numerical solution of first passage problems in random ... - CiteSeerX

NUMERICAL SOLUTION OF FIRST PASSAGE PROBLEMS IN RANDOM VIBRATIONS HANS PETTER LANGTANGEN3

The reliability of dynamic systems modeled by white noise excited, stochastic, ordinary dierential equations can be computed from deterministic backward Kolmogorov equations. This paper discusses and compares some numerical solution methods for boundary value problems involving backward Kolmogorov equations. The numerical examples concern single degree of freedom oscillators subjected to white noise and ltered white noise excitation. The eciency of various methods and the sensitivity of the solution to the choice of the numerical parameters are particularly discussed. Abstract.

1. Introduction. A frequently applied design criterion for dynamic systems subjected to uncertain input data is to require some response parameters to stay within a prescribed safe region, for a time interval, with a suciently high probability. In structural engineering the displacement of the system is often used as a critical response parameter. If the excitation is a stochastic process the displacement will be likewise and failure may occur when the displacement process exits the safe region. The time T to the rst exit of the safe region is commonly called the rst passage time of the displacement process and constitutes a measure of the life time of the structure. The reliability of the structure is then dependent upon the rst passage time statistics. Exact analytical expressions for the statistics of T are unfortunately not available even for the simplest dynamic system of engineering interest. Hence, approximation methods are required. The purpose of this paper is to present an improved numerical method for accurately computing the distribution or the moments of T and to evaluate the behavior of this method. Methods for calculating rst passage time statistics of the single degree of freedom linear oscillator subjected to white noise excitation have been discussed by Crandall [8]. A particular class of methods that can yield exact statistics for T is based on Markov process theory and commonly called diusion methods. A recent review of their application to rst passage problems has been given by Roberts [17]. One of the particularly attractive features of diusion methods is that mechanical nonlinearities and non-Gaussian excitation present, in principle, no diculties. Diusion methods result in linear, second order, partial dierential equations, frequently in high dimensions, for the distribution or moments of T . The equations are usually de ned on in nite domains. Two- and three-dimensional versions of the partial dierential equations associated with diusion methods have been solved by various techniques. For example, Toland and Yang [20] used a random walk model, Sun and Hsu [19] worked with a 3 University of Oslo, Dept. of Mathematics, Mechanics Division, P.O. Box 1053 Blindern, 0316 Oslo 3, Norway. Also at Center for Industrial Research, Dept. of Industrial Mathematics, P.O. Box 124 Blindern, 0314 Oslo 3, Norway. 1

cell-method, Langley [15] utilized a variational approach with Hermite polynomial expansions and Bergman, Spencer and their co-workers [1, 2, 3, 18] employed a fairly general nite element solution method. The present article follows the latter type of numerical approach. The method and its implementation are extended to an arbitrary number of space dimensions and parts of the solution method are signi cantly improved with respect to computational eciency and storage requirements. The main object of this paper is however to report the behavior of dierent numerical strategies in some model problems. In particular, we demonstrate the in uence of various numerical parameters on the accuracy. Two of the numerical examples concern ltered white noise excitation. To the author's knowledge these rst passage problems have not been solved previously in the literature by general diusion methods.

2. Problem description. Suppose the stochastic dynamic system can be mod-

eled by a system of rst-order, ordinary, stochastic dierential equations where the only stochastic excitation is of a white noise type: (2.1)

d X d Xi (t) = ai + Bij Nj (t); i = 1; . . . ; d: dt j =1

Here X(t) = (X1 (t); . . . ; Xd (t))T is the response vector of the system, ai and Bij are functions of X1 ; . . . ; Xd . Moreover, Ni (t) is a white noise process de ned as the generalized derivative of a normalized Wiener process [13] with E [Ni (t)] = 0 and E [Ni (t +  )Ni (t)] =  ( ), where  is the Dirac delta function and E [1] is the expectation operator. All the Ni (t) processes are assumed to be independent. One can show that X(t) governed by (2.1) is a vector Markov process [13]. Let x be a safe region for X(t). Engineering applications are frequently concerned with determining the probability that X(t) 2 x for some time interval [t0; t]. For example, a failure criterion may be formulated as X 62 x . The time T to rst exit from x , conditional on X(t0 ) = y 2 x , is commonly called the rst passage time of the process X(t). Of particular interest is the reliability function R(t j y)  Pr fT > t j X(t0 ) = yg, with y = (y1 ; . . . ; yd )T . It can be shown that R is governed by the backward Kolmogorov equation [6], [13], [18] )

(

(2.2)

d d @2R @R 1 X @R X ; y 2 x  : Crs = + ar @t r=1 @yr 2 s=1 @yr @ys

The coecients Crs are given as Crs = di=1 Bri Bsi . The functional form of ar and Brs are as de ned in (2.1), but in equation (2.2) Xi must be replaced by yi in the expressions for ar and Crs . The initial condition reads R(t0 j y) = 1. In many practical applications the matrix Brs contains mostly zeroes and the boundary conditions must then be prescribed with care in order to achieve a wellposed problem. Fichera [11] has studied the well-posed-ness of the present problem and the main results relevant for our equation (2.2), when the eigenvalues of Crs are P

2

non-negative, are as follows. Divide the boundary @ x of x into 3 non-overlapping parts, 01 , 02 and 03 de ned by (2.3)

01 =

8
0; R(t j y1 ; 6C~2) = 0:

It is also possible to prescribe R = 0 at y2 = 0, but this has little eect on the solution, except that some numerical noise on the boundary may be slightly ampli ed (cf. [1]). The above partial boundary conditions can be given a physical interpretation. At y1 = X1 the system will move out of the safe domain if the velocity is positive. Hence R = 0 when y2 > 0. In the case where y2 < 0 the system will move into the domain and no value of R can be prescribed. The same reasoning can be applied to the conditions at x = 0 X1 . For a linear oscillator we have f (x) = !02 x. This problem will be referred to as model problem 1. Model problem 2 consists of a nonlinear oscillator with a \bangbang" spring that has f (x) = !0sign(x). First passage time statistics related to this problem has been considered by Toland and Yang [20] using a random walk method. Model problem 3 concerns a Dung oscillator where the spring is expressed as f (x) = !0(x1 + "x31 ). In later sections speci cations of the safe domain are given in terms of X1 . For model problem 1 X2 1 = S0 =(2!03 ) while for model problem 2 p X1 = S0 =( 2!02 ).

5.2. Filtered white noise excitation. In model problem 4 and 5 we consider

an oscillator with the equation of motion (5.5)

Z + 2!0Z_ + !02 (Z + "Z 3 ) = Q u(Q(t))

where u is a deterministic function and Q(t) is a Gaussian process with mean Q , standard deviation Q , and autocorrelation function E [Q(t)Q(t +  )] = Q2 exp (0Q j j), that is, Q has a low frequency dominated spectrum. The process Q(t) can be obtained by ltering white noise according to the equation (5.6)

p Q_ = 0Q (Q 0 Q ) + Q 2Q N3 (t):

10

It is evident that E [X1 ] = !002Q E [u(Q)] when  = 0. Introducing X1 = Z , X2 = X_ 1 and X3 = Q, the Markov vector X = (X1 ; X2 ; X3)T is governed by a system of white noise excited stochastic dierential equations of the form (2.1). The coecients in the backward Kolmogorov equation read (5.7) (5.8) (5.9) (5.10) (5.11)

a1 a2 a3 C33 Cij

= = = = =

y1 Qu(y3 ) 0 2!0 y2 0 !02 (y1 + "y13 ) 0Q (y3 0 Q ) Q2 Q 0; i 6= 3 or j 6= 3:

Dierent choices of u give rise to dierent low frequency dominated load processes. To this author's knowledge the backward Kolmogorov equation associated with (5.7)(5.11) has not been solved elsewhere in the literature. Model problem 4 employs a Gaussian excitation process with u(Q) = Q. When " = 0 one can nd exact closed form expressions for the rst and second order moments by using e.g. the moment equations. The results are omitted here but the numerical value of X1 , which is used in the speci cations of x , will be given later. The linear oscillator with low frequency Gaussian excitation results in a boundary value problem that is particularly challenging to solve. The boundary value problems for R and Mk involves additional complexities compared to e.g. the d = 3 problem investigated by Spencer [18]. Besides general numerical diculties due to singularities and the shape of the solution it is necessary to deal properly with the prescription of boundary conditions in order to ensure a well-posed problem. De ning the domain x as jx1j  X1 , jx2j < C~2 and jx3j < C~3 , and using (2.3)-(2.5) one obtains (5.12) (5.13) (5.14) (5.15) (5.16)

R(t j X1 ; y2 ; y3) R(t j 0 X1 ; y2 ; y3) R(t j y1; C~2; y3) R(t j y1 ; 0C~2; y3) R(t j y1 ; y2; 6C~3)

= = = = =

0; 0 < y2 < C~2; jy3 j < C~3; 0; 0C~2 < y2 < 0; jy3j < C~3; 0; jy1 j  X1 ; jy3j < C~3 ; a2 > 0; 0; jy1 j  X1 ; jy3j < C~3 ; a2 < 0; 0; jy1 j < X1 ; jy2j < C~2 :

The mathematical problem we want to solve corresponds to C~2 ; C~3 ! 1, but nite values must be used in the numerical computations. Model problem 5 is related to slow-drift oscillations of moored marine structures where the excitation is often of low frequency and exponential nature. Transformation of a normalized (Q = 1, Q = 0), normally distributed Q to an exponentially distributed u(Q), with unit expectation and variance, can be carried out by u(Q) = 0 ln(1 0 8(Q)), where 8 is the normalized, univariate, cumulative, normal distribution function. The boundary conditions become dierent in this problem compared to model problem 4. The domain x is de ned as jx1 0 E [X1 ] j  X1 , jx2j < C~2

11

and C~3 < x3 < C~4 . Note that E [X1 ] = Q . Mathematically, C~2; C~4 C~3 ! 01. From (2.3)-(2.5) it follows that (5.17)R(t j Q + X1 ; y2; y3 ) (5.18)R(t j Q 0 X1 ; y2; y3 ) (5.19) R(t j y1; C~2; y3 ) (5.20) R(t j y1; 0C~2; y3 )

= = = =

! 1, while

0; 0 < y2 < C~2; C~3 < y3 < C~4; 0; 0C~2 < y2 < 0; C~3 < y3 < C~4; 0; jy1 0 Q j  X1 ; C~3 < y3 < C~4; a2 > 0; 0; jy1 0 Q j  X1 ; C~3 < y3 < C~4; a2 < 0:

As in model problem 4 one can also prescribe boundary conditions for R on y3 = C~3 and y3 = C~4. It is clear that R ! 0 as Q ! 1 so one may set (5.21)

R(t j y1 ; y2; C~4) = 0; jy1 0 Q j < X1 ; jy2 j < C~2 :

As Q ! 01 the excitation vanishes and has hence no eect on the reliability. The proper condition at y3 = C~3 is then (5.22)

@ R(t j y1; y2 ; C~3) = 0; jy1 0 Q j < X1 ; jy2j < C~2: @y3

In model problem 1-4 it is trivial to show that R(t jy) = R(t j 0 y) with a similar result also for Mk (y). Hence it is possible to reduce the number of nodal values entering the matrix systems by 50%. In the examples presented in the next section we have taken advantage of this symmetry with respect to y = 0.

6. Results. When solving rst passage problems associated with oscillating systems we will in the present work be concerned with applications to structural reliability. This implies that e.g. 0:9  R(t j y)  1 is the region of the solution of most interest. Moreover, the bounds on X1 are fairly large, e.g.,  3. The results to be presented below are computed with !0 = 1 and S0 = 1= in all the model problems 1-3. In model problems 4 and 5 we have used " = 0, Q = 10, Q = Q = 1 and Q = 0. In all problems t0 = 0. The grids consist of m1 21 1 12 md multi-linear elements of regular, non-distorted, hypercube shape. Many of the results are presented in a compact tabular form. In the tables the column text \upw." refers to the choice of upwind weighting functions. 6.1. Integration with the trapezoidal rule. When solving partial dierential equations by the nite element method in higher space dimensions nodal point integration using the trapezoidal rule on multi-linear elements will give substantial savings in storage requirements, but the accuracy may be questionable. The performance of the trapezoidal rule has been investigated in the previously described model problems. It is unfortunately evident that the trapezoidal rule is not suited in these problems. Table 1 displays a comparison between some dierent integration rules and choices of weighting functions in model problem 2. The convergence of nodal point integration is seen to be very slow compared to Gaussian quadrature. Since upwind weighting functions are always required to obtain meaningful solutions, the inferior behavior of

12

m1 m2 upw. 28 56 QUAD 28 56 SUPG 28 56 dSUPG 28 56 SUPG 70 140 QUAD 70 140 SUPG 70 140 dSUPG

integration rule M1 (0; 0) Gauss quadrature 169 Gauss quadrature 166 trapezoidal rule 65 trapezoidal rule 106 Gauss quadrature 170 Gauss quadrature 167 trapezoidal rule 82 Table 1

Nonlinear Bang-bang oscillator (model problem 2) with  = 0:01, = 1, C~2 = 2 = 0:5, except for m1 > 56 where 2 = 0:75. MCS estimated M1 (0; 0) = 170.

40, 1 = 3:0 and

the trapezoidal rule may be caused by the modi ed SUPG weighting functions. As Table 1 shows, both modi cation of SUPG and nodal point integration contribute to decrease the accuracy. Since the trapezoidal rule with dSUPG weighting gives rise to algebraic equations that are (almost) equivalent to the equations produced by a standard, rst-order, upwind nite dierence scheme, we may draw the conclusion that the standard nite dierence methodology is in general not suitable for numerical solution of rst passage problems. Nevertheless, Franklin and Rodemich [12] solved a simple Pontriagin-Vitt equation by a nite dierence method with success. Their problem has also been run with the nite element methods in the present work and nodal point integration with dSUPG worked in fact better than Gauss quadrature with full SUPG for this particular equation.

6.2. Sensitivity to numerical parameters. The backward Kolmogorov bound-

ary value problem gives rise to numerical diculties associated with in nite domains and resolution of singularities. To overcome the diculties we may use upwind weighting functions, mesh grading, smaller elements and a proper nite location of the boundaries. Some examples are given below to show to what extent various numerical parameters in uence the accuracy. One of our aims is also to recover possible problems with or shortcomings of nite element solution of the backward Kolmogorov equation. Previous contributions to the literature have mostly focused on the advantages of the nite element approach. To enable comprehensive future discussions and comparisons of dierent methods for obtaining rst passage time statistics it is important to have documentation on both advantages and possible limitations of the present solution strategy. The rst example concerns model problem 1. Table 2 displays a comparison of dierent choices of weighting functions for various grid resolutions. In this example SUPG is more accurate than QUAD on the nest grid. On coarser grids both SUPG and QUAD can lead to R > 1 at t = 14 although SUPG was more likely to produce occasions of R > 1. If one considers the complete R(t j 0; 0) curve for 0  t < 1, as has been usual in the literature, there are only very small dierences between SUPG

13

m1 m2 upw. R(14 j 0; 0) 14 28 SUPG 1.044 14 28 QUAD 0.994 28 56 SUPG 0.993 28 56 QUAD 0.983 42 84 SUPG 0.988 42 84 QUAD 0.984 Model problem 1,  R(14 j 0; 0) = 0:988.

Table 2

= 0:1, 1t = 0:5, = 3, C~2 = 90, 1 = 2:0 and 2 = 0:5.

MCS estimated

and QUAD. Generally QUAD was less sensitive than SUPG to variations in C~2, i and n. In time dependent problems a uniform grid has shown to be successful [2, 3]. However, our type of mesh grading improves the results in the present example and 1 = 2:0, 2 = 1=1 seemed to be a good choice. The Tables 3-6 concern the nonlinear oscillator in model problem 2. In these examples SUPG was clearly inferior to QUAD with respect to accuracy, and the latter was also more robust than the former. For example, SUPG showed little sensitivity to C~2 and i for = 1 while the sensitivity was very pronounced for = 4. On coarse grids QUAD was signi cantly superior to SUPG. More elements were needed in order to maintain sucient accuracy as the displacement bounds were increased. For smaller bounds, e.g. = 1, the problem is generally easy to solve and there are minor dierences between dierent numerical strategies. MCS also becomespmore expensive as increases, the execution time is typically proportional to exp ( 2 ). For e.g. = 3 the nite element approach for nding the complete M1 (y1 ; y2) was much more ecient than MCS for calculating the single value M1 (0; 0). A trial and error process utilizing plots of R or Mk in addition to experience and MCS seems to be the most eective way to determine the boundary location C~2. On coarse grids the solution may show considerable sensitivity to changes in C~2, especially if C~2 is chosen larger than strictly necessary. This sensitivity decreases rapidly as the grid is re ned. However, many problems in higher space dimensions must probably be run on a coarse grid. A proper tuning of parameters related to boundary locations and mesh grading may lead to fairly accurate solutions even on coarse grids. Experience with the present set of model problems reveals that comparison of M or R with MCS for a single point, e.g. (y = E [X]), may be sucient in the tuning process. When solving the moment equation the optimal mesh grading parameters were 1 = 1:5 and 2 = 0:75, with the exception of a few cases where 1 = 2 led to higher accuracy. In time dependent problems a uniform grid seemed to be an ecient choice for model problem 2. To demonstrate that no particular numerical strategy turned out to be \best" we

14

m1 m2 upw. M1 (0; 0) 14 28 SUPG 24.7 14 28 QUAD 26.2 56 112 SUPG 26.2 56 112 QUAD 26.3 Table 3

Model problem 2,  = 0:1, = 1, C~2 = 24, 1 = 1:5 and 2 = 0:75. MCS estimated M1 (0; 0) = 26:6.

m1 m2 upw. C~2 M1 (0; 0) 28 56 SUPG 24 940 28 56 QUAD 24 1360 28 56 SUPG 54 1170 28 56 QUAD 54 1360 56 112 SUPG 54 1210 56 112 QUAD 54 1490 Table 4

Model problem 2,  = 0:1, = 4, 1 = 1:5 and 2 = 0:75. MCS estimated M1 (0; 0) = 1660.

m1 m2 upw. C~2 M1 (0; 0) 28 56 SUPG 40 1730 28 56 QUAD 40 1830 28 56 SUPG 90 1860 28 56 QUAD 90 1890 56 112 SUPG 90 1810 56 112 QUAD 90 1890 Table 5

Model problem 2,  = 0:01, = 3, 1 = 1:5 and 2 = 0:75. MCS estimated M1 (0; 0) = 1890.

m1 m2 upw. C~2 R(15 j 0; 0) 28 56 SUPG 16 0.991 28 56 QUAD 16 0.992 28 56 SUPG 36 0.994 28 56 QUAD 36 0.992 56 112 SUPG 36 0.992 56 112 QUAD 36 0.992 Table 6

Model problem 2,  = 0:1, 1t = 0:5, = 3, 1 = 2 = 1:0. MCS estimated R(15 j 0; 0) = 0:992.

15

m1 m2 upw. M1 (0; 0) 28 56 SUPG 83.4 28 56 QUAD 65.1 56 112 SUPG 86.6 56 112 QUAD 76.7 Table 7

Model problem 3,  = 0:1, " = 2, = 1, 1 = 1:5 and 2 = 0:75. MCS estimated M1 (0; 0) = 87:4.

show results in Table 7 from model problem 3 where SUPG was signi cantly superior to QUAD. SUPG was also less sensitive to variations in C~2. Model problem 4 turned out to make greater demands to the numerical solution schemes than the other model problems. Since this particular problem has not been solved in the literature before it may be illustrating to show the typical shape of the reliability function. Figures 1-3 display R as a function of a single space coordinate in three space directions and at three dierent points of time. The other two coordinates are kept xed at zero. The parameters equaled  = 0:1, C~2 = 360, C~3 = 120, = 3, m1 = 14, m2 = 20, m3 = 26, 1t = 1:0,  = 1:0, 1 = 2, 2 = 0:667 and 3 = 0:5. Quadratic weighting functions were used. The nite element solution value for R(25 j 0; 0; 0) equaled 0.944 while MCS (10 000 samples) gave 0.954. SUPG was in general not competitive with QUAD in this problem. Numerical oscillations caused diculties and the damping inherent in the -scheme when  = 1 was useful. However, in other problems  = 1 may lead to too inaccurate results, cf. [2]. A general feature of model problem 4 was that the solution showed considerable sensitivity to numerical parameters such as mi , i, , C~2 and C~3 . Model problem 5 is closely related to model problem 4 but considerably easier to solve by the present numerical method. As plots of R in this problem neither exist in the literature examples of the typical features of the function is presented in Figures 4-6. It is seen that one can work with a much coarser mesh in y3 -direction than in the other two space directions. For the particular example in Figures 4-6 the parameters were  = 0:1, C~2 = 240, C~3 = 04, C~4 = 10, = 3, m1 = 22, m2 = 36, m3 = 12, 1t = 0:5,  = 0:75, 1 = 2, 2 = 0:667 and 3 = 1:0. Quadratic weighting functions were used. MCS (10 000 samples) for R(10 j 10; 0; 0) resulted in 0.971 while the nite element solution read 0.965 (recall that E [X1 ] = Q = 10). The sensitivity of the solution to the choice of various numerical parameters was signi cantly smaller than in model problem 4. Tables 8 and 9 show some results. The sensitivity of R to variations in the mesh grading parameters is displayed in Table 8. The results corresponds to the SUPG method. Similar, but slightly less accurate results, were obtained by QUAD. As in the previous model problems QUAD showed less sensitivity (in comparison with SUPG) to perturbations in the numerical parameters as n was increased. Table 9 displays some additional results when 1 = 2, 2 = 2=3 and 3 = 1. In

16

Fig. 1. R(t j y1 ; 0; 0) as a function of y1 for t = 20 (solid line), t = 100 (dashed line) and t = 200 (3) in model problem 4.

Fig. 2. R(t j 0; y2 ; 0) as a function of y2 for t = 20 (solid line), t = 100 (dashed line) and t = 200 (3) in model problem 4.

17

Fig. 3. R(t j 0; 0; y3 ) as a function of y3 for t = 20 (solid line), t = 100 (dashed line) and t = 200 (3) in model problem 4.

m1 m2 m3 1 2 R(10 j 10; 0; 0) 16 24 12 1 1 0.961 16 24 12 3/2 2/3 0.967 16 24 12 2 1/3 0.938 24 36 18 1 1 0.934 24 36 18 3/2 2/3 0.963 24 36 18 2 1/3 0.972 Table 8

model problem 5, other parameters are as given in the text. MCS estimated R(10 j 10; 0; 0) = 0:971.

m1 m2 m3 16 24 12 16 24 12 16 24 12 16 24 12 22 36 12 24 36 18 24 36 18

C~2 240 240 160 160 240 240 240

upw. R(10 j 10; 0; 0) QUAD 0.918 SUPG 0.938 QUAD 0.887 SUPG 0.955 QUAD 0.965 QUAD 0.954 SUPG 0.972

Table 9

Model problem 5, other parameters are as given in the text. MCS estimated R(10 j 10; 0; 0) = 0:971.

18

Fig. 4. R(t j y1 ; 0; 0) as a function of y1 for t = 1 (solid line), t = 5 (dashed line) and t = 10 (3) in model problem 5.

the present problem SUPG is slightly superior to QUAD. A proper resolution of the solution in x2 - and x3-direction in the interior parts of the domain requires a minimum number of elements to be used. Therefore C~2 and C~3 needs to be smaller on a coarse mesh compared to the optimal values on a ner mesh. If the mesh is too coarse, e.g. n = 819 in the present problem, signi cantly inaccurate results may occur and the solution is usually considerably sensitive to variations in numerical parameters such as C~i and i . For d > 3 it would be very advantageous to have numerical methods that could produce qualitatively acceptable results on coarse grids (mi  10). The value of 1t had little impact on the accuracy in model problem 5 as long as 1t  1, re ecting that the spatial approximation properties constitute the critical parts of the numerical method.

6.3. Behavior of iterative equation solvers. Utilizing iterative methods for

solving matrix systems increases the potential of nite element discretization of backward Kolmogorov equations signi cantly. This subsection reports the eciency of the approach in more detail. When solving the time dependent backward Kolmogorov equation the solution at the previous time level can be used as start vector for the iterative methods. In such cases only a few iterations, usually 1-6, are necessary to obtain a converged solution when using preconditioning. Roughly speaking, R-OMR(k) requires approximately

19

Fig. 5. R(t j 10; y2 ; 0) as a function of y2 for t = 1 (solid line), t = 5 (dashed line) and t = 10 (3) in model problem 5.

Fig. 6. R(t j 10; 0; y3 ) as a function of y3 for t = 1 (solid line), t = 5 (dashed line) and t = 10 (3) in model problem 5.

20

n 1653 6441 1653 6441 1653 6441 1653 6441 1653 6441

k it. 1 1 35 1 1 89 1 9 9 1 9 23 3 1 589 3 1 > 900 3 9 19 3 9 316 3 15 24 3 15 28 Table 10

Number of iterations for R-OMR(k) in model problem 2 (Pontriagin-Vitt equation).  = 0:1, 1 = 1:5, 2 = 0:75, C~2 = 60 ( = 1) and C~2 = 135 ( = 3).

half as many iterations as OM(k) but is also twice as expensive per iteration. The number of iterations increases only slightly with n and 1t. The work required by the equation solvers showed negligible sensitivity to other numerical parameters. If preconditioning is not applied the work per time level increases signi cantly. As an example, consider model problem 2 with  = 0:01, = 3, m2 = 2m1 = 56 and QUAD. R-OMR(5) and OM(1) required about 2500-3000 work units for solving the system of equations when no preconditioning was applied. With preconditioning the number of work units was reduced to about 170. In stationary problems associated with the solution for Mk (y) preconditioning is almost always required to avoid divergence or extremely slow convergence of the iterative solvers. Table 10 shows some selected results for the behavior of R-OMR(k) in model problem 2. It is evident that the method is sensitive to the choice of the parameter k. The optimal value of k is signi cantly larger in the present type of problems than in hydrodynamical convection-diusion problems. As is increased, k must also be increased to achieve stability and fast convergence. With a suciently large k the iterative method is robust and the work increases very slightly with increasing n. It is seen that the work increases rapidly with and n when k is small. OM(k) for small k turned out to be useless for  3. In general, R-OMR(k) was more ecient than OM(k) in the most demanding problems. The work associated with the iterative solvers also depended on the boundary locations and on the mesh grading parameters if k was not suciently large to ensure robustness. In cases where the numerical parameters led to low accuracy in the solution the iterative solver usually required a large number of iterations. Thus if the convergence of the iterative solver is slow it may indicate that k is too small or that the numerical parameters like mi , i, C~i etc. are not properly tuned.

21

m1 m2 upw. C~2 R(15 j 0; 0) 28 56 SUPG 16 0.990 28 56 QUAD 16 0.989 28 56 SUPG 36 0.991 28 56 QUAD 36 0.990 Table 11

Explicit 2nd order Runge-Kutta time integration. Model problem 2,  1 = 2 = 1:0. MCS estimated R(15 j 0; 0) = 0:992.

= 0:1, 1t = 0:05, = 3,

6.4. Implicit versus explicit time integration. Since storage requirements

and not CPU time seems to be the main limitation for solving higher dimensional backward Kolmogorov equations it may be preferable to use explicit time integration schemes like the second order Runge-Kutta method. However, there are two main disadvantages with explicit schemes. The rst is that the nite element integration and assembly process is usually much more costly than solving the matrix system, at least for the n-values relevant to the presently available computer generation. The explicit method should therefore avoid the assembly process at each time level. This is easily accomplished by storing the matrices M, L, S and Q and carrying only matrixvector products at each time steps. Nevertheless, such an approach requires about the same storage as the implicit method. The second disadvantage of an explicit scheme is that 1t is related to the size of the smallest element in the mesh due to stability requirements. In the present formulation of rst passage problems small elements are needed in the vicinity of singular points at the boundary to avoid unacceptable numerical instabilities in the spatial discretization. The explicit Runge-Kutta method has been tested on some of our model problems and found less ecient than the implicit approach. Table 11 shows the performance of the explicit method in the same problem as covered by the implicit scheme in Table 6. It is seen that the accuracy of the Runge-Kutta method is slightly inferior to the results produced by the implicit approach. At each time level the explicit method requires two matrix-vector products. Consideration of the typical work per time step in the implicit method reveals that the explicit strategy can be faster than the implicit one if the 1t required by the explicit method is not less than 1=12 of the 1t that is suitable in an implicit scheme. A central question is what happens when the mesh is re ned. If m1 = 56 and m2 = 112 are used in the problem in Table 11 the explicit method is not stable even with 1t = 0:0005 and the implicit integration strategy is extremely faster. The fact that temporal truncation errors can usually be ignored in comparison with spatial truncation errors in the present type of problems makes it practical to employ time steps which are considerably larger than what is dictated by stability requirements of explicit schemes.

7. Conclusion and discussion. The paper has presented and evaluated general

nite element solution methods for computing rst passage time statistics of oscillating

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systems. Use of preconditioned conjugate gradient-like methods for solving matrix systems was an important part of the method for obtaining eciency. In the problems treated herein the time spent on solving matrix systems was usually much smaller than the time spent on the element by element spatial integration process. The numerical examples included novel model problems, the impact of various numerical parameters on the accuracy, comparisons of explicit versus implicit time integration and veri cation by comparison with Monte Carlo simulations. The main conclusion is that the performance of the nite element method is very dependent on the type of problem being solved. In some problems, e.g. the model problems 1, 2 and 5, the method must be considered as fairly robust and easily used. Other problems recover serious sensitivity to numerical parameters. When the interest concerns conditions corresponding to low failure probabilities the present numerical formulation is particularly attractive since only a few time steps (typically 10-40) are needed and the iterative solution of matrix systems is extremely ecient in time dependent problems. The general indication of our comparison between implicit and explicit time integration reveals that pure explicit schemes are inferior to implicit methods with respect to eciency. It should be emphasized that explicit-implicit approaches may be advantageous where implicit time stepping is used for the small elements around the singular points while explicit time stepping is used for elements of larger size. The development of numerical methods with better spatial approximation properties are especially warranted. Such methods should ideally improve the damping of oscillations due to singularities at the boundaries, guarantee that 0  R  1 and give qualitatively acceptable results on coarse grids as this would be an important property when d > 3. A central question is whether the proposed methods can compete with MCS. Of course, the computational work associated with numerical solution of partial dierential equations increases exponentially with d while the increase is only linear in d for MCS methods. As a simple model one may consider T to be exponentially distributed. Then the standard deviation of the estimator for E [T ] used in MCS behaves 2 as m ~ 01=2e0 =2 , where m ~ is the number of samples of T and represents the width of the failure bounds in standard deviation units.. In the nite element method it will suce to keep the number of elements per unit length constant as increases and hence the work is roughly proportional to d . Thus for narrow bounds MCS is eective, especially for larger d values. As the bounds increase the nite element approach becomes superior. However, for large failure bounds serious numerical instabilities may arise in the nite element method. In such cases simpler approximation formulas for the rst passage time statistics are usually accurate. Another important fact is that MCS yields consistent values of the statistics while numerical solution of the backward Kolmogorov equation may give probabilities that exceed unity. The general conclusion must then be that if the nite element method is robust in the problem being solved, the bounds correspond to  3 0 4, the nite element scheme

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is more ecient than MCS. Acknowledgments.

cussions.

The author is thankful to Prof. Bill Spencer for valuable dis-

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