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<li class="toctree-l1 current"><a class="current reference internal" href="#">References and acknowledgements</a><ul>
<li class="toctree-l2"><a class="reference internal" href="#citing-tlm-adjoint">Citing tlm_adjoint</a></li>
<li class="toctree-l2"><a class="reference internal" href="#references">References</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#dolfin-adjoint">dolfin-adjoint</a></li>
<li class="toctree-l3"><a class="reference internal" href="#taylor-remainder-convergence-testing">Taylor remainder convergence testing</a></li>
<li class="toctree-l3"><a class="reference internal" href="#differentiating-fixed-point-problems">Differentiating fixed-point problems</a></li>
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<section id="references-and-acknowledgements">
<h1>References and acknowledgements<a class="headerlink" href="#references-and-acknowledgements" title="Link to this heading"></a></h1>
<section id="citing-tlm-adjoint">
<h2>Citing tlm_adjoint<a class="headerlink" href="#citing-tlm-adjoint" title="Link to this heading"></a></h2>
<p>tlm_adjoint is described in</p>
<ul class="simple">
<li><p>James R. Maddison, Daniel N. Goldberg, and Benjamin D. Goddard, ‘Automated
calculation of higher order partial differential equation constrained
derivative information’, SIAM Journal on Scientific Computing, 41(5), pp.
C417–C445, 2019, doi: 10.1137/18M1209465</p></li>
</ul>
<p>The automated assembly and linear solver caching applied by tlm_adjoint is
based on the approach described in</p>
<ul class="simple">
<li><p>J. R. Maddison and P. E. Farrell, ‘Rapid development and adjoining of
transient finite element models’, Computer Methods in Applied Mechanics and
Engineering, 276, 95–121, 2014, doi: 10.1016/j.cma.2014.03.010</p></li>
</ul>
<p>Checkpointing with tlm_adjoint is described in</p>
<ul class="simple">
<li><p>James R. Maddison, ‘Step-based checkpointing with high-level algorithmic
differentiation’, Journal of Computational Science 82, 102405, 2024,
doi: 10.1016/j.jocs.2024.102405</p></li>
</ul>
</section>
<section id="references">
<h2>References<a class="headerlink" href="#references" title="Link to this heading"></a></h2>
<section id="dolfin-adjoint">
<h3>dolfin-adjoint<a class="headerlink" href="#dolfin-adjoint" title="Link to this heading"></a></h3>
<p>tlm_adjoint implements high-level algorithmic differentiation using an
approach based on that used by dolfin-adjoint, described in</p>
<ul class="simple">
<li><p>P. E. Farrell, D. A. Ham, S. W. Funke, and M. E. Rognes, ‘Automated
derivation of the adjoint of high-level transient finite element programs’,
SIAM Journal on Scientific Computing 35(4), pp. C369–C393, 2013,
doi: 10.1137/120873558</p></li>
</ul>
<p>tlm_adjoint was developed from a custom extension to dolfin-adjoint.</p>
</section>
<section id="taylor-remainder-convergence-testing">
<h3>Taylor remainder convergence testing<a class="headerlink" href="#taylor-remainder-convergence-testing" title="Link to this heading"></a></h3>
<p>The functions in <a class="reference external" href="autoapi/tlm_adjoint/verification/index.html">tlm_adjoint/verification.py</a> implement Taylor remainder
convergence testing using the approach described in</p>
<ul class="simple">
<li><p>P. E. Farrell, D. A. Ham, S. W. Funke, and M. E. Rognes, ‘Automated
derivation of the adjoint of high-level transient finite element programs’,
SIAM Journal on Scientific Computing 35(4), pp. C369–C393, 2013,
doi: 10.1137/120873558</p></li>
</ul>
</section>
<section id="differentiating-fixed-point-problems">
<h3>Differentiating fixed-point problems<a class="headerlink" href="#differentiating-fixed-point-problems" title="Link to this heading"></a></h3>
<p>The <cite>FixedPointSolver</cite> class in <a class="reference external" href="autoapi/tlm_adjoint/fixed_point/index.html">tlm_adjoint/fixed_point.py</a> derives tangent-linear and
adjoint information using the approach described in</p>
<ul class="simple">
<li><p>Jean Charles Gilbert, ‘Automatic differentiation and iterative processes’,
Optimization Methods and Software, 1(1), pp. 13–21, 1992,
doi: 10.1080/10556789208805503</p></li>
<li><p>Bruce Christianson, ‘Reverse accumulation and attractive fixed points’,
Optimization Methods and Software, 3(4), pp. 311–326, 1994,
doi: 10.1080/10556789408805572</p></li>
</ul>
</section>
<section id="binomial-checkpointing">
<h3>Binomial checkpointing<a class="headerlink" href="#binomial-checkpointing" title="Link to this heading"></a></h3>
<p>The <cite>MultistageCheckpointSchedule</cite> class in
<a class="reference external" href="autoapi/tlm_adjoint/checkpoint_schedules/binomial/index.html">tlm_adjoint/checkpoint_schedules/binomial.py</a> implements the
binomial checkpointing strategy described in</p>
<ul class="simple">
<li><p>Andreas Griewank and Andrea Walther, ‘Algorithm 799: revolve: an
implementation of checkpointing for the reverse or adjoint mode of
computational differentiation’, ACM Transactions on Mathematical Software,
26(1), pp. 19–45, 2000, doi: 10.1145/347837.347846</p></li>
</ul>
<p>The <cite>MultistageCheckpointSchedule</cite> class determines a memory/disk storage
distribution using an initial run of the checkpoint schedule, leading to a
distribution equivalent to that in</p>
<ul class="simple">
<li><p>Philipp Stumm and Andrea Walther, ‘MultiStage approaches for optimal offline
checkpointing’, SIAM Journal on Scientific Computing, 31(3), pp. 1946–1967,
2009, doi: 10.1137/080718036</p></li>
</ul>
<p>The <cite>TwoLevelCheckpointSchedule</cite> class in
<a class="reference external" href="autoapi/tlm_adjoint/checkpoint_schedules/binomial/index.html">tlm_adjoint/checkpoint_schedules/binomial.py</a> implements the
two-level mixed periodic/binomial checkpointing approach described in</p>
<ul class="simple">
<li><p>Gavin J. Pringle, Daniel C. Jones, Sudipta Goswami, Sri Hari Krishna
Narayanan, and Daniel Goldberg, ‘Providing the ARCHER community with adjoint
modelling tools for high-performance oceanographic and cryospheric
computation’, version 1.1, EPCC, 2016</p></li>
</ul>
<p>and in the supporting information for</p>
<ul class="simple">
<li><p>D. N. Goldberg, T. A. Smith, S. H. K. Narayanan, P. Heimbach, and M.
Morlighem,, ‘Bathymetric influences on Antarctic ice-shelf melt rates’,
Journal of Geophysical Research: Oceans, 125(11), e2020JC016370, 2020,
doi: 10.1029/2020JC016370</p></li>
</ul>
</section>
<section id="pytorch">
<h3>PyTorch<a class="headerlink" href="#pytorch" title="Link to this heading"></a></h3>
<p>The PyTorch interface in <a class="reference external" href="autoapi/tlm_adjoint/torch/index.html">tlm_adjoint/torch.py</a> follows the same principles as
described in</p>
<blockquote>
<div><ul class="simple">
<li><p>Nacime Bouziani and David A. Ham, ‘Physics-driven machine learning models
coupling PyTorch and Firedrake’, 2023, arXiv:2303.06871v3</p></li>
</ul>
</div></blockquote>
</section>
</section>
<section id="funding">
<h2>Funding<a class="headerlink" href="#funding" title="Link to this heading"></a></h2>
<p>Early development work leading to tlm_adjoint was conducted as part of a U.K.
Natural Environment Research Council funded project (NE/L005166/1). Further
development has been conducted as part of a U.K. Engineering and Physical
Sciences Research Council funded project (EP/R021600/1) and a Natural
Environment Research Council funded project (NE/T001607/1).</p>
</section>
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