<div dir="ltr"><div dir="ltr"><br></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Thu, 22 Dec 2022 at 12:08, Matteo Semplice <<a href="mailto:matteo.semplice@uninsubria.it">matteo.semplice@uninsubria.it</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex">

  
  <div>
    <p><br>
    </p>
    <div>Il 22/12/22 20:06, Dave May ha scritto:<br>
    </div>
    <blockquote type="cite">
      
      <div><br>
      </div>
      <div><br>
        <div class="gmail_quote">
          <div dir="ltr" class="gmail_attr">On Thu 22. Dec 2022 at
            10:27, Matteo Semplice <<a href="mailto:matteo.semplice@uninsubria.it" target="_blank">matteo.semplice@uninsubria.it</a>>
            wrote:<br>
          </div>
          <blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex">
            <div>
              <p>Dear Dave and Matt,</p>
              <p>    I am really dealing with two different use cases in
                a code that will compute a levelset function passing
                through a large set of points. If I had
                DMSwarmSetMigrateType() and if it were safe to switch
                the migration mode back and forth in the same swarm,
                this would cover all my use cases here. Is it safe to
                add it back to petsc? Details below if you are curious.<br>
              </p>
              <p>1) During preprocessing I am loading a point cloud from
                disk (in whatever order it comes) and need to send the
                particles to the right ranks. Since the background DM is
                a DMDA I can easily figure out the destination rank.
                This would be covered by your suggestion not to attach
                the DM, except that later I need to locate these points
                with respect to the background cells in order to
                initialize data on the Vecs associated to the DMDA.<br>
              </p>
              <p>2) Then I need to implement a semilagrangian time
                evolution scheme. For this I'd like to send particles
                around at the "foot of characteristic", collect data
                there and then send them back to the originating point.
                The first migration would be based on particle
                coordinates (DMSwarmMigrate_DMNeighborScatter and the
                restriction to only neighbouring ranks is perfect),
                while for the second move it would be easier to just
                send them back to the originating rank, which I can
                easily store in an Int field in the swarm. Thus at each
                timestep I'd need to swap migrate types in this swarm
                (DMScatter for moving them to the feet and BASIC to send
                them back).</p>
            </div>
          </blockquote>
          <div dir="auto"><br>
          </div>
          <div dir="auto">When you use BASIC, you would have to
            explicitly call the point location routine from your code as
            BASIC does not interact with the DM. </div>
          <div dir="auto"><br>
          </div>
          <div dir="auto">Based on what I see in the code, switching
             migrate modes between basic and dmneighbourscatter should
            be safe.</div>
          <div dir="auto"><br>
          </div>
          <div dir="auto">If you are fine calling the point location
            from your side then what you propose should work.</div>
        </div>
      </div>
    </blockquote>
    <p>If I understood the code correctly, BASIC will just migrate
      particles sending them to what is stored in DMSwarmField_rank,
      right?</p></div></blockquote><div><br></div><div>Correct.</div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><div><p> That'd be easy since I can create a SWARM with all the data
      I need and an extra int field (say "original_rank") and copy those
      values into DMSwarmField_rank before calling migrate for the
      "going back" step. After this backward migration I do not need to
      locate particles again (e.g. I do not need DMSwarmSortGetAccess
      after the BASIC migration, but only after the DMNeighborScatter
      one).</p></div></blockquote><div><br></div><div>Okay </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><div>
    <p>Thus having back DMSwarmSetMigrateType() should be enough for me.<br></p></div></blockquote><div><br></div><div>Okay. Thanks for clarifying.</div><div><br></div><div>Cheers,</div><div>Dave</div><div><br></div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><div><p>
    </p>
    <p>Thanks</p>
    <p>    Matteo<br>
    </p>
    <blockquote type="cite">
      <div>
        <div class="gmail_quote">
          <div dir="auto"><br>
          </div>
          <div dir="auto">Cheers</div>
          <div dir="auto">Dave</div>
          <div dir="auto"><br>
          </div>
          <div dir="auto"><br>
          </div>
          <blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex">
            <div>
              <p dir="auto"><br>
              </p>
              Thanks
              <p>    Matteo<br>
              </p>
              <div>Il 22/12/22 18:40, Dave May ha scritto:<br>
              </div>
            </div>
            <div>
              <blockquote type="cite">
                <div dir="auto">Hey Matt,</div>
                <div style="background-color:rgba(0,0,0,0);border-color:rgb(0,0,0)"><br>
                  <div class="gmail_quote" style="background-color:rgba(0,0,0,0);border-color:rgb(0,0,0)">
                    <div dir="ltr" class="gmail_attr">On Thu 22. Dec
                      2022 at 05:02, Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>>
                      wrote:<br>
                    </div>
                    <blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)">
                      <div dir="ltr">
                        <div dir="ltr">On Thu, Dec 22, 2022 at 6:28 AM
                          Matteo Semplice <<a href="mailto:matteo.semplice@uninsubria.it" target="_blank">matteo.semplice@uninsubria.it</a>>
                          wrote:<br>
                        </div>
                        <div class="gmail_quote">
                          <blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)">
                            <div>
                              <p>Dear all</p>
                              <p>    please ignore my previous email and
                                read this one: I have better localized
                                the problem. Maybe DMSwarmMigrate is
                                designed to migrate particles only to
                                first neighbouring ranks?<br>
                              </p>
                            </div>
                          </blockquote>
                          <div>Yes, I believe that was the design.</div>
                          <div><br>
                          </div>
                          <div>Dave, is this correct?</div>
                        </div>
                      </div>
                    </blockquote>
                    <div dir="auto"><br>
                    </div>
                    <div dir="auto">Correct. <span style="font-size:21px;word-spacing:1px;border-color:rgb(49,49,49);color:rgb(49,49,49)">DMSwarmMigrate_</span><span style="font-size:21px;word-spacing:1px;border-color:rgb(49,49,49);color:rgb(49,49,49)">DMNeighborScatter()
                        only scatter points to the neighbour ranks -
                        where neighbours are defined by the DM provided
                        to represent the mesh.</span><br>
                    </div>
                    <div dir="auto"><span style="font-size:21px;word-spacing:1px;border-color:rgb(49,49,49);color:rgb(49,49,49)"><br>
                      </span></div>
                    <div dir="auto"><span style="font-size:21px;word-spacing:1px;border-color:rgb(49,49,49);color:rgb(49,49,49)">DMSwarmMigrate_</span><span style="font-size:21px;word-spacing:1px;border-color:rgb(49,49,49);color:rgb(49,49,49)">DMNeighborScatter()
                        Is selected by default if you attach a DM.</span><br>
                    </div>
                    <div dir="auto"><span style="font-size:21px;word-spacing:1px;border-color:rgb(49,49,49);color:rgb(49,49,49)"><br>
                      </span></div>
                    <div dir="auto"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px">The
                          scatter method should be over ridden with </span></font></div>
                    <div dir="auto"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px"><span style="word-spacing:0px;font-size:medium;color:rgb(0,0,0)"><br>
                          </span></span></font></div>
                    <div dir="auto"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px"><span style="word-spacing:0px;font-size:medium;color:rgb(0,0,0)">DMSwarmSetMigrateType()</span><br>
                        </span></font></div>
                    <div dir="auto"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px"><span style="word-spacing:0px;font-size:medium;color:rgb(0,0,0)"><br>
                          </span></span></font></div>
                    <div dir="auto" style="background-color:rgba(0,0,0,0);border-color:rgb(0,0,0)"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px">however
                          it appears this method no longer exists.</span></font></div>
                    <div dir="auto"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px">
                          <div><br>
                          </div>
                          If one can determine the exact rank where
                          points should should be sent and it is not
                          going to be the neighbour rank (given by the
                          DM), I would suggest not attaching the DM at
                          all.</span></font></div>
                    <div dir="auto"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px"><br>
                        </span></font></div>
                    <div dir="auto"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px">However
                          if this is not possible and one wanted to
                          scatter to say the neighbours neighbours, we
                          will have to add a new interface and refactor
                          things a little bit.</span></font></div>
                    <div dir="auto"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px"><br>
                        </span></font></div>
                    <div dir="auto"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px">Cheers</span></font></div>
                    <div dir="auto"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px">Dave</span></font></div>
                    <div dir="auto"><font style="color:rgb(49,49,49)"><span style="font-size:21px;word-spacing:1px"><br>
                        </span></font></div>
                    <div dir="auto"><br>
                    </div>
                    <blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)">
                      <div dir="ltr">
                        <div class="gmail_quote">
                          <div><br>
                          </div>
                          <div>  Thanks,</div>
                          <div><br>
                          </div>
                          <div>    Matt</div>
                        </div>
                      </div>
                      <div dir="ltr">
                        <div class="gmail_quote">
                          <div> </div>
                          <blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)">
                            <div>
                              <div>Il 22/12/22 11:44, Matteo Semplice ha
                                scritto:<br>
                              </div>
                              <blockquote type="cite">
                                <p>Dear everybody,</p>
                                <p>    I have bug a bit into the code
                                  and I am able to add more information.<br>
                                </p>
                                <div>Il 02/12/22 12:48, Matteo Semplice
                                  ha scritto:<br>
                                </div>
                                <blockquote type="cite">
                                  <div>Hi.</div>
                                  <div>I am sorry to take this up again,
                                    but further tests show that it's not
                                    right yet.<br>
                                  </div>
                                  <div><br>
                                  </div>
                                  <div>Il 04/11/22 12:48, Matthew
                                    Knepley ha scritto:<br>
                                  </div>
                                  <blockquote type="cite">
                                    <div dir="ltr">
                                      <div dir="ltr">On Fri, Nov 4, 2022
                                        at 7:46 AM Matteo Semplice <<a href="mailto:matteo.semplice@uninsubria.it" target="_blank">matteo.semplice@uninsubria.it</a>>
                                        wrote:<br>
                                      </div>
                                      <div class="gmail_quote">
                                        <blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)">
                                          <div> On 04/11/2022 02:43,
                                            Matthew Knepley wrote:<br>
                                            <blockquote type="cite">
                                              <div dir="ltr">
                                                <div dir="ltr">On Thu,
                                                  Nov 3, 2022 at 8:36 PM
                                                  Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>>
                                                  wrote:<br>
                                                </div>
                                                <div class="gmail_quote">
                                                  <blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)">
                                                    <div dir="ltr">
                                                      <div dir="ltr">On
                                                        Thu, Oct 27,
                                                        2022 at 11:57 AM
                                                        Semplice Matteo
                                                        <<a href="mailto:matteo.semplice@uninsubria.it" target="_blank">matteo.semplice@uninsubria.it</a>>
                                                        wrote:<br>
                                                      </div>
                                                      <div class="gmail_quote">
                                                        <blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)">
                                                          <div>
                                                          <div dir="ltr">
                                                          <div style="font-family:Calibri,Arial,Helvetica,sans-serif;font-size:12pt;background-color:rgb(255,255,255);color:rgb(0,0,0)">Dear
                                                          Petsc
                                                          developers,</div>
                                                          <div style="font-family:Calibri,Arial,Helvetica,sans-serif;font-size:12pt;background-color:rgb(255,255,255);color:rgb(0,0,0)">   
                                                          I am trying to
                                                          use a DMSwarm
                                                          to locate a
                                                          cloud of
                                                          points with
                                                          respect to a
                                                          background
                                                          mesh. In the
                                                          real
                                                          application
                                                          the points
                                                          will be loaded
                                                          from disk, but
                                                          I have created
                                                          a small demo
                                                          in which</div>
                                                          <div style="font-family:Calibri,Arial,Helvetica,sans-serif;font-size:12pt;background-color:rgb(255,255,255);color:rgb(0,0,0)">
                                                          <ul style="font-family:Calibri,Arial,Helvetica,sans-serif">
                                                          <li style="font-family:Calibri,Arial,Helvetica,sans-serif"><span style="font-family:Calibri,Arial,Helvetica,sans-serif">each processor
                                                          creates Npart
                                                          particles, all
                                                          within the
                                                          domain covered
                                                          by the mesh,
                                                          but not all in
                                                          the local
                                                          portion of the
                                                          mesh</span></li>
                                                          <li style="font-family:Calibri,Arial,Helvetica,sans-serif"><span style="font-family:Calibri,Arial,Helvetica,sans-serif">migrate the
                                                          particles</span></li>
                                                          </ul>
                                                          <div style="font-family:Calibri,Arial,Helvetica,sans-serif">After
                                                          migration most
                                                          particles are
                                                          not any more
                                                          in the DMSwarm
                                                          (how many and
                                                          which ones
                                                          seems to
                                                          depend on the
                                                          number of
                                                          cpus, but it
                                                          never happens
                                                          that all
                                                          particle
                                                          survive the
                                                          migration
                                                          process).</div>
                                                          <div style="font-family:Calibri,Arial,Helvetica,sans-serif"><br>
                                                          </div>
                                                          </div>
                                                          </div>
                                                          </div>
                                                        </blockquote>
                                                        <div>Thanks for
                                                          sending this.
                                                          I found the
                                                          problem.
                                                          Someone has
                                                          some overly
                                                          fancy code
                                                          inside DMDA to
                                                          figure out the
                                                          local bounding
                                                          box from the
                                                          coordinates.</div>
                                                        <div>It is
                                                          broken for
                                                          DM_BOUNDARY_GHOSTED,
                                                          but we never
                                                          tested with
                                                          this. I will
                                                          fix it.</div>
                                                      </div>
                                                    </div>
                                                  </blockquote>
                                                  <div><br>
                                                  </div>
                                                  <div>Okay, I think
                                                    this fix is correct</div>
                                                  <div><br>
                                                  </div>
                                                  <div>  <a href="https://eur01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fgitlab.com%2Fpetsc%2Fpetsc%2F-%2Fmerge_requests%2F5802&data=05%7C01%7Cmatteo.semplice%40uninsubria.it%7C94569ac2a32a4838103608dae44f9c46%7C9252ed8bdffc401c86ca6237da9991fa%7C0%7C0%7C638073327837332509%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=9yKNH0%2FbepSLU2PO5S%2B2GkYSmjjFIDGHpapy%2FUb43D4%3D&reserved=0" target="_blank">https://gitlab.com/petsc/petsc/-/merge_requests/5802</a></div>
                                                  <div><br>
                                                  </div>
                                                  <div>I incorporated
                                                    your test as
                                                    src/dm/impls/da/tests/ex1.c.
                                                    Can you take a look
                                                    and see if this
                                                    fixes your issue?</div>
                                                </div>
                                              </div>
                                            </blockquote>
                                            <p>Yes, we have tested 2d
                                              and 3d, with various
                                              combinations of
                                              DM_BOUNDARY_* along
                                              different directions and
                                              it works like a charm.</p>
                                            <p>On a side note, neither <span style="font-family:monospace"><span style="font-family:monospace;background-color:rgb(255,255,255);color:rgb(0,0,0)">DMSwarmViewXDMF</span></span>
                                              nor <span style="font-family:monospace"><span style="font-family:monospace;background-color:rgb(255,255,255);color:rgb(0,0,0)">DMSwarmMigrate</span></span>
                                              seem to be implemented for
                                              1d: I get</p>
                                            <p><span style="font-family:monospace"><span style="font-family:monospace;background-color:rgb(255,255,255);color:rgb(0,0,0)">[0]PETSC
                                                  ERROR: No support for
                                                  this operation for
                                                  this object type</span><span style="font-family:monospace;background-color:rgb(255,255,255);color:rgb(178,24,24)">
                                                                                                                                           </span><span style="font-family:monospace;background-color:rgb(255,255,255);color:rgb(0,0,0)">
                                                </span>[0]PETSC ERROR:
                                                Support not provided for
                                                1D<br>
                                              </span></p>
                                            <p>However, currently I have
                                              no need for this feature.<br>
                                              <span style="font-family:monospace"></span></p>
                                            <p>Finally, if the test is
                                              meant to stay in the
                                              source, you may remove the
                                              call to <span style="font-family:monospace"><span style="font-family:monospace;background-color:rgb(255,255,255);color:rgb(0,0,0)">DMSwarmRegisterPetscDatatypeField</span></span>
                                              as in the attached patch.<br>
                                            </p>
                                            <p>Thanks a lot!!</p>
                                          </div>
                                        </blockquote>
                                        <div>Thanks! Glad it works.</div>
                                        <div>
                                          <p>   Matt</p>
                                        </div>
                                      </div>
                                    </div>
                                  </blockquote>
                                  <p cols="72">There are still problems
                                    when not using 1,2 or 4 cpus. Any
                                    other number of cpus that I've
                                    tested does not work corectly.<br>
                                  </p>
                                </blockquote>
                                <p>I have now modified
                                  private_DMDALocatePointsIS_2D_Regular
                                  to print out some debugging
                                  information. I see that this is called
                                  twice during migration, once before
                                  and once after
                                  DMSwarmMigrate_DMNeighborScatter. If I
                                  understand correctly, the second call
                                  to
                                  private_DMDALocatePointsIS_2D_Regular
                                  should be able to locate all particles
                                  owned by the rank but it fails for
                                  some of them because they have been
                                  sent to the wrong rank (despite being
                                  well away from process boundaries).<br>
                                </p>
                                <p>For example, running the example
                                  src/dm/impls/da/tests/ex1.c with Nx=21
                                  (20x20 Q1 elements on [-1,1]X[-1,1])
                                  with 3 processors,</p>
                                <p>- the particles (-0.191,-0.462) and
                                  (0.191,-0.462) are sent cpu2 instead
                                  of cpu0</p>
                                <p>- those at (-0.287,-0.693)and
                                  (0.287,-0.693) are sent to cpu1
                                  instead of cpu0</p>
                                <p>- those at (0.191,0.462) and
                                  (-0.191,0.462) are sent to cpu0
                                  instead of cpu2</p>
                                <p>(This is 2d and thus not affected by
                                  the 3d issue mentioned yesterday on
                                  petsc-dev. Tests were made based on
                                  the release branch pulled out this
                                  morning, i.e. on commit <span style="font-family:monospace"><span style="font-family:monospace;background-color:rgb(255,255,255);color:rgb(178,104,24)">bebdc8d016f</span></span>).<br>
                                </p>
                              </blockquote>
                              <p>I see: particles are sent "all around"
                                and not only to the destination rank.</p>
                              <p>Still however, running the example
                                src/dm/impls/da/tests/ex1.c with Nx=21
                                (20x20 Q1 elements on [-1,1]X[-1,1])
                                with 3 processors, there are 2 particles
                                initially owned by rank2 (at y=-0.6929
                                and x=+/-0.2870) that are sent only to
                                rank1 and never make it to rank0 and are
                                thus lost in the end since rank1,
                                correctly, discards them.</p>
                              <p>Thanks</p>
                              <p>    Matteo<br>
                              </p>
                            </div>
                          </blockquote>
                        </div>
                        <br clear="all">
                        <div><br>
                        </div>
                        -- <br>
                        <div dir="ltr">
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                                    <div>What most experimenters take
                                      for granted before they begin
                                      their experiments is infinitely
                                      more interesting than any results
                                      to which their experiments lead.<br>
                                      -- Norbert Wiener</div>
                                    <div><br>
                                    </div>
                                    <div><a href="https://eur01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.cse.buffalo.edu%2F~knepley%2F&data=05%7C01%7Cmatteo.semplice%40uninsubria.it%7C94569ac2a32a4838103608dae44f9c46%7C9252ed8bdffc401c86ca6237da9991fa%7C0%7C0%7C638073327837332509%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=LEmGbachok2Z3m9SyCASEcTWytkpxw%2FnHCIXjYXPTa8%3D&reserved=0" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><br>
                                    </div>
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                    </blockquote>
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Professore Associato in Analisi Numerica
Dipartimento di Scienza e Alta Tecnologia
Università degli Studi dell'Insubria
<a href="https://eur01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.google.com%2Fmaps%2Fsearch%2FVia%2BValleggio%2C%2B11%2B-%2BComo%3Fentry%3Dgmail%26source%3Dg&data=05%7C01%7Cmatteo.semplice%40uninsubria.it%7C94569ac2a32a4838103608dae44f9c46%7C9252ed8bdffc401c86ca6237da9991fa%7C0%7C0%7C638073327837332509%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=0MwkPJb5hslcrFhqrLCaN4OKs%2FoylMR3dOib2MYsNOY%3D&reserved=0" target="_blank">Via Valleggio, 11 - Como</a></pre>
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    <pre cols="72">-- 
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Professore Associato in Analisi Numerica
Dipartimento di Scienza e Alta Tecnologia
Università degli Studi dell'Insubria
Via Valleggio, 11 - Como</pre>
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