<html xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns="http://www.w3.org/TR/REC-html40">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<meta name="Generator" content="Microsoft Word 15 (filtered medium)">
<style><!--
/* Font Definitions */
@font-face
{font-family:"Cambria Math";
panose-1:2 4 5 3 5 4 6 3 2 4;}
@font-face
{font-family:Calibri;
panose-1:2 15 5 2 2 2 4 3 2 4;}
@font-face
{font-family:Consolas;
panose-1:2 11 6 9 2 2 4 3 2 4;}
/* Style Definitions */
p.MsoNormal, li.MsoNormal, div.MsoNormal
{margin:0in;
font-size:11.0pt;
font-family:"Calibri",sans-serif;}
a:link, span.MsoHyperlink
{mso-style-priority:99;
color:blue;
text-decoration:underline;}
span.apple-converted-space
{mso-style-name:apple-converted-space;}
span.EmailStyle19
{mso-style-type:personal-reply;
font-family:"Calibri",sans-serif;
color:windowtext;}
.MsoChpDefault
{mso-style-type:export-only;
font-size:10.0pt;}
@page WordSection1
{size:8.5in 11.0in;
margin:1.0in 1.0in 1.0in 1.0in;}
div.WordSection1
{page:WordSection1;}
--></style>
</head>
<body lang="EN-US" link="blue" vlink="purple" style="word-wrap:break-word">
<div class="WordSection1">
<p class="MsoNormal">Thanks, Hong. Now it works! I can work with backwards Euler for now. With regards to the other two options, I think -ts_trajectory_solution_only is ok because backwards euler does not have intermediate stage. With respect to -ts_trajectory_type
memory, can I still do checkpointing to be able to solve larger problems?<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">I have also noticed that TSComputeIJacobianP() is only used by the theta methods. Are there plans to support higher order methods?<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Miguel <o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<div style="border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0in 0in 0in">
<p class="MsoNormal" style="margin-left:.5in"><b><span style="font-size:12.0pt;color:black">From:
</span></b><span style="font-size:12.0pt;color:black">"Zhang, Hong" <hongzhang@anl.gov><br>
<b>Date: </b>Monday, December 21, 2020 at 8:16 PM<br>
<b>To: </b>"Salazar De Troya, Miguel" <salazardetro1@llnl.gov><br>
<b>Cc: </b>Satish Balay via petsc-users <petsc-users@mcs.anl.gov><br>
<b>Subject: </b>Re: [petsc-users] Support for full jacobianP in TSSetIJacobianP<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
</div>
<p class="MsoNormal" style="margin-left:.5in">Thank you for providing the example. This is very helpful. Sorry that I was not accurate about what should be in IJacobianP. With current API, a little hack is needed to get it work.
<o:p></o:p></p>
<div>
<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in">In IJacobianP, we have to provide shift*M_P*dt if we expand the formula in the paper to accommodate parameters mass matrices. So I changed your code as follows: <o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
</div>
<div>
<div>
<p class="MsoNormal" style="margin-left:.5in"> if self.deriv == "c":<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in"> dt = ts.getTimeStep() # dt is negative in backward run<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in"> Jp[0, 0] = -shift*udot[0]*dt<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in">I noticed that there is some problem with the input variable Xdot and have been working on a fix. But as a quick workaround, you can use backward Euler with the following options before the fix is ready:<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in">-ts_type beuler -ts_trajectory_type memory -ts_trajectory_solution_only <o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in">Thanks,<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in">Hong<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in"><br>
<br>
<o:p></o:p></p>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<div>
<p class="MsoNormal" style="margin-left:.5in">On Dec 20, 2020, at 3:28 PM, Salazar De Troya, Miguel <<a href="mailto:salazardetro1@llnl.gov">salazardetro1@llnl.gov</a>> wrote:<o:p></o:p></p>
</div>
<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
<div>
<div>
<p class="MsoNormal" style="margin-left:.5in">Hello Hong,<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in">Thank you. My apologies for rushing to blame the API instead of looking at my own code. I’ve put together a minimum example in petsc4py that I am attaching to this email. Here I am solving the simple ODE: c * xdot
= b * x(t) with initial condition x(0) = a and the cost function J equal to the solution at the final time “T”, i.e. J = x(T). The analytical solution is x(t) = a * exp(b/c *t). In the example, there is the option to calculate the derivatives w.r.t “b” or
“c” in the keyword argument “deriv” passed to “SimpleODE”. For “b”, the solver returns the correct derivatives (checked with the analytical expression), but this does not work for “c”. I might be building the wrong jacobian that I pass to “setIJacobianP”.
Could you please take a look at it?<span class="apple-converted-space"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in">Thanks.<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in">Miguel<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
<div style="border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0in 0in 0in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><b><span style="font-size:12.0pt">From:<span class="apple-converted-space"> </span></span></b><span style="font-size:12.0pt">"Zhang, Hong" <<a href="mailto:hongzhang@anl.gov">hongzhang@anl.gov</a>><br>
<b>Date:<span class="apple-converted-space"> </span></b>Saturday, December 19, 2020 at 10:02 AM<br>
<b>To:<span class="apple-converted-space"> </span></b>"Salazar De Troya, Miguel" <<a href="mailto:salazardetro1@llnl.gov">salazardetro1@llnl.gov</a>><br>
<b>Cc:<span class="apple-converted-space"> </span></b>Satish Balay via petsc-users <<a href="mailto:petsc-users@mcs.anl.gov">petsc-users@mcs.anl.gov</a>><br>
<b>Subject:<span class="apple-converted-space"> </span></b>Re: [petsc-users] Support for full jacobianP in TSSetIJacobianP</span><o:p></o:p></p>
</div>
</div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">On Dec 18, 2020, at 6:35 PM, Salazar De Troya, Miguel <<a href="mailto:salazardetro1@llnl.gov">salazardetro1@llnl.gov</a>> wrote:<o:p></o:p></p>
</div>
<div>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
<div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Ok, I was not able to get such case to work in my firedrake-ts implementation. Maybe I am missing something in my code. I looked at the TSAdjoint paper<span class="apple-converted-space"> </span><a href="https://arxiv.org/pdf/1912.07696.pdf">https://arxiv.org/pdf/1912.07696.pdf</a><span class="apple-converted-space"> </span>Equation
2.1 and at the adjoint method for the theta method (Equation 2.15) where the mass matrix is not differentiated w.r.t. the design parameter “p” and decided to ask the question.<o:p></o:p></p>
</div>
</div>
</div>
</blockquote>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">For notational brevity, the formula used in the paper does not assume that the mass matrix depends on the parameters p. But it can be easily extended for this case.<o:p></o:p></p>
</div>
</div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><br>
<br>
<br>
<o:p></o:p></p>
</div>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Is the actual implementation different from what is in the paper?<o:p></o:p></p>
</div>
</div>
</div>
</blockquote>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">The actual implementation is more general than the formula presented in the paper. Note that PETSc TS takes the ODE problem as F(U_t,U,P,t) = G(U,P,t) and does not ask for a mass matrix explicitly from users. When
users provide IFunction, which is F(Udot,U,P,t), IJacobian (dF/dU) and IJacobianP (dF/dP) are needed by TSAdjoint to compute the sensitivities. Differentiating the mass matrix (more precisely, the term M*U_t ) is needed when you prepare the call back function
IJacobianP. So if we have M(P)*U_t - f(t,U,P) in IFunction, IJacobianP should be M_P*U_t - f_P where U_t is provided by PETSc as an input argument.<o:p></o:p></p>
</div>
</div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Thanks,<o:p></o:p></p>
</div>
</div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Hong<o:p></o:p></p>
</div>
</div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><br>
<br>
<br>
<o:p></o:p></p>
</div>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Thanks<o:p></o:p></p>
</div>
</div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Miguel<o:p></o:p></p>
</div>
</div>
<div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
<div style="border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0in 0in 0in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><b><span style="font-size:12.0pt">From:<span class="apple-converted-space"> </span></span></b><span style="font-size:12.0pt">"Zhang, Hong" <<a href="mailto:hongzhang@anl.gov">hongzhang@anl.gov</a>><br>
<b>Date:<span class="apple-converted-space"> </span></b>Friday, December 18, 2020 at 3:11 PM<br>
<b>To:<span class="apple-converted-space"> </span></b>"Salazar De Troya, Miguel" <<a href="mailto:salazardetro1@llnl.gov">salazardetro1@llnl.gov</a>><br>
<b>Cc:<span class="apple-converted-space"> </span></b>Satish Balay via petsc-users <<a href="mailto:petsc-users@mcs.anl.gov">petsc-users@mcs.anl.gov</a>><br>
<b>Subject:<span class="apple-converted-space"> </span></b>Re: [petsc-users] Support for full jacobianP in TSSetIJacobianP</span><o:p></o:p></p>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
</div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">The current interface is general and should be applicable to this case as soon as users can provide IJacobianP, which is dF(Udot,U,P,t)/dP. Were you able to generate it in firedrake? If so, could you provide an
example that I can test?<span class="apple-converted-space"> </span><o:p></o:p></p>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Thanks,<o:p></o:p></p>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Hong<o:p></o:p></p>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><br>
<br>
<br>
<br>
<o:p></o:p></p>
</div>
</div>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">On Dec 18, 2020, at 10:58 AM, Salazar De Troya, Miguel <<a href="mailto:salazardetro1@llnl.gov">salazardetro1@llnl.gov</a>> wrote:<o:p></o:p></p>
</div>
</div>
</div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
<div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Yes, that is the case I am considering. The special case I am concerned about is as following: the heat equation in variational form and in firedrake/UFL notation is as follows: p*u_t*v*dx + inner(grad(u), grad(v))*dx
= 0, where u is the temperature, u_t is its time derivative, v is just the test function, dx is the integration domain and p is the design parameter. If “p” were discontinuous, one can’t just factor “p” into the second term due to the divergence theorem. Meaning
that p*u_t*v*dx + inner(grad(u), grad(v))*dx = 0 is different than u_t*v*dx + inner(1.0 / p * grad(u), grad(v))*dx = 0, which is what ideally one would obtain in order to adapt to the current interface in TSAdjoint.<o:p></o:p></p>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Thanks<o:p></o:p></p>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Miguel<o:p></o:p></p>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
</div>
<div style="border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0in 0in 0in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><b><span style="font-size:12.0pt">From:<span class="apple-converted-space"> </span></span></b><span style="font-size:12.0pt">"Zhang, Hong" <<a href="mailto:hongzhang@anl.gov">hongzhang@anl.gov</a>><br>
<b>Date:<span class="apple-converted-space"> </span></b>Thursday, December 17, 2020 at 7:25 PM<br>
<b>To:<span class="apple-converted-space"> </span></b>"Salazar De Troya, Miguel" <<a href="mailto:salazardetro1@llnl.gov">salazardetro1@llnl.gov</a>><br>
<b>Cc:<span class="apple-converted-space"> </span></b>Satish Balay via petsc-users <<a href="mailto:petsc-users@mcs.anl.gov">petsc-users@mcs.anl.gov</a>><br>
<b>Subject:<span class="apple-converted-space"> </span></b>Re: [petsc-users] Support for full jacobianP in TSSetIJacobianP</span><o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
</div>
</div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Hi Miguel,<span class="apple-converted-space"> </span><o:p></o:p></p>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Thank you for the nice work. I do not understand what you propose to do here. What is the obstacle to using current TSSetIJacobianP() for the corner case you mentioned? Are you considering a case in which the mass
matrix is parameterized, e.g. M(p) udot - f(t,u) = g(t,u) ?<o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Thanks,<o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Hong<span class="apple-converted-space"> </span><o:p></o:p></p>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><br>
<br>
<br>
<br>
<br>
<o:p></o:p></p>
</div>
</div>
</div>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">On Dec 17, 2020, at 3:38 PM, Salazar De Troya, Miguel via petsc-users <<a href="mailto:petsc-users@mcs.anl.gov">petsc-users@mcs.anl.gov</a>> wrote:<o:p></o:p></p>
</div>
</div>
</div>
</div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
</div>
<div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><span lang="ES">Hello,</span><o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><span lang="ES"> </span><o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">I am working on hooking up TSAdjoint with pyadjoint through the firedrake-ts interface (<a href="https://github.com/IvanYashchuk/firedrake-ts"><span style="color:#0563C1">https://github.com/IvanYashchuk/firedrake-ts</span></a>).
I have done most of the implementation and now I am just testing for corner cases. One of them is when the design variable is multiplying the first derivative term. It would be the case ofF(Udot,U,P,t) = G(U,P,t) in<span class="apple-converted-space"> </span><a href="https://www.mcs.anl.gov/petsc/petsc-master/docs/manualpages/Sensitivity/TSSetIJacobianP.html"><span style="color:#0563C1">https://www.mcs.anl.gov/petsc/petsc-master/docs/manualpages/Sensitivity/TSSetIJacobianP.html</span></a><span class="apple-converted-space"> </span>.
I imagine that one could think of refactoring the “P” in the left hand side to the right hand side, but this is not trivial when “P” is a discontinuous field over the domain. I think it would be better to include the case of F(Udot,U,P,t) = G(U,P,t) in TSSetIJacobianP
and I am volunteering to do it. Given the current implementation of TSAdjoint, is this something feasible?<o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Thanks<o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in">Miguel<o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><span lang="ES" style="font-size:9.0pt;font-family:Consolas">Miguel A. Salazar de Troya</span><o:p></o:p></p>
</div>
</div>
</div>
</div>
<div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><span style="font-size:9.0pt;font-family:Consolas">Postdoctoral Researcher, Lawrence Livermore National Laboratory</span><o:p></o:p></p>
</div>
</div>
</div>
</div>
</div>
<div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><span style="font-size:9.0pt;font-family:Consolas">B141</span><o:p></o:p></p>
</div>
</div>
</div>
</div>
</div>
<div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><span style="font-size:9.0pt;font-family:Consolas">Rm: 1085-5</span><o:p></o:p></p>
</div>
</div>
</div>
</div>
</div>
<div>
<div>
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><span style="font-size:9.0pt;font-family:Consolas">Ph: 1(925) 422-6411</span><o:p></o:p></p>
</div>
</div>
</div>
</div>
</div>
</div>
</blockquote>
</div>
</div>
</div>
</blockquote>
</div>
</div>
</div>
</blockquote>
</div>
<div style="margin-left:.5in">
<p class="MsoNormal" style="margin-left:.5in"><br>
<br>
<br>
<o:p></o:p></p>
</div>
<p class="MsoNormal" style="margin-left:.5in"><simple-ode.py><o:p></o:p></p>
</div>
</blockquote>
</div>
<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
</div>
</div>
</body>
</html>