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<div class="section" id="performing-sensitivity-analysis">
<span id="chapter-sa"></span><h1>Performing sensitivity analysis<a class="headerlink" href="#performing-sensitivity-analysis" title="Permalink to this headline">¶</a></h1>
<p>The <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span></code> library provides a framework based on discrete adjoint models
for sensitivity analysis for ODEs and DAEs. The ODE/DAE solution process
(henceforth called the forward run) can be obtained by using either
explicit or implicit solvers in <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span></code>, depending on the problem
properties. Currently supported method types are <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSRK.html#TSRK">TSRK</a></span></code> (Runge-Kutta)
explicit methods and <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSTHETA.html#TSTHETA">TSTHETA</a></span></code> implicit methods, which include
<code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSBEULER.html#TSBEULER">TSBEULER</a></span></code> and <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSCN.html#TSCN">TSCN</a></span></code>.</p>
<div class="section" id="using-the-discrete-adjoint-methods">
<h2>Using the discrete adjoint methods<a class="headerlink" href="#using-the-discrete-adjoint-methods" title="Permalink to this headline">¶</a></h2>
<p>Consider the ODE/DAE</p>
<div class="math">
\[F(t,y,\dot{y},p) = 0, \quad y(t_0)=y_0(p) \quad t_0 \le t \le t_F
\]</div>
<p>and the cost function(s)</p>
<div class="math">
\[\Psi_i(y_0,p) = \Phi_i(y_F,p) + \int_{t_0}^{t_F} r_i(y(t),p,t)dt \quad i=1,...,n_\text{cost}.
\]</div>
<p>The <code class="docutils literal notranslate"><span class="pre">TSAdjoint</span></code> routines of PETSc provide</p>
<div class="math">
\[\frac{\partial \Psi_i}{\partial y_0} = \lambda_i
\]</div>
<p>and</p>
<div class="math">
\[\frac{\partial \Psi_i}{\partial p} = \mu_i + \lambda_i (\frac{\partial y_0}{\partial p}).
\]</div>
<p>To perform the discrete adjoint sensitivity analysis one first sets up
the <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span></code> object for a regular forward run but with one extra function
call</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSetSaveTrajectory.html#TSSetSaveTrajectory">TSSetSaveTrajectory</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">),</span>
</pre></div>
</div>
<p>then calls <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSolve.html#TSSolve">TSSolve</a>()</span></code> in the usual manner.</p>
<p>One must create two arrays of <span class="math">\(n_\text{cost}\)</span> vectors
<span class="math">\(\lambda\)</span> and<span class="math">\(\mu\)</span> (if there are no parameters <span class="math">\(p\)</span>
then one can use <code class="docutils literal notranslate"><span class="pre">NULL</span></code> for the <span class="math">\(\mu\)</span> array.) The
<span class="math">\(\lambda\)</span> vectors are the same dimension and parallel layout as
the solution vector for the ODE, the <span class="math">\(mu\)</span> vectors are of dimension
<span class="math">\(p\)</span>; when <span class="math">\(p\)</span> is small usually all its elements are on the
first MPI process, while the vectors have no entries on the other
processes. <span class="math">\(\lambda_i\)</span> and <span class="math">\(mu_i\)</span> should be initialized with
the values <span class="math">\(d\Phi_i/dy|_{t=t_F}\)</span> and <span class="math">\(d\Phi_i/dp|_{t=t_F}\)</span>
respectively. Then one calls</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSSetCostGradients.html#TSSetCostGradients">TSSetCostGradients</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscInt.html#PetscInt">PetscInt</a></span> <span class="n">numcost</span><span class="p">,</span> <span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span> <span class="o">*</span><span class="n">lambda</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span> <span class="o">*</span><span class="n">mu</span><span class="p">);</span>
</pre></div>
</div>
<p>If <span class="math">\(F()\)</span> is a function of <span class="math">\(p\)</span> one needs to also provide the
Jacobian <span class="math">\(-F_p\)</span> with</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSSetRHSJacobianP.html#TSSetRHSJacobianP">TSSetRHSJacobianP</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/Mat.html#Mat">Mat</a></span> <span class="n">Amat</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscErrorCode.html#PetscErrorCode">PetscErrorCode</a></span> <span class="p">(</span><span class="o">*</span><span class="n">fp</span><span class="p">)(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscReal.html#PetscReal">PetscReal</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/Mat.html#Mat">Mat</a></span><span class="p">,</span><span class="kt">void</span><span class="o">*</span><span class="p">),</span><span class="kt">void</span> <span class="o">*</span><span class="n">ctx</span><span class="p">)</span>
</pre></div>
</div>
<p>The arguments for the function <code class="docutils literal notranslate"><span class="pre">fp()</span></code> are the timestep context,
current time, <span class="math">\(y\)</span>, and the (optional) user-provided context.</p>
<p>If there is an integral term in the cost function, i.e. <span class="math">\(r\)</span> is
nonzero, it can be transformed into another ODE that is augmented to the
original ODE. To evaluate the integral, one needs to create a child
<code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span></code> objective by calling</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSCreateQuadratureTS.html#TSCreateQuadratureTS">TSCreateQuadratureTS</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscBool.html#PetscBool">PetscBool</a></span> <span class="n">fwd</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="o">*</span><span class="n">quadts</span><span class="p">);</span>
</pre></div>
</div>
<p>and provide the ODE RHS function (which evaluates the integrand
<span class="math">\(r\)</span>) with</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSetRHSFunction.html#TSSetRHSFunction">TSSetRHSFunction</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">quadts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span> <span class="n">R</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscErrorCode.html#PetscErrorCode">PetscErrorCode</a></span> <span class="p">(</span><span class="o">*</span><span class="n">rf</span><span class="p">)(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscReal.html#PetscReal">PetscReal</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span><span class="p">,</span><span class="kt">void</span><span class="o">*</span><span class="p">),</span><span class="kt">void</span> <span class="o">*</span><span class="n">ctx</span><span class="p">)</span>
</pre></div>
</div>
<p>Similar to the settings for the original ODE, Jacobians of the integrand
can be provided with</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSetRHSJacobian.html#TSSetRHSJacobian">TSSetRHSJacobian</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">quadts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span> <span class="n">DRDU</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span> <span class="n">DRDU</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscErrorCode.html#PetscErrorCode">PetscErrorCode</a></span> <span class="p">(</span><span class="o">*</span><span class="n">drdyf</span><span class="p">)(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscReal.html#PetscReal">PetscReal</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span><span class="o">*</span><span class="p">,</span><span class="kt">void</span><span class="o">*</span><span class="p">),</span><span class="kt">void</span> <span class="o">*</span><span class="n">ctx</span><span class="p">)</span>
<span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSSetRHSJacobianP.html#TSSetRHSJacobianP">TSSetRHSJacobianP</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">quadts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span> <span class="n">DRDU</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span> <span class="n">DRDU</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscErrorCode.html#PetscErrorCode">PetscErrorCode</a></span> <span class="p">(</span><span class="o">*</span><span class="n">drdyp</span><span class="p">)(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscReal.html#PetscReal">PetscReal</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span><span class="o">*</span><span class="p">,</span><span class="kt">void</span><span class="o">*</span><span class="p">),</span><span class="kt">void</span> <span class="o">*</span><span class="n">ctx</span><span class="p">)</span>
</pre></div>
</div>
<p>where <span class="math">\(\mathrm{drdyf}= dr /dy\)</span>, <span class="math">\(\mathrm{drdpf} = dr /dp\)</span>.
Since the integral term is additive to the cost function, its gradient
information will be included in <span class="math">\(\lambda\)</span> and <span class="math">\(\mu\)</span>.</p>
<p>Lastly, one starts the backward run by calling</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSAdjointSolve.html#TSAdjointSolve">TSAdjointSolve</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">).</span>
</pre></div>
</div>
<p>One can obtain the value of the integral term by calling</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSGetCostIntegral.html#TSGetCostIntegral">TSGetCostIntegral</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span> <span class="o">*</span><span class="n">q</span><span class="p">).</span>
</pre></div>
</div>
<p>or accessing directly the solution vector used by quadts.</p>
<p>The second argument of <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSCreateQuadratureTS.html#TSCreateQuadratureTS">TSCreateQuadratureTS</a>()</span></code> allows one to choose
if the integral term is evaluated in the forward run (inside
<code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSolve.html#TSSolve">TSSolve</a>()</span></code>) or in the backward run (inside <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSAdjointSolve.html#TSAdjointSolve">TSAdjointSolve</a>()</span></code>) when
<code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSSetCostGradients.html#TSSetCostGradients">TSSetCostGradients</a>()</span></code> and <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSSetCostIntegrand.html#TSSetCostIntegrand">TSSetCostIntegrand</a>()</span></code> are called before
<code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSolve.html#TSSolve">TSSolve</a>()</span></code>. Note that this also allows for evaluating the integral
without having to use the adjoint solvers.</p>
<p>To provide a better understanding of the use of the adjoint solvers, we
introduce a simple example, corresponding to
<a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/power_grid/ex3adj.c.html">TS Power Grid Tutorial ex3adj</a>.
The problem is to study dynamic security of power system when there are
credible contingencies such as short-circuits or loss of generators,
transmission lines, or loads. The dynamic security constraints are
incorporated as equality constraints in the form of discretized
differential equations and inequality constraints for bounds on the
trajectory. The governing ODE system is</p>
<div class="math">
\[\begin{aligned}
\phi' &= &\omega_B (\omega - \omega_S) \\
2H/\omega_S \, \omega' & =& p_m - p_{max} sin(\phi) -D (\omega - \omega_S), \quad t_0 \leq t \leq t_F,\end{aligned}\]</div>
<p>where <span class="math">\(\phi\)</span> is the phase angle and <span class="math">\(\omega\)</span> is the
frequency.</p>
<p>The initial conditions at time <span class="math">\(t_0\)</span> are</p>
<div class="math">
\[\begin{aligned}
\phi(t_0) &=& \arcsin \left( p_m / p_{max} \right), \\
w(t_0) & =& 1.\end{aligned}\]</div>
<p><span class="math">\(p_{max}\)</span> is a positive number when the system operates normally.
At an event such as fault incidence/removal, <span class="math">\(p_{max}\)</span> will change
to <span class="math">\(0\)</span> temporarily and back to the original value after the fault
is fixed. The objective is to maximize <span class="math">\(p_m\)</span> subject to the above
ODE constraints and <span class="math">\(\phi<\phi_S\)</span> during all times. To accommodate
the inequality constraint, we want to compute the sensitivity of the
cost function</p>
<div class="math">
\[\Psi(p_m,\phi) = -p_m + c \int_{t_0}^{t_F} \left( \max(0, \phi - \phi_S ) \right)^2 dt
\]</div>
<p>with respect to the parameter <span class="math">\(p_m\)</span>. <span class="math">\(numcost\)</span> is <span class="math">\(1\)</span>
since it is a scalar function.</p>
<p>For ODE solution, PETSc requires user-provided functions to evaluate the
system <span class="math">\(F(t,y,\dot{y},p)\)</span> (set by <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSetIFunction.html#TSSetIFunction">TSSetIFunction</a>()</span></code> ) and its
corresponding Jacobian <span class="math">\(F_y + \sigma F_{\dot y}\)</span> (set by
<code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSetIJacobian.html#TSSetIJacobian">TSSetIJacobian</a>()</span></code>). Note that the solution state <span class="math">\(y\)</span> is
<span class="math">\([ \phi \; \omega ]^T\)</span> here. For sensitivity analysis, we need to
provide a routine to compute <span class="math">\(\mathrm{f}_p=[0 \; 1]^T\)</span> using
<code class="docutils literal notranslate"><span class="pre">TSASetRHSJacobianP()</span></code>, and three routines corresponding to the
integrand <span class="math">\(r=c \left( \max(0, \phi - \phi_S ) \right)^2\)</span>,
<span class="math">\(r_p = [0 \; 0]^T\)</span> and
<span class="math">\(r_y= [ 2 c \left( \max(0, \phi - \phi_S ) \right) \; 0]^T\)</span> using
<code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSSetCostIntegrand.html#TSSetCostIntegrand">TSSetCostIntegrand</a>()</span></code>.</p>
<p>In the adjoint run, <span class="math">\(\lambda\)</span> and <span class="math">\(\mu\)</span> are initialized as
<span class="math">\([ 0 \; 0 ]^T\)</span> and <span class="math">\([-1]\)</span> at the final time <span class="math">\(t_F\)</span>.
After <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sensitivity/TSAdjointSolve.html#TSAdjointSolve">TSAdjointSolve</a>()</span></code>, the sensitivity of the cost function w.r.t.
initial conditions is given by the sensitivity variable <span class="math">\(\lambda\)</span>
(at time <span class="math">\(t_0\)</span>) directly. And the sensitivity of the cost function
w.r.t. the parameter <span class="math">\(p_m\)</span> can be computed (by users) as</p>
<div class="math">
\[\frac{\mathrm{d} \Psi}{\mathrm{d} p_m} = \mu(t_0) + \lambda(t_0) \frac{\mathrm{d} \left[ \phi(t_0) \; \omega(t_0) \right]^T}{\mathrm{d} p_m} .
\]</div>
<p>For explicit methods where one does not need to provide the Jacobian
<span class="math">\(F_u\)</span> for the forward solve one still does need it for the
backward solve and thus must call</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSetRHSJacobian.html#TSSetRHSJacobian">TSSetRHSJacobian</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/Mat.html#Mat">Mat</a></span> <span class="n">Amat</span><span class="p">,</span> <span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/Mat.html#Mat">Mat</a></span> <span class="n">Pmat</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscErrorCode.html#PetscErrorCode">PetscErrorCode</a></span> <span class="p">(</span><span class="o">*</span><span class="n">f</span><span class="p">)(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscReal.html#PetscReal">PetscReal</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/Mat.html#Mat">Mat</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/Mat.html#Mat">Mat</a></span><span class="p">,</span><span class="kt">void</span><span class="o">*</span><span class="p">),</span><span class="kt">void</span> <span class="o">*</span><span class="n">fP</span><span class="p">);</span>
</pre></div>
</div>
<p>Examples include:</p>
<ul class="simple">
<li><p>a discrete adjoint sensitivity using explicit time stepping methods
<a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex16adj.c.html">TS Tutorial ex16adj</a>,</p></li>
<li><p>a discrete adjoint sensitivity using implicit time stepping methods
<a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex20adj.c.html">TS Tutorial ex20adj</a>,</p></li>
<li><p>an optimization using the discrete adjoint models of ERK
<a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex16opt_ic.c.html">TS Tutorial ex16opt_ic</a>
and
<cite>TS Tutorial ex16opt_p</cite> <<a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex16opt_p.c.html">https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex16opt_p.c.html</a>>`__,</p></li>
<li><p>an optimization using the discrete adjoint models of Theta methods
for stiff DAEs
<a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex20opt_ic.c.html">TS Tutorial ex20opt_ic</a>
and
<a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex20opt_p.c.html">TS Tutorial ex20opt_p</a>,</p></li>
<li><p>an ODE-constrained optimization using the discrete adjoint models of
Theta methods for cost function with an integral term
<a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/power_grid/ex3opt.c.html">TS Power Grid Tutorial ex3opt</a>,</p></li>
<li><p>a discrete adjoint sensitivity using <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSCN.html#TSCN">TSCN</a></span></code> (Crank-Nicolson)
methods for DAEs with discontinuities
<a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/power_grid/stability_9bus/ex9busadj.c.html">TS Power Grid Stability Tutorial ex9busadj.c</a>,</p></li>
<li><p>a DAE-constrained optimization using the discrete adjoint models of
<code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSCN.html#TSCN">TSCN</a></span></code> (Crank-Nicolson) methods for cost function with an integral
term
<a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/power_grid/stability_9bus/ex9busopt.c.html">TS Power Grid Tutorial ex9busopt.c</a>,</p></li>
<li><p>a discrete adjoint sensitivity using <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSCN.html#TSCN">TSCN</a></span></code> methods for a PDE
problem
<a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/advection-diffusion-reaction/ex5adj.c.html">TS Advection-Diffusion-Reaction Tutorial ex5adj</a>.</p></li>
</ul>
</div>
<div class="section" id="checkpointing">
<h2>Checkpointing<a class="headerlink" href="#checkpointing" title="Permalink to this headline">¶</a></h2>
<p>The discrete adjoint model requires the states (and stage values in the
context of multistage timestepping methods) to evaluate the Jacobian
matrices during the adjoint (backward) run. By default, PETSc stores the
whole trajectory to disk as binary files, each of which contains the
information for a single time step including state, time, and stage
values (optional). One can also make PETSc store the trajectory to
memory with the option <code class="docutils literal notranslate"><span class="pre">-ts_trajectory_type</span> <span class="pre">memory</span></code>. However, there
might not be sufficient memory capacity especially for large-scale
problems and long-time integration.</p>
<p>A so-called checkpointing scheme is needed to solve this problem. The
scheme stores checkpoints at selective time steps and recomputes the
missing information. The <code class="docutils literal notranslate"><span class="pre">revolve</span></code> library is used by PETSc
<code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSTrajectory.html#TSTrajectory">TSTrajectory</a></span></code> to generate an optimal checkpointing schedule that
minimizes the recomputations given a limited number of available
checkpoints. One can specify the number of available checkpoints with
the option
<code class="docutils literal notranslate"><span class="pre">-ts_trajectory_max_cps_ram</span> <span class="pre">[maximum</span> <span class="pre">number</span> <span class="pre">of</span> <span class="pre">checkpoints</span> <span class="pre">in</span> <span class="pre">RAM]</span></code>.
Note that one checkpoint corresponds to one time step.</p>
<p>The <code class="docutils literal notranslate"><span class="pre">revolve</span></code> library also provides an optimal multistage
checkpointing scheme that uses both RAM and disk for storage. This
scheme is automatically chosen if one uses both the option
<code class="docutils literal notranslate"><span class="pre">-ts_trajectory_max_cps_ram</span> <span class="pre">[maximum</span> <span class="pre">number</span> <span class="pre">of</span> <span class="pre">checkpoints</span> <span class="pre">in</span> <span class="pre">RAM]</span></code>
and the option
<code class="docutils literal notranslate"><span class="pre">-ts_trajectory_max_cps_disk</span> <span class="pre">[maximum</span> <span class="pre">number</span> <span class="pre">of</span> <span class="pre">checkpoints</span> <span class="pre">on</span> <span class="pre">disk]</span></code>.</p>
<p>Some other useful options are listed below.</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">-ts_trajectory_view</span></code> prints the total number of recomputations,</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">-ts_monitor</span></code> and <code class="docutils literal notranslate"><span class="pre">-ts_adjoint_monitor</span></code> allow users to monitor
the progress of the adjoint work flow,</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">-ts_trajectory_type</span> <span class="pre">visualization</span></code> may be used to save the whole
trajectory for visualization. It stores the solution and the time,
but no stage values. The binary files generated can be read into
MATLAB via the script
<code class="docutils literal notranslate"><span class="pre">${PETSC_DIR}/share/petsc/matlab/PetscReadBinaryTrajectory.m</span></code>.</p></li>
</ul>
</div>
</div>
<div class="section" id="solving-steady-state-problems-with-pseudo-timestepping">
<h1>Solving Steady-State Problems with Pseudo-Timestepping<a class="headerlink" href="#solving-steady-state-problems-with-pseudo-timestepping" title="Permalink to this headline">¶</a></h1>
<p><strong>Simple Example:</strong> <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span></code> provides a general code for performing pseudo
timestepping with a variable timestep at each physical node point. For
example, instead of directly attacking the steady-state problem</p>
<div class="math">
\[G(u) = 0,
\]</div>
<p>we can use pseudo-transient continuation by solving</p>
<div class="math">
\[u_t = G(u).
\]</div>
<p>Using time differencing</p>
<div class="math">
\[u_t \doteq \frac{{u^{n+1}} - {u^{n}} }{dt^{n}}
\]</div>
<p>with the backward Euler method, we obtain nonlinear equations at a
series of pseudo-timesteps</p>
<div class="math">
\[\frac{1}{dt^n} B (u^{n+1} - u^{n} ) = G(u^{n+1}).
\]</div>
<p>For this problem the user must provide <span class="math">\(G(u)\)</span>, the time steps
<span class="math">\(dt^{n}\)</span> and the left-hand-side matrix <span class="math">\(B\)</span> (or optionally,
if the timestep is position independent and <span class="math">\(B\)</span> is the identity
matrix, a scalar timestep), as well as optionally the Jacobian of
<span class="math">\(G(u)\)</span>.</p>
<p>More generally, this can be applied to implicit ODE and DAE for which
the transient form is</p>
<div class="math">
\[F(u,\dot{u}) = 0.
\]</div>
<p>For solving steady-state problems with pseudo-timestepping one proceeds
as follows.</p>
<ul>
<li><p>Provide the function <code class="docutils literal notranslate"><span class="pre">G(u)</span></code> with the routine</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSetRHSFunction.html#TSSetRHSFunction">TSSetRHSFunction</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span> <span class="n">r</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscErrorCode.html#PetscErrorCode">PetscErrorCode</a></span> <span class="p">(</span><span class="o">*</span><span class="n">f</span><span class="p">)(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscReal.html#PetscReal">PetscReal</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span><span class="p">,</span><span class="kt">void</span><span class="o">*</span><span class="p">),</span><span class="kt">void</span> <span class="o">*</span><span class="n">fP</span><span class="p">);</span>
</pre></div>
</div>
<p>The arguments to the function <code class="docutils literal notranslate"><span class="pre">f()</span></code> are the timestep context, the
current time, the input for the function, the output for the function
and the (optional) user-provided context variable <code class="docutils literal notranslate"><span class="pre">fP</span></code>.</p>
</li>
<li><p>Provide the (approximate) Jacobian matrix of <code class="docutils literal notranslate"><span class="pre">G(u)</span></code> and a function
to compute it at each Newton iteration. This is done with the command</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSetRHSJacobian.html#TSSetRHSJacobian">TSSetRHSJacobian</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/Mat.html#Mat">Mat</a></span> <span class="n">Amat</span><span class="p">,</span> <span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/Mat.html#Mat">Mat</a></span> <span class="n">Pmat</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscErrorCode.html#PetscErrorCode">PetscErrorCode</a></span> <span class="p">(</span><span class="o">*</span><span class="n">f</span><span class="p">)(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscReal.html#PetscReal">PetscReal</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Vec/Vec.html#Vec">Vec</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/Mat.html#Mat">Mat</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/Mat.html#Mat">Mat</a></span><span class="p">,</span><span class="kt">void</span><span class="o">*</span><span class="p">),</span><span class="kt">void</span> <span class="o">*</span><span class="n">fP</span><span class="p">);</span>
</pre></div>
</div>
<p>The arguments for the function <code class="docutils literal notranslate"><span class="pre">f()</span></code> are the timestep context, the
current time, the location where the Jacobian is to be computed, the
(approximate) Jacobian matrix, an alternative approximate Jacobian
matrix used to construct the preconditioner, and the optional
user-provided context, passed in as <code class="docutils literal notranslate"><span class="pre">fP</span></code>. The user must provide the
Jacobian as a matrix; thus, if using a matrix-free approach, one must
create a <code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/MATSHELL.html#MATSHELL">MATSHELL</a></span></code> matrix.</p>
</li>
</ul>
<p>In addition, the user must provide a routine that computes the
pseudo-timestep. This is slightly different depending on if one is using
a constant timestep over the entire grid, or it varies with location.</p>
<ul>
<li><p>For location-independent pseudo-timestepping, one uses the routine</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSPseudoSetTimeStep.html#TSPseudoSetTimeStep">TSPseudoSetTimeStep</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscInt.html#PetscInt">PetscInt</a></span><span class="p">(</span><span class="o">*</span><span class="n">dt</span><span class="p">)(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscReal.html#PetscReal">PetscReal</a></span><span class="o">*</span><span class="p">,</span><span class="kt">void</span><span class="o">*</span><span class="p">),</span><span class="kt">void</span><span class="o">*</span> <span class="n">dtctx</span><span class="p">);</span>
</pre></div>
</div>
<p>The function <code class="docutils literal notranslate"><span class="pre">dt</span></code> is a user-provided function that computes the
next pseudo-timestep. As a default one can use
<code class="docutils literal notranslate"><span class="pre"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSPseudoTimeStepDefault.html#TSPseudoTimeStepDefault">TSPseudoTimeStepDefault</a>(<a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a>,<a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscReal.html#PetscReal">PetscReal</a>*,void*)</span></code> for <code class="docutils literal notranslate"><span class="pre">dt</span></code>. This
routine updates the pseudo-timestep with one of two strategies: the
default</p>
<div class="math">
\[dt^{n} = dt_{\mathrm{increment}}*dt^{n-1}*\frac{|| F(u^{n-1}) ||}{|| F(u^{n})||}
\]</div>
<p>or, the alternative,</p>
<div class="math">
\[dt^{n} = dt_{\mathrm{increment}}*dt^{0}*\frac{|| F(u^{0}) ||}{|| F(u^{n})||}
\]</div>
<p>which can be set with the call</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSPseudoIncrementDtFromInitialDt.html#TSPseudoIncrementDtFromInitialDt">TSPseudoIncrementDtFromInitialDt</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">);</span>
</pre></div>
</div>
<p>or the option <code class="docutils literal notranslate"><span class="pre">-ts_pseudo_increment_dt_from_initial_dt</span></code>. The value
<span class="math">\(dt_{\mathrm{increment}}\)</span> is by default <span class="math">\(1.1\)</span>, but can be
reset with the call</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSPseudoSetTimeStepIncrement.html#TSPseudoSetTimeStepIncrement">TSPseudoSetTimeStepIncrement</a></span><span class="p">(</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TS.html#TS">TS</a></span> <span class="n">ts</span><span class="p">,</span><span class="n"><a href="https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Sys/PetscReal.html#PetscReal">PetscReal</a></span> <span class="n">inc</span><span class="p">);</span>
</pre></div>
</div>
<p>or the option <code class="docutils literal notranslate"><span class="pre">-ts_pseudo_increment</span> <span class="pre"><inc></span></code>.</p>
</li>
<li><p>For location-dependent pseudo-timestepping, the interface function
has not yet been created.</p></li>
</ul>
</div>
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