussed in detail in [2]. Here, weonly provide the basic equations of variational formulation of thestationary FPE with this method. The PUFEM approximation of thepdf in the nth iteration, Wn, can be written as:

Wn(x) =P∑s=1

Qs∑k=1

c ′skϕs(x)ζsk(x), (4.1)

where, ϕs(x) are node-centered overlapping compactly supportedpositive weight functions which bring about an implicit discretiza-tion [2] of the solution domainΩ and satisfy the property of parti-tion of unity, i.e.

∑s ϕs(x) = 1,∀x ∈ Ω . The functions ζsk(x), k =

1, . . . ,Qs are basis functions used in the approximation space, de-fined locally on the compact support of the corresponding parti-tion of unity function ϕs. The product ϕsζsk is called the shapefunction, and the set forms a conforming approximation space over

M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277 271

the global solution domainΩ . It is important to note here that thecoefficients c ′sk represent the amplitudes of the local shape func-tions inside compactly supported domains. Additional details canbe obtained from [2]. Using this approximation space, theweightedGalerkin variational equations for the stationary FPE are given by:∫Ωs

QsP∑i=1

LF P (c ′iφi)φjWn−1dΩ + α∫Γs∩Γ

QsP∑i=1

c ′iφiφjWn−1dΓ

= α

∫Γs∩Γ

Ψ ?φjWn−1dΓ , j = 1, . . . ,QsP (4.2)

where, φi = ϕs(x)ψsk(x), Ωs is the local domain of influence ofthe shape functions ϕsψsk, i.e. the compact support of ϕs. Similarly,Γs ∩ Γ represents the intersection of the local element boundarywith the global domain boundary. The resulting elements of thematrices involved in the linear system of Eq. (3.5) are:

B′ij =∫Ωs

LF P (ϕkψkl)ϕpψpqWn−1dΩ, (4.3)

B′Γ ij = α∫Γs∩Γ

ϕkψklϕpψpqWn−1dΓ , (4.4)

F ′Γ j = α∫Γs∩Γ

Ψ ?ϕpψpqWn−1dΓ , (4.5)

where, i =(∑k−1

s=1 Qs + l)and j =

(∑p−1s=1 Qs + q

).

4.1. Conditioning of the stiffness matrix

As seen in Section 3.1, the condition number of the stiffnessmatrix BG is an important issue in the stability of the aboveapproach. Unfortunately, if a local approximation scheme suchas PUFEM is used for either the weighted Galerkin or Hilbertapproaches, the stiffness matrix invariably turns out to be ill-conditioned. This is because of the following reason—the pdf usedas the weight gives relative weightage to different regions of thedomain, 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 alocal scheme, the shape functions and their coefficients (ci orc ′i ) have local influence. In other words, the integrals associatedwith the shape functions close to the boundary are evaluated onlocal domains only near the boundary region. By virtue of theexponentially low weight given to these regions by the weightingpdf , these integrals get nearly ‘‘washed out’’ in comparison withthe integrals evaluated on local domains close to the mean.Consequently, the entries in the stiffness matrix B correspondingto coefficients of the local shape functions near the boundarydomains diminish severely in comparison with the entries forcoefficients of the shape functions in the interior. This effect makesthe boundary coefficients unobservable, and the resulting stiffnessmatrix numerically ill-conditioned.

4.1.1. A numerical fixWe thus conclude that using a pdf as the weight for the

modification of the inner product causes ill-conditioning ofthe stiffness matrix in local approximation techniques, becauseit renders the local coefficients near the boundary regionsunobservable. A natural solution to this problem is to extract theportion of the stiffness matrix which has acceptable conditioningfor inversion, and to retain the solution for the remainingcoefficients from the previous iteration. In this manner, not allcoefficients are modified in going from one iteration to another, asthe coefficients close to the boundary do not change. This methodalso gives a simple way of trimming the domain of the solutionfrom one iteration to the next—by identifying and pruning regionswhich receive weightage below a specified tolerance from the

weighting pdf . However, we mention that selective modificationof the coefficients usually leads to discontinuities and/or rippleformation in and around the concerned local domains. To counterthis, the two sets of coefficients (from the current and the previousiterations) are patched together to produce a smooth surface. This‘‘patching’’ procedure can be done using the PUFEMalgorithmwiththe help of blending functions, such as thosementioned in [2]. Thisapproach provided highly acceptable results as illustrated in thenext section.

5. Results

In this section, we present numerical examples to illustratethe theoretical ideas presented above. We show that with theweighted norm approach, it is possible to obtain high accuracy,while using a small number of degrees of freedom.

5.1. Solution refinement of the stationary FPE: Results

We first consider two nonlinear systems residing in 2D state-space described below:

5.1.1. System 1Consider the following 2-D damped Duffing oscillator:

x+ ηx+ αx+ βx3 = gG(t). (5.1)

We assign the parameters appearing above the following values:α = −15, β = 30, η = 10, g = 1 (soft-spring case). Theanalytical solution of the stationary FPE for the above system isgiven by the following expression:

Ws(x, x) = C exp−2

η

g2Q

[αx2

2+βx4

4+x2

2

], (5.2)

where, C is a normalization constant. Fig. 1(a) shows the truestationary distribution for this system, which is a bimodal pdf .

5.1.2. System 2Consider the following 2-D nonlinear oscillator [11]:

x+ β x+ x+ α(x2 + x2)x = gG(t). (5.3)

We set the following values: α = 0.125, β = −0.5, g = 0.86.The analytical solution of the stationary FPE for the above systemis known, and given by the following expression:

Ws(x, x) = C exp−12g2

[α2(x2 + x2)2 + β(x2 + x2)

]. (5.4)

From Fig. 2(a), we see that from the top-view, the truestationary distribution for this system looks like a ring. Notice thatthe stationary distributions for both systems are exponentials of apolynomial function.As the first exercise, we evaluate the various norms involved

to ensure that the systems described above conform with thetheory presented in Section 3. In particular, we demonstrate thatthe numerical fix suggested in Section 4.1.1 to tackle the issue ofunobservability of the boundary nodes indeed causes the constantK appearing in Lemma 3.7 to be less than unity, hence leadingto a contraction mapping, which in turn implies convergence.Table 1 contains the various quantities that appear in Lemmas 3.4and 3.7. These are ballpark numbers and give order of magnitudeestimates. The constants K1–K3 appearing in Lemma 3.4 havebeen computed by evaluating the various domain and boundaryintegrals. The operator norm, ‖LF P ‖ has been computed viadiscretization. Notice that using a pdf as the weight to modifythe L2 norm causes the stiffness matrix to be nearly singular.

272 M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277

(a) True solution: damped duffing oscillator. (b) Error surface: standard L2 approach.

(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 ofthe transition region between low and high weightage regions ofthe weighting pdf .

(f) Smoothing of the ripples by patching solutions from adjacentiterations.

Fig. 1. Simulation results for the damped duffing oscillator.

However, the numerical fix suggested in Section 4.1.1 brings downthe ill-conditioning significantly, enough to make the constant Kappearing in Lemma 3.7 less than unity. Notice however,that thevalue of K suggests that convergence is expected to be faster forsystem1 than system2. Thiswas indeed observed and is illustratedin the convergence plots presented below.We now proceed to the actual results of solution refinement

for the described systems. Fig. 1(a)–(f) show results for system 1

(soft-spring Duffing oscillator). Fig. 1(b) shows the error surfaceusing the PUFEM algorithm with the standard L2 inner productapproach on a 16 × 16 grid equipped with local quadratic basisfunctions. This error surface serves as a reference for the standardL2 approach. We next perform the iterative refinement process,starting with the L2 solution computed on a much coarser grid(12 × 12). This solution is also the weight for inner-productmodification in the first iteration. Upon using the modified inner-

M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277 273

(a) True solution: 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.

Table 1Approximate estimates of various norms and constants appearing in the theory, forsystems 1 and 2

Quantity/Norm System 1 System 2

K1 10−9 10−9

K2 10−5 10−5K3 1 1‖LF P ‖ = M 41.13 7.87‖B−1G ‖ = P , before fix 2.25× 108 7.92× 109

‖B−1G ‖fixed = Pfixed 1.96× 103 1.04× 104

‖F′Γ ‖ = Q 2.44× 10−9 3.57× 10−9K (Lemma 3.7) 0.835 0.961

The numerical fix proposed in Section 4.1.1 ensures convergence by improving theconditioning of the stiffness matrix.

product approach in conjunction with patching of neighboringiteration approximations, the accuracy improves significantly,which is evident in the error surface shown in Fig. 1(c). The truepower of this approach is illustrated in Fig. 1(d), in which theconvergence characteristics for three comparable methods havebeen shown. The graph corresponding to the iterative PUFEMshows that the process is commenced on a coarse 12 × 12 grid,and the use of the pdf obtained after every iteration to improvethe inner-product space for the subsequent iteration, leads tosignificant drops in error. Once no further accuracy is possiblewiththe 12 × 12 mesh, a switch is made to a finer grid (14 × 14),beginning with the last pdf obtained from the previous (12 × 12)grid as theweight for the first iteration on thenewgrid. The spacingbetween circles on the iterative PUFEM graph illustrates the drop

in error after individual iterations. Thus, huddling of the circlessignifies saturation on a particular grid, and a switch to a finer gridis made following such behavior. In the graph shown, iterationshave been terminated after saturation of the (16 × 16) grid, andthe final error surface is shown in Fig. 1(c). The most significantcontribution of this result is that it shows that it is possible toachieve extremely accurate approximations with a small numberof degrees of freedom. For example, compare in Fig. 1(d) the errorafter the final iteration on the 16 × 16 grid (≡ 1536 PUFEM DOFsusing quadratic bases)with the error of the L2 approachon a30×30grid (≡ 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 bypruning out unobservable coefficients which are weighted outby the pdf . As expected, ripples form on either side of the twoweighty modes, where the pdf drops off suddenly to extremelysmall values on either side. However, the process of patching thecurrent solution with the previous iteration smoothes out theseripples, 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 ofapproximation nodes, as compared with the standard L2 approach.However, the results are not as drastic here, because for thissystem, it is possible to obtain fairly accurate results with eventhe standard L2 approach. Also, the convergence rate is slower asvisible in Fig. 2(c), which is also evident from the numerical valueof constant K in Table 1.Wemention thatwe have obtained similarencouraging results for several other 2-D oscillators.

274 M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277

(a) Computed x1 − x2 marginal distribution for system 3 at the end ofthe iterative process.

(b) True x1 − x2 marginal probability density function for system 3.

(c) Error surface resulting from standard L2 error projection using2800 DOFs.

(d) Error surface at the end of the iterative process.

Fig. 3. Simulation results for system 3.

Fig. 4. Comparative convergence characteristics for the three dimensional system.

5.1.3. System 3: Example in 3D state-spaceConsider the following dynamical systems studied in

Wojtkiewicz et al. [22]:

x =

[ 0 1 0−ω0 −2ζω0 10 0 −α

]x+

[001

]w(t). (5.5)

The constants appearing in the above equation have the followingvalues [22]: α = ω0 = 1, and ζ = 0.2. The stationary FPEfor the system above was solved in [22] using the traditionalFEM approach with ‘‘brick’’ elements. RMS error (defined as e2 ,

√1r−1

∑ri=1(W(xi)− W(xi))2, r = number of test points) reported

with this method was e2(FEM) = 1.133 × 10−4, using 125,000bricks. The same problem was solved in the current work on a6 × 6 × 6 grid, utilizing the local p-refinement feature of thePUFEM algorithm. Cubic polynomials were allocated to nodesin the interior region of the domain and quadratic polynomialsto the boundary nodes. This polynomial assignment results ina problem size of 2800 DOFs. The standard L2-norm approachresults in an RMS error of e2(6 × 6 × 6, cubic+ quadratic, L2) =1.336 × 10−3. This approximation was used as the first weight tocommence solution refinementwith themodified-norm approach.The obtained results are shown in Fig. 3. Fig. 3(a) and (b) show theconverged x1 − x2 marginal surface alongside the true marginaldistribution for the system. The iterative process was found toreduce the above stated RMS error of L2 approach with 2800DOFs by about one order of magnitude, down to e2(6 × 6 ×6, cubic+ quadratic,modified-norm) = 1.984 × 10−4. Fig. 3(c)and (d) show the error surfaces before and after the iterativesolution refinement process. The error reduction is clearly visiblein these plots. In Fig. 4we compare the convergence characteristicsof the L2-norm approach with the modified-norm method. On thex-axis, we show the size of the discretized problem (i.e. numberof degrees of freedom used in the approximation) and the y-axis shows the RMS error in the obtained approximation. Theunfeasible region in the right section of the figure demarcatedby the dash-dot line depicts problem sizes that are beyond thecapacity of the computational resources available to the authors.The dashed horizontal line clearly shows that the modified-norm approach provides slightly better accuracy than the best

M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277 275

(a) Variation of the x-coordinate of the stable equilibrium with thehomotopy parameter (p = ε).

(b) The progression of iterations from a known dynamicalsystem,D0 , to the unknown,D1 .

(c) Movement of the domain of solution along the iterative procedure. (d) A comparison of the final iteration with the true solution.

Fig. 5. Illustration of space homotopy by variation of dynamical systems,D0 −D1 .

approximation obtained with the standard L2 approach, withabout half the number of DOFs. (the standard L2 approach ona 7 × 7 × 7 grid with quartic polynomials in the interiornodes and linear polynomials in the boundary nodes results inan RMS error of e2(7 × 7 × 7, quartic+ linear, L2) = 2.823 ×10−4 and the resulting problem size is 5247 DOFs) Furthermore,both approaches (standard L2 PUFEM and modified-norm PUFEM)require three orders ofmagnitude less DOFs than the standard FEMapproach for the same order of accuracy. We mention that similarresults can be obtained for systems with 4 and higher dimensionalstate-space. The results shown in this work were obtained on asmall computer (PC), and each iteration of the 3Dproblem requiredabout 45 min of computational time. If attempted on an advancedcomputational platform, the current approach can be utilized tosolve problems in higher dimensions.

5.2. Space homotopy: Results

In order to illustrate the use of the homotopic approach fordomain determination of the stationary FPE, we consider thefollowing Duffing oscillator which is a modified version of the oneused in the previous section (Eq. (5.1)):

x = −αx− β x+ ε(x3 + σ)+ w. (5.6)

The homotopy parameter p in the above system is ε, variationin which generates a family of dynamical systems of varying

nonlinearity. The role of the parameter σ is to shift the domainof the significant portion of the pdf as the homotopy parameter isvaried. Its presence allows us to validate the fact that the proposedmethod can successfully track changes in the domain as ε changesfrom 0 to 1. Also, α is assumed to be positive (correspondingto a hard spring with a solitary stable equilibrium point). Fig. 5shows the results for the above system. In Fig. 5(a), the variationof the x-coordinate of the stable equilibrium point of the systemis shown with ε. The marked values (stars) on this curve depictthe values of the parameter used enroute to the desired dynamicalsystem, corresponding to ε = 1. It was found necessary to initiallytake small steps (see figure) in order to satisfy the assumptionsstated in Lemma 3.7. In general, the nature of this variationwill depend on the particular manner in which the homotopicparameter influences the systemunder consideration.Wementionthat the execution of space homotopy also involves the solutionrefinement process described above. Before we can proceed froma particular value of ε to the next, it must be ensured that theapproximation obtained for the current ε has converged to withinthe acceptable tolerance, which requires the refinement iterationsillustrated above. This is also apparent from the algorithm detailedin Section 3.2.Fig. 5(b) shows the smooth variation of the converged

solutions obtained for each ε (=p). In Fig. 5(c), we see thatthe domain inside which the dynamical system D1 is solved, iscompletely disconnected from the domain for the initially knownsystem, D0. However, through the iterative process of homotopic

276 M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277

Fig. 6. A schematic of the combined process of homotopic domain tracking and iterative solution refinement.

approximations, (of which only 4 are shown in this figure) thedesired result is achievable. Finally, Fig. 5(d) shows the closenessof the final iteration on the dynamical system corresponding top = ε = 1 (drawn surface) with the analytical result (shown withcrosses), which is known in this case for p = ε = 1.The results presented in this paper are summarized using a

schematic of the homotopic recursive algorithm in Fig. 6. Thetop-left plot in this schematic illustrates the issue of domaindetermination for the FPE. For a general nonlinear dynamicalsystem, it is difficult to determine the appropriate location and sizeof the finite sized domain on which to solve the FPE numerically.Using an extremely large sized conservative domain can leadto wastage of computational resources, possibly making higherdimensional problems unfeasible. At the same time, too smalla domain can result in significant errors because it may notaccommodate the entire probability density function. The methodof modified norms discussed in this paper has been shownto resolve this issue, while also improving the approximationaccuracy. In other words, the current approach obtains the desiredsolution shown in the top-left plot in Fig. 6 starting with thesolution for a known system. This involves setting up an iterativeprocedure using the best available solution for the current valueof the homotopic parameter, p. Starting with p = 0, theavailable solution is successively refined using modified-normerror projection as shown in the top-right plot. This constitutesthe inner loop of the algorithm described in Section 3.2. This partof the recursion reduces approximation error, while keeping thesize of the discretized problem small. Once the desired accuracy ismet, the value of the homotopic parameter is increased, moving onto the next dynamical system, progressively towards the desired

system, i.e. p = 1. (bottom-left plot) This constitutes the outer loopof the algorithm. The final result is a highly accurate approximationof the stationary distribution for the nonlinear dynamical systemof interest. The bottom-right plot shows the error-reduction pathtaken by the current approach, as compared with the standard L2approach. Note that the only way to reduce the error with the L2approach is to change the size of the discretized problem for givendomain size, whereas the modified-norm approach can reduceapproximation error with a fixed problem size by changing thedefinition of the norm. The resulting advantage for the modified-norm approach is clearly apparent from the bottom-right plotin Fig. 6.

6. Conclusion

A homotopic, iterative approach to solution refinement anddomain determination for the stationary Fokker–Planck equationhas been presented. A modification of the standard L2 innerproduct by using the most recent approximation of the actualpdf has been shown to improve solution accuracy beyond thatachievable by the standard inner product, using the same numberof degrees of freedom. This approach also provides a naturalway to determine the solution domain for nonlinear systems, byworking through a one parameter family of dynamical systems.It has been shown that the use of the modified inner productleads to a stable iterative process (barring pathological caseslike pdf s with δ-function like singularities) for a finite numberof iterations, and convergence can be guaranteed for certaindynamical systems (although not in general). However, with theuse of patching of adjacent approximations, superior convergence

M. Kumar et al. / Probabilistic Engineering Mechanics 24 (2009) 265–277 277

has been illustrated for significantly nonlinear systems. Therefore,it is possible with the current approach to obtain high accuracywith a small number of degrees of freedom — which is asignificant advantage, when the problem of interest resides inhigher dimensions.

References

[1] Risken H. The Fokker Planck equation: Methods of solution and applications.Springer series in synergetics, Springer-Verlag; 1989.

[2] Kumar M, Singla P, Chakravorty S, Junkins JL. The partition of unity finiteelement approach to the staitonary Fokker–Planck equation. Journal ofGuidance and Control Design, also Paper 2006-6285 at the 2006 AIAA/AASastrodynamics specialist conf. [submitted for publication].

[3] Fuller AT. Analysis of nonlinear stochastic systems by means of theFokker–Planck equation. Int J Control 1969;9(6):603–55.

[4] Johnson EA, Wojtkiewicz SF, Bergman LA, Spencer BF. Observations withregard to massively parallel computation for Monte Carlo simulation ofstochastic dynamical systems. Int J Non-linear Mech 1997;32(4):721–34.

[5] Harnpornchai N, Pradlwarter HJ, Schnëller GI. Stochastic analysis of dynamicalsystems by phase-space-controlled Monte Carlo simulation. Comput MethodsAppl Mech Engr 1999;168:273–83.

[6] Roberts JB, Spanos PD. Random vibration and statistical linearization. DoverPublications; 2003.

[7] Wu WF, Lin YK. Cumulant-neglect closure for nonlinear oscillators underparametric and external excitations. Int J Non-linear Mech 1984;19(4):349–62.

[8] Pradlwarter HJ, VastaM. Numerical solution of the Fokker–Planck equation viaGaussian superposition representation. In: Shairaishi N, ShinozukaM,WenYK,editors. Structural safety and reliability: Proc. ICOSSAR’97 — 7th int. conf. onstructural safety and reliability, vol. 2, 1997, p. 917–23.

[9] Park RS, Scheers D. Nonlinear mapping of Gaussian state uncertainties:Theory and applications to spacecraft control and navigation. In: AAS/AIAAastrodynamics specialist conference, AAS paper no. 05-404.

[10] Pradlwarter HJ. Nonlinear stochastic response distributions by local statisticallinearization. Int J Non-linear Mech 2001;36(7):1135–51.

[11] Muscolino G, Ricciardi G, Vasta M. Stationary and non stationary probabilitydensity function for non-linear oscillators. Int J Non-linear Mech 1997;32(6):1051–64.

[12] Paola MDi, Sofi A. Approximate solution of the Fokker–Planck–Kolmogorovequation. Probabilist Eng Mech 2002;17:369–84.

[13] Wojtkiewicz SF, Bergman LA, Spencer BF Jr. High fidelity numerical solutionsof the Fokker–Planck equation. In: Bazzani A, Ellison J, Mais H, Turchetti G,editors. Proc. of ICOSSAR’97: The 7th international conference on structuralsafety and reliability, 1997.

[14] Johnson EA, Wojtkiewicz SF, Bergman LA, SpencerJr BF. Finite element andfinite difference solutions to the transient Fokker–Planck equation. In: BazzaniA, Ellison J, Mais H Turchetti G, editors. Proc. of a workshop: Nonlinear andstochastic beam dynamics in accelerators — a challenge to theoretical andnumerical physics, 1997, p. 290–306.

[15] Masud A, Bergman LA. Application of multi-scale finite element methods tothe solution of the Fokker–Planck equation. Comput Methods Appl Mech Engr[in press].

[16] Langley RS. A finite element method for the statistics of random nonlinearvibration. J Sound Vib 1985;101(1):41–54.

[17] Kumar M, Singla P, Junkins JL, Chakravorty S. A multi-resolution meshlessapproach to steady state uncertainty determination in nonlinear dynamicalsystems. In: 38th IEEE southeastern symp. on sys. theory. 2006.

[18] Chakravorty S, KumarM, Singla P. A quasi-GaussianKalman filter. In: Americancontrol conference. 2006.

[19] Singla P. Multi-resolution methods for high fidelity modeling and controlallocation in large-scale dynamical systems. Ph.D. dissertation. College Station(USA) : Department of Aerospace Engineering. Texas A&M University, 2006.

[20] Babuška I, Melenk J. The partition of unity finite element method. Int J NumMeth 1997;40:727–58.

[21] Atluri SN, Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach inComputational Mechanics. Comput Mech 1998;22(2):117–27.

[22] Wojtkiewicz SF, Bergman LA, Spencer BF Jr. Numerical solution of some threestate random vibration problems. In: Bergman LA, Spencer Jr BF, editors.Vibration of nonlinear, random and time-varying system. 1995.