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Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory

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  • 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.

    Mathematics Subject Classification: Primary:65C30, 60H10;Secondary:68P05.

    Citation:

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  • Figure 1.  Outline of the adaptive SODE algorithm based on Rejection Sampling with Memory. This depicts the general schema for the Rejection Sampling with Memory algorithm (RSwM). The green arrow depicts the path taken when the step is rejected, whereas the red arrow depicts the path that is taken when the step is accepted. The blue text denotes steps which are specific to the stochastic systems in order to ensure correct sampling properties which are developed in section 4.

    Figure 2.  Adaptive algorithm Kolmogorov-Smirnov Tests. Equation 31 was solved from $t=0$ to $t=2$. Scatter plots of the p-values from Kolmogorov-Smirnov tests against the normal distribution. At the end of each run, a Kolmogorov-Smirnov test was performed on the values at end of the Brownian path for 200 simulations. The $x$-axis is the absolute tolerance (with relative tolerance set to zero) and the $y$-axis is the p-value of the Kolmogorov Smirnov tests.

    Figure 3.  Adaptive algorithm correctness checks. QQplots of the distribution of the Brownian path at the end time $T=2$ over 10,000 paths. The $x$-axis is the quantiles of the standard normal distribution while the $y$-axis is the estimated quantiles for the distribution of $W_2 / \sqrt{2}$. Each row is for a example equation for Examples 1-3 respectively, and each column is for the algorithm RSwM1-3 respectively. $\epsilon=10^{-4}$. The red dashed line represents $x=y$, meaning the quantiles of a 10,000 standard normal random variables equate with the quantiles of the sample. The blue circles represent the quantile estimates for $W(2)/\sqrt{2}$ which should be distributed as a standard normal.

    Figure 4.  Adaptive Algorithm Error Comparison on Equation 31, Equation 33, and Equation 35. Comparison of the rejection sampling with memory algorithms on examples 1-3. Along the $x$-axis is $\epsilon$ which is the user chosen local error tolerance. The $y$-axis is the average $l^{2}$ error along the path. These values are taken as the mean for 100 runs. Both axes are log-scaled.

    Figure 5.  Adaptive Algorithm Timing Comparison. Comparison of the rejection sampling with memory algorithms. Along the $x$-axis is $\epsilon$ which is the user chosen local error tolerance. In the $y$-axis time is plotted (in seconds). The values are the elapsed time for a 1000 Both axes are log-scaled.

    Figure 6.  Solution to the Lorenz system Equation 39 on $t\in[0,10]$ with additive noise and parameters $\alpha=10$, $\rho=28$, $\sigma =3$, and $\beta=8/3$ at varying tolerances. The system was solved using ESRK+RSwM3 with the relative tolerance fixed at zero and varying absolute tolerances.

    Figure 7.  Stochastic cell differentiation model solutions. (A) Timeseries of the concentration of $[Ecad]$. The solution is plotted once every 100 accepted steps due to memory limitations. (B) Timeseries of the concentration of $[Vim]$. The solution is plotted once every 100 accepted steps due to memory limitations. (C) Accepted $h$ over time. The $h$ values were taken every 100 accepted steps due to memory limitations. (D) Elapsed time of the Euler-Maruyama and ESRK+RSwM3 algorithms on the stochastic cell model. Each algorithm was used to solve the model on $t\in[0,1]$ 10,000 times. The elapsed time for the fixed timestep methods for given $h$'s are shown as the filled lines, while the dashed line is the elapsed time for the adaptive method. The red circles denote the minimal $h$ and times for which the method did not show numerical instability in the ensemble.

    Table 1.  ESRK1. Table (a) shows the legend for how the numbers in in Table (b) correspond to the coefficient arrays/matrices $c^{(i)}, A^{(i)}, B^{(i)}, \alpha, \beta^{(i)}, \tilde{\alpha}$, and $\tilde{\beta^{(i)}}$. For example, these tables show that $\alpha^T = (\frac{1}{3},\frac{2}{3},0,0)$. Note that the matrices $A^{(i)}$ and $B^{(i)}$ are lower triangular since the method is explicit.

     | Show Table
    DownLoad: CSV
    Algorithm 1 RSwM1
    1: Set the values $\epsilon$, $h_{max}$, $T$
    2: Set $t=0$, $W=0$, $Z=0$, $X=X_0$
    3: Take an initial $h$, $\Delta Z,\Delta W \sim N(0,h)$
    4: While $t<T$ do
    5:     Attempt a step with $h$, $\Delta W$, $\Delta Z$ to calculate $X_{temp}$ according to (2)
    6:     Calculate E according to (9)
    7:     Update $q$ using (21)
    8:     If $(q<1)$ then            ▷% Reject the Step
    9:         Take $\Delta\tilde{W}\sim N\left(q\Delta W,(1-q)qh\right)$ and $\Delta\tilde{Z}\sim N\left(q\Delta Z,(1-q)qh\right)$
    10:         Calculate $\overline{\Delta W}=\Delta W-\Delta\tilde{W}$ and $\overline{\Delta Z}=\Delta Z-\Delta\tilde{Z}$
    11:         Push $\left(\left(1-q\right)h,\overline{\Delta W},\overline{\Delta Z}\right)$ into stack $S$
    12:         Update $h := qh$
    13:         Update $\Delta W := \Delta \tilde{W}$, $\Delta Z := \Delta \tilde{Z}$
    14:     else            ▷% Accept the Step
    15:         Update $t:=t+h$, $W:=W+\Delta W$, $Z:=Z+\Delta Z$, $X = X + X_{temp}$
    16:         if ($S$ is empty) then
    17:             Update $c:=\min(h_{max},qh)$, $h:=\min(c,T-t)$
    18:             Take $\Delta W, \Delta Z \sim N(0,h)$
    19:         else
    20:             Pop the top of $S$ as $L$
    21:             Update $h:=L_1$, $\Delta W := L_2$, $\Delta Z := L_3$
    22:         end if
    23:     end if
    24: end while
     | Show Table
    DownLoad: CSV
    Algorithm 2 RSwM3
    1: Set the values $\epsilon$, $h_{max}$, $T$
    2: Set $t=0$, $W=0$, $Z=0$, $X=X_0$
    3: Take an initial $h$, $\Delta Z,\Delta W \sim N(0,h)$
    4: while $t<T$ do
    5:     Attempt a step with $h$, $\Delta W$, $\Delta Z$ to calculate $X_{temp}$ according to (2)
    6:     Calculate E according to (9)
    7:     Update $q$ using (21)
    8:     if $(q<1)$ then            ▷% Reject the Step
    9:         Set $h_{s}=0$, $\Delta W =0$, $\Delta Z =0$
    10:         while $S_2$ is not empty do
    11:             Pop the top of $S_2$ as $L$
    12:             if $h_{s} + L_{1}<(1-q)h$ then
    13:                 Update $h_{s} := h_{s} +L_{1}$
    14:                 Update $\Delta W_{tmp} := \Delta W_{tmp} + L_{2}$, $\Delta Z_{tmp} := \Delta Z_{tmp} + L_{3}$
    15:             else
    16:                 Push $L$ onto $S_2$ and break
    17:             end if
    18:         end while
    19:         Set $h_{K} = h - h_s$, $K_{2} = \Delta W - \Delta W_{tmp}$, $K_{3} = \Delta Z - \Delta Z_{tmp}$
    20:         Set $q_{K} = \frac{qh}{h_K}$
    21:         Take $\Delta \tilde{W} \sim N(q_K K_2,(1-q_K)q_K L_1)$
    22:         Take $\Delta \tilde{Z} \sim N(q_K K_3,(1-q_K)q_K L_1)$
    23:         Pop $((1-q_K)h_{K},K_2 - \Delta \tilde{W}, K_3 - \Delta \tilde{Z})$ onto $S_1$
    24:         Pop $(q_K L_1, \Delta \tilde{W}, \Delta \tilde{Z})$ onto $S_2$
    25:         Update $\Delta W = \Delta \tilde{W}$, $\Delta Z = \Delta \tilde{Z}$, $h = qh$
    26:     else            ▷% Accept the Step
    27:         Update $t:=t+h$, $W:=W+\Delta W$, $Z:=Z+\Delta Z$, $X = X + X_{temp}$
    28:         Empty $S_2$
    29:         Update $c:=\min(h_{max},qh)$, $h:=\min(c,T-t)$
    30:         Set $h_{s}=0$, $\Delta W =0$, $\Delta Z =0$
    31:         while $S$ is not empty do
    32:             Pop the top of $S_1$ as $L$
    33:             if ($h_s + L_1 < h$) then            ▷% Temporary not far enough
    34:                 Update $h_{s} := h_{s} + L_{1}$, $\Delta W := \Delta W + L_{2}$, $\Delta Z := \Delta Z + L_{3}$
    35:                 Push a copy of $L$ onto $S_2$
    36:             else            ▷% Final part of step from stack
    37:                 Set $q_{tmp}=\frac{h-h_{s}}{L_{1}}$
    38:                 Let $\Delta \tilde{W} \sim N(q_{tmp}L_{2},(1-q_{tmp})q_{tmp}L_1)$
    39:                 Let $\Delta \tilde{Z} \sim N(q_{tmp}L_{3},(1-q_{tmp})q_{tmp}L_1)$
    40:                 Push $((1-q_{tmp})L_1,L_2 - \Delta \tilde{W}, L_3 - \Delta \tilde{Z})$ onto $S_1$
    41:                 Update $\Delta W := \Delta W + \Delta \tilde{W}$, $\Delta Z := \Delta Z + \Delta \tilde{Z}$
    42:             end if
    43:         end while
                                    ▷% Update for last portion to step. Note zero if final part is from stack
    44:         if ($h-h_s$ is not zero) then
    45:             Let $\eta_W, \eta_{Z} \sim N(0,h-h_s)$
    46:             Update $\Delta W = \Delta W + \eta_{W}$, $\Delta Z = \Delta Z + \eta_{Z}$
    47:             Push $(h-h_s,\eta_{W},\eta_{Z})$ onto $S_2$
    48:         end if
    49:     end if
    50: end while
     | Show Table
    DownLoad: CSV

    Table 2.  Fixed timestep method fails and runtimes. The fixed timestep algorithms and ESRK+RSwM3 algorithms were used to solve the stochastic cell model on $t\in[0,1]$ 10,000 times. Failures were detected by checking if the solution contained any NaN values. During a run, if any NaNs were detected, the solver would instantly end the simulations and declare a failure. The runtime for the adaptive algorithm (with no failures) was 186.81 seconds.

    Euler-Maruyama Runge-Kutta Milstein Rößler SRI
    $\Delta t$ Fails (/10,000) Time (s) Fails (/10,000) Time (s) Fails (/10,000) Time (s)
    $2^{-16}$ 137 133.35 131 211.92 78 609.27
    $2^{-17}$ 39 269.09 26 428.28 17 1244.06
    $2^{-18}$ 3 580.14 6 861.01 0 2491.37
    $2^{-19}$ 1 1138.41 1 1727.91 0 4932.70
    $2^{-20}$ 0 2286.35 0 3439.90 0 9827.16
    $2^{-21}$ 0 4562.20 0 6891.35 0 19564.16
     | Show Table
    DownLoad: CSV

    Table 3.  $qmax$ determination tests. Equation 31, Equation 33, and Equation 35 were solved using the ESRK+RSwM3 algorithm with a relative tolerance of 0 and absolute tolerance of $2^{-14}$. The elapsed time to solve a Monte Carlo simulation of 100,000 simulations to $T=1$ was saved and the mean error at $T=1$ was calculated. The final column shows timing results for using ESRK+RSwM3 on the stochastic cell model from F solved with the same tolerance settings as in subsection 6.2 to solve a Monte Carlo simulation of 10,000 simulations.

    Example 1 Example 2 Example 3 Cell Model
    qmax Time (s) Error Time (s) Error Time (s) Error Time (s)
    $1+2^{-5}$ 37.00 2.57e-8 60.87 2.27e-7 67.71 3.42e-9 229.83
    $1+2^{-4}$ 34.73 2.82e-8 32.40 3.10e-7 66.68 3.43e-9 196.36
    $1+2^{-3}$ 49.14 3.14e-8 132.33 8.85e-7 65.94 3.44e-9 186.81
    $1+2^{-2}$ 39.33 3.59e-8 33.90 1.73e-6 66.33 3.44e-9 205.57
    $1+2^{-1}$ 38.22 3.82e-8 159.94 2.58e-6 68.16 3.44e-9 249.77
    $1+2^{0}$ 82.76 4.41e-8 34.41 3.58e-6 568.22 3.44e-9 337.99
    $1+2^{1}$ 68.16 9.63e-8 33.98 6.06e-6 87.50 3.22e-9 418.78
    $1+2^{2}$ 48.23 1.01e-7 33.97 9.74e-6 69.78 3.44e-9 571.59
     | Show Table
    DownLoad: CSV
    Algorithm 3 Initial $h$ Determination
    1: Let $d_0 = \Vert X_0 \Vert$
    2: Calculate $f_0 = f(X_0,t)$ and $\sigma_0 = 3g(X_0,t)$
    3: Let $d_1 = \Vert \mathrm{max}(|f_0 +\sigma_0|,|f_0 -\sigma_0|)\Vert$
    4: if $d_0 < 10^{-5}$ or $d_{1} < 10^{-5}$ then
    5:     Let $h_0 = 10^{-6}$
    6: else
    7:     Let $h_0 = 0.01(d_0/d_1)$
    8: end if
    9: Calculate an Euler step: $X_1 = X_0 + h_0f_0$
    10: Calculate new estimates: $f_1 = f(X_1,t)$ and $\sigma_0 = 3g(X_1,t)$
    11: Determine $\sigma_1^M = \mathrm{max}(|\sigma_0 +\sigma_1|),|\sigma_0 -\sigma_1|)$
    12: Let $d_2 = \Vert \mathrm{max}(|f_1-f_0 + \sigma_1^M|,|f_1-f_0 - \sigma_1^M|)\Vert/h_0$
    13: if $\mathrm{max}(d_1,d_2)<10^{-15}$ then
    14:     Let $h_1 = \mathrm{max}(10^{-6},10^{-3}h_0)$
    15: else
    16:     Let $h_1 = 10^{-(2+\log_10(\mathrm{max}(d_1,d_2))/(order+0.5)}$
    17: end if
    18: Let $h=\mathrm{min}(100h_0,h_1)$
     | Show Table
    DownLoad: CSV

    Table 4.  SRA1. Table (a) shows the legend for how the numbers in in Table (b) correspond to the coefficient arrays/matrices $c^{(i)}, A^{(i)}, B^{(i)}, \alpha,$ and $\beta^{(i)}$. Note that the matrices $A^{(i)}$ and $B^{(i)}$ are lower triangular since the method is explicit.

     | Show Table
    DownLoad: CSV
    Algorithm 4 RSwM2
    1: Set the values $\epsilon$, $h_{max}$, $T$
    2: Set $t=0$, $W=0$, $Z=0$, $X=X_0$
    3: Take an initial $h$, $\Delta Z,\Delta W \sim N(0,h)$
    4: while $t<T$ do
    5:     Attempt a step with $h$, $\Delta W$, $\Delta Z$ to calculate $X_{temp}$ according to (2)
    6:     Calculate E according to (9)
    7:     Update $q$ using (21)
    8:     if $(q<1)$ then            ▷% Reject the Step
    9:         Take $\Delta\tilde{W}\sim N\left(q\Delta W,(1-q)qh\right)$ and $\Delta\tilde{Z}\sim N\left(q\Delta Z,(1-q)qh\right)$
    10:         Calculate $\overline{\Delta W}=\Delta W-\Delta\tilde{W}$ and $\overline{\Delta Z}=\Delta Z-\Delta\tilde{Z}$
    11:         Push $\left(\left(1-q\right)h,\overline{\Delta W},\overline{\Delta Z}\right)$ into stack $S$
    12:         Update $h := qh$
    13:         Update $\Delta W := \Delta \tilde{W}$, $\Delta Z := \Delta \tilde{Z}$
    14:     else            ▷% Accept the Step
    15:         Update $t:=t+h$, $W:=W+\Delta W$, $Z:=Z+\Delta Z$, $X = X + X_{temp}$
    16:         Update $c:=\min(h_{max},qh)$, $h:=\min(c,T-t_n)$
    17:         Set $h_{s}=0$, $\Delta W =0$, $\Delta Z =0$
    18:         while $S$ is not empty do
    19:             Pop the top of $S$ as $L$
    20:             if ($h_s + L_1 < h$) then            ▷% Temporary not far enough
    21:                 Update $h_{s} := h_{s} + L_{1}$, $\Delta W := \Delta W + L_{2}$, $\Delta Z := \Delta Z + L_{3}$
    22:             else            ▷% Final part of step from stack
    23:                 Set $q_{tmp}=\frac{h-h_{s}}{L_{1}}$
    24:                 Let $\Delta \tilde{W} \sim N(q_{tmp}L_{2},(1-q_{tmp})q_{tmp}L_1)$
    25:                 Let $\Delta \tilde{Z} \sim N(q_{tmp}L_{3},(1-q_{tmp})q_{tmp}L_1)$
    26:                 Push $((1-q_{tmp})L_1,L_2 - \Delta \tilde{W}, L_3 - \Delta \tilde{Z})$ onto $S$
    27:                 Update $\Delta W := \Delta W + \Delta \tilde{W}$, $\Delta Z := \Delta \tilde{Z}$
    28:             end if
    29:         end while
            ▷% Update for last portion to step. Note zero if final part is from stack
    30:         if ($h-h_s$ is not zero) then
    31:             Let $\eta_W, \eta_{Z} \sim N(0,h-h_s)$
    32:             Update $\Delta W = \Delta W + \eta_{W}$, $\Delta Z = \Delta Z + \eta_{Z}$
    33:         end if
    34:     end if
    35: end while
     | Show Table
    DownLoad: CSV

    Table 5.  Table of Parameter Values for the Stochastic Cell Model.

    Parameter Value Parameter Value Parameter Value Parameter Value
    $J1_{200}$ 3 $J1_{E}$ 0.1 $K_{2}$ 1 $k0_{O}$ 0.35
    $J2_{200}$ 0.2 $J2_{E}$ 0.3 $K_{3}$ 1 $kO_{200}$ 0.0002
    $J1_{34}$ 0.15 $J1_{V}$ 0.4 $K_{4}$ 1 $kO_{34}$ 0.001
    $J2_{34}$ 0.35 $J2_{V}$ 0.4 $K_{5}$ 1 $kd_{snail}$ 0.09
    $J_{O}$ 0.9 $J3_{V}$ 2 $K_{TR}$ 20 $kd_{tgf}$ 0.1
    $J0_{snail}$ 0.6 $J1_{zeb}$ 3.5 $K_{SR}$ 100 $kd_{zeb}$ 0.1
    $J1_{snail}$ 0.5 $J2_{zeb}$ 0.9 $TGF0$ 0 $kd_{TGF}$ 0.9
    $J2_{snail}$ 1.8 $K_{1}$ 1 $Tk$ 1000 $kd_{ZEB}$ 1.66
    $k0_{snail}$ 0.0005 $k0_{zeb}$ 0.003 $\lambda_{1}$ 0.5 $k0_{TGF}$ 1.1
    $n1_{200}$ 3 $n1_{snail}$ 2 $\lambda_{2}$ 0.5 $k0_{E}$ 5
    $n2_{200}$ 2 $n1_{E}$ 2 $\lambda_{3}$ 0.5 $k0_{V}$ 5
    $n1_{34}$ 2 $n2_{E}$ 2 $\lambda_{4}$ 0.5 $k_{E1}$ 15
    $n2_{34}$ 2 $n1_{V}$ 2 $\lambda_{5}$ 0.5 $k_{E2}$ 5
    $n_{O}$ 2 $n2_{V}$ 2 $\lambda_{SR}$ 0.5 $k_{V1}$ 2
    $n0_{snail}$ 2 $n2_{zeb}$ 6 $\lambda_{TR}$ 0.5 $k_{V2}$ 5
    $k_{O}$ 1.2 $k_{200}$ 0.02 $k_{34}$ 0.01 $k_{tgf}$ 0.05
    $k_{zeb}$ 0.06 $k_{TGF}$ 1.5 $k_{SNAIL}$ 16 $k_{ZEB}$ 16
    $kd_{ZR_{1}}$ 0.5 $kd_{ZR_{2}}$ 0.5 $kd_{ZR_{3}}$ 0.5 $kd_{ZR_{4}}$ 0.5
    $kd_{ZR_{5}}$ 0.5 $kd_{O}$ 1.0 $kd_{200}$ 0.035 $kd_{34}$ 0.035
    $kd_{SR}$ 0.9 $kd_{E}$ 0.05 $kd_{V}$ 0.05
     | Show Table
    DownLoad: CSV
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