How to speed up the simulation time of a big system of ODEs?

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Gabriele Galli
Gabriele Galli 2020-9-10
编辑： Gabriele Galli 2020-9-11
Hi Everyone,
I have a big system of equations (378 diff. equations) and I have to simulate it using an ODE solver (I am using ODE23s) but the simulation time is too long.
Is there a way to improve my code in order to have a small simulation time?
Attached you can find the code.
Thank you!
Gabri

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回答（1 个）

J. Alex Lee
J. Alex Lee 2020-9-11
don't use syms
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J. Alex Lee
J. Alex Lee 2020-9-11
well, if you insist on the symbolics approach, i'm afraid i can't help because i don't know how to use the symbolic toolbox (and don't have it).
if you want to step back, the only anyone can help you is if you show the original 18 equations, and why/how you are taking partial derivatives w.r.t. parameters.
Gabriele Galli
Gabriele Galli 2020-9-11
编辑：Gabriele Galli 2020-9-11
Yes, sure. I mean maybe we can find a better solution. I am not familiar either with the symbolic toolbox. The equations are contained in the f matrix below:
r1 = k1*TGFb_TGFbR;
r2 = k2*TGFbR*TGFb;
r3 = k3*TGFb_TGFbR*(1-exp(-((t-k20)/(k21))^10));
r4 = k4*TGFb_TGFbR_P;
r5 = k5*TGFb_TGFbR_P*I_Smad;
r6 = k6*I_Smad_TGFb_TGFbR_P;
r7 = k7*Smad*TGFb_TGFbR_P;
r8 = k8*Smad;
r9 = k9*Smad_N;
r10 = k10*2*Smad_P*Smad_P;
r11 = k11*Smad_P_Smad_P;
r12 = k10*Smad_P*CoSmad;
r13 = k11*Smad_P_CoSmad;
r14 = k8*CoSmad;
r15 = k9*CoSmad_N;
r16 = k12*k8*Smad_P_Smad_P;
r17 = k8*Smad_P;
r18 = k9*Smad_P_N;
r19 = k12*k8*Smad_P_CoSmad;
r20 = k13*Smad_P_N;
r21 = k10*2*Smad_P_N*Smad_P_N;
r22 = k11*Smad_P_Smad_P_N;
r23 = k10*Smad_P_N*CoSmad_N;
r24 = k11*Smad_P_CoSmad_N;
r25 = k14*Smad_P_CoSmad_N^2/(Smad_P_CoSmad_N^2 + k15^2);
r26 = k16*I_Smad_mRNA1;
r27 = k17*I_Smad_mRNA2;
r28 = k18*I_Smad_mRNA2;
r29 = k19*I_Smad;
f=[r1 - r2;
r1 - r2;
- r1 + r2 - r3 + r4 + r6;
r3 - r4 - r5;
r5 - r6;
- r7 - r8 + r9;
r7 - r10 + r11 - r12 + r13 - r17 + r18;
-r12 + r13 - r14 + r15;
r10 - r11 - r16;
r12 - r13 - r19;
r8 - r9 + r20;
r16 + r21 - r22;
r17 - r18 - r20 - r21 + r22 - r23 + r24;
r19 + r23 - r24;
r14 - r15 - r23 + r24;
r25 - r26;
r26 - r27;
r28 - r29 - r5 + r6];
In particular, k1 through k21 are the parameters with nominal values known and TGFb_TGFbR through I_Smad are the variables, for semplicity I called them x1 thorough x18.
What I want to find is the sensitivity matrix Sxd:
Sxd=dfdx*Sx+dfdk;
%dfdx is the Jacobian of f wrt x(=vector contained the 18 parameters), dim(dfdx=18x18)
%dfdk= is the Jacobian of f wrt k(=vector contained the 21 parameters), dim(dfdk=18x21)
%Sx=sensitivity matrix=dx(t,k)/dk. This is what I want to calculate using the ODE...
... I for simplicity I created I called each of these x1, x2, ..., x378 and reshape...
... them in a matrix 18x21. So basically, each line of the matrix contains the sensitivity...
... of a variable x wrt to each of the 21 parameters.
%Sxd is the first derivative wrt t of Sx. dim(Sxd)=18x21. For this reason, I reshape...
... this matrix in a column vector (dim=378x1) in order to be a suitable input for the ODE solver.
The time span I want to simulate is [0, 180] and we can assume all the 378 initial conditions = 0 except for:
IC(1)=1; IC(2)=1; IC(6)=40.98; IC(8)=34.15; IC(11)=19.02; IC(15)=15.85;
Please, let me know if you need more information.
Thank you!

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