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Boundedness in a two-species chemotaxis parabolic system with two chemicals
Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory
Department of Mathematics, Center for Complex Biological Systems, University of California, Irvine, CA 92697, USA |
Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.
References:
[1] |
R. L. Burden and J. D. Faires,
Numerical Analysis 9th ed. , Brooks/Cole, Cengage Learning, Boston, MA, 2011. |
[2] |
K. Burrage and P. Burrage,
General order conditions for stochastic runge-kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems, Appl. Numer. Math., 28 (1998), 161-177.
doi: 10.1016/S0168-9274(98)00042-7. |
[3] |
K. Burrage and P. M. Burrage,
High strong order explicit runge-kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics, 22 (1996), 81-101.
doi: 10.1016/S0168-9274(96)00027-X. |
[4] |
P. M. Burrage,
Runge-Kutta Methods for Stochastic Differential Equations Thesis, The University of Queensland Brisbane, 1999. |
[5] |
P. M. Burrage and K. Burrage,
A variable stepsize implementation for stochastic differential equations, SIAM Journal on Scientific Computing, 24 (2002), 848-864.
doi: 10.1137/S1064827500376922. |
[6] |
J. R. Cash and A. H. Karp,
A variable order runge-kutta method for initial value problems with rapidly varying right-hand sides, ACM Transactions on Mathematical Software, 16 (1990), 201-222.
doi: 10.1145/79505.79507. |
[7] |
F. Ceschino,
Modification de la longueur du pas dans l'integration numerique parles methodes a pas lies, Chiffres, 4 (1961), 101-106.
|
[8] |
J. R. Dormand and P. J. Prince,
A family of embedded runge kutta formulae, Journal of Computational and Applied Mathematics, 6 (1980), 19-26.
doi: 10.1016/0771-050X(80)90013-3. |
[9] |
J. G. Gaines and T. J. Lyons,
Variable step size control in the numerical solution of stochastic differential equations, SIAM Journal on Applied Mathematics, 57 (1997), 1455-1484.
doi: 10.1137/S0036139995286515. |
[10] |
A. Ghasemi and S. Zahediasl,
Normality tests for statistical analysis: A guide for non-statisticians, Int J Endocrinol Metab, 10 (2012), 486-489.
doi: 10.5812/ijem.3505. |
[11] |
E. Hairer, S. P. N orsett and G. Wanner,
Solving Ordinary Differential Equations I 2nd rev. edn, Springer series in computational mathematics. Springer-Verlag, Berlin, New York, 1993. |
[12] |
T. Hong, K. Watanabe, C. H. Ta, A. Villarreal-Ponce, Q. Nie and X. Dai, An ovol2-zeb1 mutual inhibitory circuit governs bidirectional and multi-step transition between epithelial and mesenchymal states PLoS Comput Biol 11 (2015), e1004569.
doi: 10.1371/journal. pcbi. 1004569. |
[13] |
S. M. Iacus,
Simulation and Inference for Stochastic Differential Equations: With R Examples (Springer Series in Statistics), Springer Publishing Company, Incorporated, 2008. |
[14] |
J. Kaneko,
Explicit order 1.5 runge kutta scheme for solutions of ito stochastic differential equations, S/urikaisekikenky/usho K/oky/uroku, 932 (1995), 46-60.
|
[15] |
P. E. Kloeden and E. Platen,
Numerical Solution of Stochastic Differential Equations Springer Berlin Heidelberg, 1992.
doi: 10.1007/978-3-662-12616-5. |
[16] |
H. Lamba,
An adaptive timestepping algorithm for stochastic differential equations, Journal of Computational and Applied Mathematics, 161 (2003), 417-430.
doi: 10.1016/j.cam.2003.05.001. |
[17] |
T. Liggett,
Continuous Time Markov Processes: An Introduction American Mathematical Society, 2010.
doi: 10.1090/gsm/113. |
[18] |
X. Mao and C. Yuan,
Stochastic Differential Equations with Markovian Switching Imperial College Press, London, 2006.
doi: 10.1142/p473. |
[19] |
N. Hofmann, T. Muller-Gronbach and K. Ritter,
Optimal approximation of stochastic differential equations by adaptive step-size control, Mathematics of Computation, 69 (2000), 1017-1034.
doi: 10.1090/S0025-5718-99-01177-1. |
[20] |
B. Oksendal,
Stochastic Differential Equations: An Introduction with Applications Springer, 2003.
doi: 10.1007/978-3-642-14394-6. |
[21] |
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery,
Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press, 2007. |
[22] |
A. Rößler,
Runge kutta methods for the strong approximation of solutions of stochastic differential equations, SIAM Journal on Numerical Analysis, 48 (2010), 922-952.
doi: 10.1137/09076636X. |
[23] |
T. Ryden and M. Wiktorsson,
On the simulation of iterated ito integrals, Stochastic Processes and their Applications, 91 (2001), 151-168.
doi: 10.1016/S0304-4149(00)00053-3. |
[24] |
L. F. Shampine,
Some practical runge-kutta formulas, Mathematics of Computation, 46 (1986), 135-150.
doi: 10.1090/S0025-5718-1986-0815836-3. |
[25] |
M. Wiktorsson,
Joint characteristic function and simultaneous simulation of iterated ito integrals for multiple independent brownian motions, The Annals of Applied Probability, 11 (2001), 470-487.
doi: 10.1214/aoap/1015345301. |
show all references
References:
[1] |
R. L. Burden and J. D. Faires,
Numerical Analysis 9th ed. , Brooks/Cole, Cengage Learning, Boston, MA, 2011. |
[2] |
K. Burrage and P. Burrage,
General order conditions for stochastic runge-kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems, Appl. Numer. Math., 28 (1998), 161-177.
doi: 10.1016/S0168-9274(98)00042-7. |
[3] |
K. Burrage and P. M. Burrage,
High strong order explicit runge-kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics, 22 (1996), 81-101.
doi: 10.1016/S0168-9274(96)00027-X. |
[4] |
P. M. Burrage,
Runge-Kutta Methods for Stochastic Differential Equations Thesis, The University of Queensland Brisbane, 1999. |
[5] |
P. M. Burrage and K. Burrage,
A variable stepsize implementation for stochastic differential equations, SIAM Journal on Scientific Computing, 24 (2002), 848-864.
doi: 10.1137/S1064827500376922. |
[6] |
J. R. Cash and A. H. Karp,
A variable order runge-kutta method for initial value problems with rapidly varying right-hand sides, ACM Transactions on Mathematical Software, 16 (1990), 201-222.
doi: 10.1145/79505.79507. |
[7] |
F. Ceschino,
Modification de la longueur du pas dans l'integration numerique parles methodes a pas lies, Chiffres, 4 (1961), 101-106.
|
[8] |
J. R. Dormand and P. J. Prince,
A family of embedded runge kutta formulae, Journal of Computational and Applied Mathematics, 6 (1980), 19-26.
doi: 10.1016/0771-050X(80)90013-3. |
[9] |
J. G. Gaines and T. J. Lyons,
Variable step size control in the numerical solution of stochastic differential equations, SIAM Journal on Applied Mathematics, 57 (1997), 1455-1484.
doi: 10.1137/S0036139995286515. |
[10] |
A. Ghasemi and S. Zahediasl,
Normality tests for statistical analysis: A guide for non-statisticians, Int J Endocrinol Metab, 10 (2012), 486-489.
doi: 10.5812/ijem.3505. |
[11] |
E. Hairer, S. P. N orsett and G. Wanner,
Solving Ordinary Differential Equations I 2nd rev. edn, Springer series in computational mathematics. Springer-Verlag, Berlin, New York, 1993. |
[12] |
T. Hong, K. Watanabe, C. H. Ta, A. Villarreal-Ponce, Q. Nie and X. Dai, An ovol2-zeb1 mutual inhibitory circuit governs bidirectional and multi-step transition between epithelial and mesenchymal states PLoS Comput Biol 11 (2015), e1004569.
doi: 10.1371/journal. pcbi. 1004569. |
[13] |
S. M. Iacus,
Simulation and Inference for Stochastic Differential Equations: With R Examples (Springer Series in Statistics), Springer Publishing Company, Incorporated, 2008. |
[14] |
J. Kaneko,
Explicit order 1.5 runge kutta scheme for solutions of ito stochastic differential equations, S/urikaisekikenky/usho K/oky/uroku, 932 (1995), 46-60.
|
[15] |
P. E. Kloeden and E. Platen,
Numerical Solution of Stochastic Differential Equations Springer Berlin Heidelberg, 1992.
doi: 10.1007/978-3-662-12616-5. |
[16] |
H. Lamba,
An adaptive timestepping algorithm for stochastic differential equations, Journal of Computational and Applied Mathematics, 161 (2003), 417-430.
doi: 10.1016/j.cam.2003.05.001. |
[17] |
T. Liggett,
Continuous Time Markov Processes: An Introduction American Mathematical Society, 2010.
doi: 10.1090/gsm/113. |
[18] |
X. Mao and C. Yuan,
Stochastic Differential Equations with Markovian Switching Imperial College Press, London, 2006.
doi: 10.1142/p473. |
[19] |
N. Hofmann, T. Muller-Gronbach and K. Ritter,
Optimal approximation of stochastic differential equations by adaptive step-size control, Mathematics of Computation, 69 (2000), 1017-1034.
doi: 10.1090/S0025-5718-99-01177-1. |
[20] |
B. Oksendal,
Stochastic Differential Equations: An Introduction with Applications Springer, 2003.
doi: 10.1007/978-3-642-14394-6. |
[21] |
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery,
Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press, 2007. |
[22] |
A. Rößler,
Runge kutta methods for the strong approximation of solutions of stochastic differential equations, SIAM Journal on Numerical Analysis, 48 (2010), 922-952.
doi: 10.1137/09076636X. |
[23] |
T. Ryden and M. Wiktorsson,
On the simulation of iterated ito integrals, Stochastic Processes and their Applications, 91 (2001), 151-168.
doi: 10.1016/S0304-4149(00)00053-3. |
[24] |
L. F. Shampine,
Some practical runge-kutta formulas, Mathematics of Computation, 46 (1986), 135-150.
doi: 10.1090/S0025-5718-1986-0815836-3. |
[25] |
M. Wiktorsson,
Joint characteristic function and simultaneous simulation of iterated ito integrals for multiple independent brownian motions, The Annals of Applied Probability, 11 (2001), 470-487.
doi: 10.1214/aoap/1015345301. |







![]() |
![]() |
Algorithm 1 RSwM1 | |
1: | Set the values |
2: | Set |
3: | Take an initial |
4: | While |
5: | Attempt a step with |
6: | Calculate E according to (9) |
7: | Update |
8: | If |
9: | Take |
10: | Calculate |
11: | Push |
12: | Update |
13: | Update |
14: | else ▷% Accept the Step |
15: | Update |
16: | if ( |
17: | Update |
18: | Take |
19: | else |
20: | Pop the top of |
21: | Update |
22: | end if |
23: | end if |
24: | end while |
Algorithm 1 RSwM1 | |
1: | Set the values |
2: | Set |
3: | Take an initial |
4: | While |
5: | Attempt a step with |
6: | Calculate E according to (9) |
7: | Update |
8: | If |
9: | Take |
10: | Calculate |
11: | Push |
12: | Update |
13: | Update |
14: | else ▷% Accept the Step |
15: | Update |
16: | if ( |
17: | Update |
18: | Take |
19: | else |
20: | Pop the top of |
21: | Update |
22: | end if |
23: | end if |
24: | end while |
Algorithm 2 RSwM3 | |
1: | Set the values |
2: | Set |
3: | Take an initial |
4: | while |
5: | Attempt a step with |
6: | Calculate E according to (9) |
7: | Update |
8: | if |
9: | Set |
10: | while |
11: | Pop the top of |
12: | if |
13: | Update |
14: | Update |
15: | else |
16: | Push |
17: | end if |
18: | end while |
19: | Set |
20: | Set |
21: | Take |
22: | Take |
23: | Pop |
24: | Pop |
25: | Update |
26: | else ▷% Accept the Step |
27: | Update |
28: | Empty |
29: | Update |
30: | Set |
31: | while |
32: | Pop the top of |
33: | if ( |
34: | Update |
35: | Push a copy of |
36: | else ▷% Final part of step from stack |
37: | Set |
38: | Let |
39: | Let |
40: | Push |
41: | Update |
42: | end if |
43: | end while |
▷% Update for last portion to step. Note zero if final part is from stack | |
44: | if ( |
45: | Let |
46: | Update |
47: | Push |
48: | end if |
49: | end if |
50: | end while |
Algorithm 2 RSwM3 | |
1: | Set the values |
2: | Set |
3: | Take an initial |
4: | while |
5: | Attempt a step with |
6: | Calculate E according to (9) |
7: | Update |
8: | if |
9: | Set |
10: | while |
11: | Pop the top of |
12: | if |
13: | Update |
14: | Update |
15: | else |
16: | Push |
17: | end if |
18: | end while |
19: | Set |
20: | Set |
21: | Take |
22: | Take |
23: | Pop |
24: | Pop |
25: | Update |
26: | else ▷% Accept the Step |
27: | Update |
28: | Empty |
29: | Update |
30: | Set |
31: | while |
32: | Pop the top of |
33: | if ( |
34: | Update |
35: | Push a copy of |
36: | else ▷% Final part of step from stack |
37: | Set |
38: | Let |
39: | Let |
40: | Push |
41: | Update |
42: | end if |
43: | end while |
▷% Update for last portion to step. Note zero if final part is from stack | |
44: | if ( |
45: | Let |
46: | Update |
47: | Push |
48: | end if |
49: | end if |
50: | end while |
Euler-Maruyama | Runge-Kutta Milstein | Rößler SRI | ||||
|
Fails (/10,000) | Time (s) | Fails (/10,000) | Time (s) | Fails (/10,000) | Time (s) |
|
137 | 133.35 | 131 | 211.92 | 78 | 609.27 |
|
39 | 269.09 | 26 | 428.28 | 17 | 1244.06 |
|
3 | 580.14 | 6 | 861.01 | 0 | 2491.37 |
|
1 | 1138.41 | 1 | 1727.91 | 0 | 4932.70 |
|
0 | 2286.35 | 0 | 3439.90 | 0 | 9827.16 |
|
0 | 4562.20 | 0 | 6891.35 | 0 | 19564.16 |
Euler-Maruyama | Runge-Kutta Milstein | Rößler SRI | ||||
|
Fails (/10,000) | Time (s) | Fails (/10,000) | Time (s) | Fails (/10,000) | Time (s) |
|
137 | 133.35 | 131 | 211.92 | 78 | 609.27 |
|
39 | 269.09 | 26 | 428.28 | 17 | 1244.06 |
|
3 | 580.14 | 6 | 861.01 | 0 | 2491.37 |
|
1 | 1138.41 | 1 | 1727.91 | 0 | 4932.70 |
|
0 | 2286.35 | 0 | 3439.90 | 0 | 9827.16 |
|
0 | 4562.20 | 0 | 6891.35 | 0 | 19564.16 |
Example 1 | Example 2 | Example 3 | Cell Model | |||||
qmax | Time (s) | Error | Time (s) | Error | Time (s) | Error | Time (s) | |
|
37.00 | 2.57e-8 | 60.87 | 2.27e-7 | 67.71 | 3.42e-9 | 229.83 | |
|
34.73 | 2.82e-8 | 32.40 | 3.10e-7 | 66.68 | 3.43e-9 | 196.36 | |
|
49.14 | 3.14e-8 | 132.33 | 8.85e-7 | 65.94 | 3.44e-9 | 186.81 | |
|
39.33 | 3.59e-8 | 33.90 | 1.73e-6 | 66.33 | 3.44e-9 | 205.57 | |
|
38.22 | 3.82e-8 | 159.94 | 2.58e-6 | 68.16 | 3.44e-9 | 249.77 | |
|
82.76 | 4.41e-8 | 34.41 | 3.58e-6 | 568.22 | 3.44e-9 | 337.99 | |
|
68.16 | 9.63e-8 | 33.98 | 6.06e-6 | 87.50 | 3.22e-9 | 418.78 | |
|
48.23 | 1.01e-7 | 33.97 | 9.74e-6 | 69.78 | 3.44e-9 | 571.59 |
Example 1 | Example 2 | Example 3 | Cell Model | |||||
qmax | Time (s) | Error | Time (s) | Error | Time (s) | Error | Time (s) | |
|
37.00 | 2.57e-8 | 60.87 | 2.27e-7 | 67.71 | 3.42e-9 | 229.83 | |
|
34.73 | 2.82e-8 | 32.40 | 3.10e-7 | 66.68 | 3.43e-9 | 196.36 | |
|
49.14 | 3.14e-8 | 132.33 | 8.85e-7 | 65.94 | 3.44e-9 | 186.81 | |
|
39.33 | 3.59e-8 | 33.90 | 1.73e-6 | 66.33 | 3.44e-9 | 205.57 | |
|
38.22 | 3.82e-8 | 159.94 | 2.58e-6 | 68.16 | 3.44e-9 | 249.77 | |
|
82.76 | 4.41e-8 | 34.41 | 3.58e-6 | 568.22 | 3.44e-9 | 337.99 | |
|
68.16 | 9.63e-8 | 33.98 | 6.06e-6 | 87.50 | 3.22e-9 | 418.78 | |
|
48.23 | 1.01e-7 | 33.97 | 9.74e-6 | 69.78 | 3.44e-9 | 571.59 |
Algorithm 3 Initial |
|
1: | Let |
2: | Calculate |
3: | Let |
4: | if |
5: | Let |
6: | else |
7: | Let |
8: | end if |
9: | Calculate an Euler step: |
10: | Calculate new estimates: |
11: | Determine |
12: | Let |
13: | if |
14: | Let |
15: | else |
16: | Let |
17: | end if |
18: | Let |
Algorithm 3 Initial |
|
1: | Let |
2: | Calculate |
3: | Let |
4: | if |
5: | Let |
6: | else |
7: | Let |
8: | end if |
9: | Calculate an Euler step: |
10: | Calculate new estimates: |
11: | Determine |
12: | Let |
13: | if |
14: | Let |
15: | else |
16: | Let |
17: | end if |
18: | Let |
![]() |
![]() |
Algorithm 4 RSwM2 | |
1: | Set the values |
2: | Set |
3: | Take an initial |
4: | while |
5: | Attempt a step with |
6: | Calculate E according to (9) |
7: | Update |
8: | if |
9: | Take |
10: | Calculate |
11: | Push |
12: | Update |
13: | Update |
14: | else ▷% Accept the Step |
15: | Update |
16: | Update |
17: | Set |
18: | while |
19: | Pop the top of |
20: | if ( |
21: | Update |
22: | else ▷% Final part of step from stack |
23: | Set |
24: | Let |
25: | Let |
26: | Push |
27: | Update |
28: | end if |
29: | end while |
▷% Update for last portion to step. Note zero if final part is from stack | |
30: | if ( |
31: | Let |
32: | Update |
33: | end if |
34: | end if |
35: | end while |
Algorithm 4 RSwM2 | |
1: | Set the values |
2: | Set |
3: | Take an initial |
4: | while |
5: | Attempt a step with |
6: | Calculate E according to (9) |
7: | Update |
8: | if |
9: | Take |
10: | Calculate |
11: | Push |
12: | Update |
13: | Update |
14: | else ▷% Accept the Step |
15: | Update |
16: | Update |
17: | Set |
18: | while |
19: | Pop the top of |
20: | if ( |
21: | Update |
22: | else ▷% Final part of step from stack |
23: | Set |
24: | Let |
25: | Let |
26: | Push |
27: | Update |
28: | end if |
29: | end while |
▷% Update for last portion to step. Note zero if final part is from stack | |
30: | if ( |
31: | Let |
32: | Update |
33: | end if |
34: | end if |
35: | end while |
Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value |
|
3 | |
0.1 | |
1 | |
0.35 |
|
0.2 | |
0.3 | |
1 | |
0.0002 |
|
0.15 | |
0.4 | |
1 | |
0.001 |
|
0.35 | |
0.4 | |
1 | |
0.09 |
|
0.9 | |
2 | |
20 | |
0.1 |
|
0.6 | |
3.5 | |
100 | |
0.1 |
|
0.5 | |
0.9 | |
0 | |
0.9 |
|
1.8 | |
1 | |
1000 | |
1.66 |
|
0.0005 | |
0.003 | |
0.5 | |
1.1 |
|
3 | |
2 | |
0.5 | |
5 |
|
2 | |
2 | |
0.5 | |
5 |
|
2 | |
2 | |
0.5 | |
15 |
|
2 | |
2 | |
0.5 | |
5 |
|
2 | |
2 | |
0.5 | |
2 |
|
2 | |
6 | |
0.5 | |
5 |
|
1.2 | |
0.02 | |
0.01 | |
0.05 |
|
0.06 | |
1.5 | |
16 | |
16 |
|
0.5 | |
0.5 | |
0.5 | |
0.5 |
|
0.5 | |
1.0 | |
0.035 | |
0.035 |
|
0.9 | |
0.05 | |
0.05 |
Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value |
|
3 | |
0.1 | |
1 | |
0.35 |
|
0.2 | |
0.3 | |
1 | |
0.0002 |
|
0.15 | |
0.4 | |
1 | |
0.001 |
|
0.35 | |
0.4 | |
1 | |
0.09 |
|
0.9 | |
2 | |
20 | |
0.1 |
|
0.6 | |
3.5 | |
100 | |
0.1 |
|
0.5 | |
0.9 | |
0 | |
0.9 |
|
1.8 | |
1 | |
1000 | |
1.66 |
|
0.0005 | |
0.003 | |
0.5 | |
1.1 |
|
3 | |
2 | |
0.5 | |
5 |
|
2 | |
2 | |
0.5 | |
5 |
|
2 | |
2 | |
0.5 | |
15 |
|
2 | |
2 | |
0.5 | |
5 |
|
2 | |
2 | |
0.5 | |
2 |
|
2 | |
6 | |
0.5 | |
5 |
|
1.2 | |
0.02 | |
0.01 | |
0.05 |
|
0.06 | |
1.5 | |
16 | |
16 |
|
0.5 | |
0.5 | |
0.5 | |
0.5 |
|
0.5 | |
1.0 | |
0.035 | |
0.035 |
|
0.9 | |
0.05 | |
0.05 |
[1] |
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