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    <p>Hi Hong,</p>
    <p>It works like a charm. I really appreciate your help.<br>
    </p>
    <p>Regards,</p>
    <p>Danyang<br>
    </p>
    <br>
    <div class="moz-cite-prefix">On 17-05-25 07:49 AM, Hong wrote:<br>
    </div>
    <blockquote
cite="mid:CAGCphBu7jAaqSA_j8wreS_XRoPpD+vJYt-hiua1Tkp4q=Bf8Jg@mail.gmail.com"
      type="cite">
      <div dir="ltr">
        <div class="gmail_extra">
          <div class="gmail_quote">Danyang:</div>
          <div class="gmail_quote">You must access inner pc, then set
            shift. See </div>
          <div class="gmail_quote">petsc/src/ksp/ksp/examples/tutorials/ex7.c</div>
          <div class="gmail_quote"><br>
          </div>
          <div class="gmail_quote">For example, I add following to
            petsc/src/ksp/ksp/examples/tutorials/ex2.c, line 191:</div>
          <div class="gmail_quote">
            <div class="gmail_quote">  PetscBool isbjacobi;</div>
            <div class="gmail_quote">  PC        pc;</div>
            <div class="gmail_quote">  ierr =
              KSPGetPC(ksp,&pc);CHKERRQ(ierr);</div>
            <div class="gmail_quote">  ierr =
PetscObjectTypeCompare((PetscObject)pc,PCBJACOBI,&isbjacobi);CHKERRQ(ierr);</div>
            <div class="gmail_quote">  if (isbjacobi) {</div>
            <div class="gmail_quote">    PetscInt nlocal;</div>
            <div class="gmail_quote">    KSP      *subksp;</div>
            <div class="gmail_quote">    PC       subpc;</div>
            <div class="gmail_quote"><br>
            </div>
            <div class="gmail_quote">    ierr =
              KSPSetUp(ksp);CHKERRQ(ierr);</div>
            <div class="gmail_quote">    ierr =
              KSPGetPC(ksp,&pc);CHKERRQ(ierr);</div>
            <div class="gmail_quote"><br>
            </div>
            <div class="gmail_quote">    /* Extract the array of KSP
              contexts for the local blocks */</div>
            <div class="gmail_quote">    ierr =
              PCBJacobiGetSubKSP(pc,&nlocal,NULL,&subksp);CHKERRQ(ierr);</div>
            <div class="gmail_quote">    printf("isbjacobi, nlocal %D,
              set option to subpc...\n",nlocal);</div>
            <div class="gmail_quote">    for (i=0; i<nlocal; i++) {</div>
            <div class="gmail_quote">      ierr =
              KSPGetPC(subksp[i],&subpc);CHKERRQ(ierr);</div>
            <div class="gmail_quote">      ierr =
              PCFactorSetShiftType(subpc,MAT_SHIFT_NONZERO);CHKERRQ(ierr);</div>
            <div class="gmail_quote">    }</div>
            <div class="gmail_quote">  }</div>
          </div>
          <div class="gmail_quote"><br>
          </div>
          <div class="gmail_quote"><br>
            <blockquote class="gmail_quote" style="margin:0px 0px 0px
              0.8ex;border-left:1px solid
              rgb(204,204,204);padding-left:1ex">
              <div bgcolor="#FFFFFF">
                <p>Dear Hong and Barry,</p>
                <p>I have implemented this option in the code, as we
                  also need to use configuration from file for
                  convenience. When I run the code using options, it
                  works fine, however, when I run the code using
                  configuration file, it does not work. The code has two
                  set of equations, flow and reactive, with prefix been
                  set to "flow_" and "react_". When I run the code using</p>
                <p>mpiexec -n 4 ../executable
                  -flow_sub_pc_factor_shift_type nonzero
                  -react_sub_pc_factor_shift_<wbr>type nonzero</p>
                <p>it works. However, if I run using</p>
                <p>mpiexec -n 4 ../executable <br>
                </p>
                <p>and let the executable file read the options from
                  file, it just does not work at "call
                  PCFactorSetShiftType(pc_flow,<wbr>MAT_SHIFT_NONZERO,
                  ierr)  or none, positive_definite ...". Do I miss
                  something here?<br>
                </p>
                <p>Below is the pseudo code I have used for flow
                  equations, similar for reactive equations. <br>
                </p>
                <p>          call MatCreateAIJ(Petsc_Comm_World,<wbr>nndof,nndof,nngbldof,    
                  &
                  <br>
                                              nngbldof,d_nz,PETSC_NULL_<wbr>INTEGER,o_nz,    
                  &
                  <br>
                                              PETSC_NULL_INTEGER,a_flow,<wbr>ierr)
                  <br>
                            CHKERRQ(ierr)<br>
                  <br>
                          call MatSetFromOptions(a_flow,ierr)
                  <br>
                          CHKERRQ(ierr)<br>
                  <br>
                          call KSPCreate(Petsc_Comm_World, ksp_flow,
                  ierr)
                  <br>
                          CHKERRQ(ierr)<br>
                  <br>
                          call KSPAppendOptionsPrefix(ksp_<wbr>flow,"flow_",ierr)
                  <br>
                          CHKERRQ(ierr)<br>
                  <br>
                          call KSPSetInitialGuessNonzero(ksp_<wbr>flow,                      
                  &
                  <br>
                                  b_initial_guess_nonzero_flow, ierr)
                  <br>
                          CHKERRQ(ierr)<br>
                  <br>
                          call KSPSetInitialGuessNonzero(ksp_<wbr>flow,                      
                  &
                  <br>
                                  b_initial_guess_nonzero_flow, ierr)
                  <br>
                          CHKERRQ(ierr)<br>
                  <br>
                          call KSPSetDM(ksp_flow,dmda_flow%<wbr>da,ierr)
                  <br>
                          CHKERRQ(ierr)
                  <br>
                          call KSPSetDMActive(ksp_flow,PETSC_<wbr>FALSE,ierr)
                  <br>
                          CHKERRQ(ierr)<br>
                </p>
                <p>        !!!!*********CHECK IF READ OPTION FROM
                  FILE*********!!!!<br>
                          if (read_option_from_file) then<br>
                  <br>
                            call KSPSetType(ksp_flow, KSPGMRES, ierr)   
                  !or KSPBCGS or others...
                  <br>
                            CHKERRQ(ierr)<br>
                  <br>
                            call KSPGetPC(ksp_flow, pc_flow, ierr)
                  <br>
                            CHKERRQ(ierr)<br>
                  <br>
                            call PCSetType(pc_flow,PCBJACOBI,
                  ierr)       !or PCILU or PCJACOBI or PCHYPRE ...
                  <br>
                            CHKERRQ(ierr)<br>
                  <br>
                            call PCFactorSetShiftType(pc_flow,<wbr>MAT_SHIFT_NONZERO,
                  ierr)  or none, positive_definite ...
                  <br>
                            CHKERRQ(ierr)<br>
                  <br>
                          end if<br>
                   <br>
                          call PCFactorGetMatSolverPackage(<wbr>pc_flow,solver_pkg_flow,ierr)
                  <br>
                          CHKERRQ(ierr)<br>
                  <br>
                          call compute_jacobian(rank,dmda_<wbr>flow%da,                      
                  &
                  <br>
                                               
                  a_flow,a_in,ia_in,ja_in,nngl_<wbr>in,         &
                  <br>
                                               
                  row_idx_l2pg,col_idx_l2pg,    <wbr>           &
                  <br>
                                                b_non_interlaced)<br>
                          call KSPSetFromOptions(ksp_flow,<wbr>ierr)
                  <br>
                          CHKERRQ(ierr)
                  <br>
                   <br>
                          call KSPSetUp(ksp_flow,ierr)
                  <br>
                          CHKERRQ(ierr)
                  <br>
                   <br>
                          call KSPSetUpOnBlocks(ksp_flow,<wbr>ierr)
                  <br>
                          CHKERRQ(ierr)
                  <br>
                   <br>
                          call KSPSolve(ksp_flow,b_flow,x_<wbr>flow,ierr)
                  <br>
                          CHKERRQ(ierr) <br>
                </p>
                <p>        <br>
                  Thanks and Regards,<br>
                </p>
                <p>Danyang</p>
                <div class="gmail-m_-8252057521010818115moz-cite-prefix">On
                  17-05-24 06:32 PM, Hong wrote:<br>
                </div>
                <blockquote type="cite">
                  <div dir="ltr"><span style="font-size:12.8px">Remove
                      your option '</span><span style="color:rgb(0,0,0);font-family:"Courier New",Courier,monospace,arial,sans-serif;font-size:14px;white-space:pre-wrap">-vecload_block_size 10'.</span><br>
                    <div><span style="color:rgb(0,0,0);font-family:"Courier New",Courier,monospace,arial,sans-serif;font-size:14px;white-space:pre-wrap">
</span></div>
                    <div><font color="#000000" face="Courier New,
                        Courier, monospace, arial, sans-serif"><span style="font-size:14px;white-space:pre-wrap">Hong</span></font></div>
                  </div>
                  <div class="gmail_extra"><br>
                    <div class="gmail_quote">On Wed, May 24, 2017 at
                      3:06 PM, Danyang Su <span dir="ltr"><<a
                          moz-do-not-send="true"
                          href="mailto:danyang.su@gmail.com"
                          target="_blank">danyang.su@gmail.com</a>></span>
                      wrote:<br>
                      <blockquote class="gmail_quote" style="margin:0px
                        0px 0px 0.8ex;border-left:1px solid
                        rgb(204,204,204);padding-left:1ex">
                        <div bgcolor="#FFFFFF">
                          <p>Dear Hong,</p>
                          <p>I just tested with different number of
                            processors for the same matrix. It sometimes
                            got "ERROR: Arguments are incompatible" for
                            different number of processors. It works
                            fine using 4, 8, or 24 processors, but
                            failed with "ERROR: Arguments are
                            incompatible" using 16 or 48 processors. The
                            error information is attached. I tested this
                            on my local computer with 6 cores 12
                            threads. Any suggestion on this?</p>
                          <p>Thanks,<br>
                          </p>
                          <p>Danyang<br>
                          </p>
                          <div>
                            <div class="gmail-m_-8252057521010818115h5">
                              <br>
                              <div
                                class="gmail-m_-8252057521010818115m_9041936809186897311moz-cite-prefix">On
                                17-05-24 12:28 PM, Danyang Su wrote:<br>
                              </div>
                              <blockquote type="cite">
                                <p>Hi Hong,</p>
                                <p>Awesome. Thanks for testing the case.
                                  I will try your options for the code
                                  and get back to you later.</p>
                                <p>Regards,</p>
                                <p>Danyang<br>
                                </p>
                                <br>
                                <div
                                  class="gmail-m_-8252057521010818115m_9041936809186897311moz-cite-prefix">On
                                  17-05-24 12:21 PM, Hong wrote:<br>
                                </div>
                                <blockquote type="cite">
                                  <div dir="ltr">
                                    <div class="gmail_extra">
                                      <div class="gmail_quote">Danyang :</div>
                                      <div class="gmail_quote">I tested
                                        your data.</div>
                                      <div class="gmail_quote">Your
                                        matrices encountered zero
                                        pivots, e.g.</div>
                                      <div class="gmail_quote">
                                        <div class="gmail_quote">petsc/src/ksp/ksp/examples/tut<wbr>orials
                                          (master)</div>
                                        <div class="gmail_quote">$
                                          mpiexec -n 24 ./ex10 -f0
                                          a_react_in_2.bin -rhs
                                          b_react_in_2.bin -ksp_monitor
                                          -ksp_error_if_not_converged</div>
                                        <div class="gmail_quote"><br>
                                        </div>
                                        <div class="gmail_quote">
                                          <div class="gmail_quote">[15]PETSC
                                            ERROR: Zero pivot in LU
                                            factorization: <a
                                              moz-do-not-send="true"
                                              href="http://www.mcs.anl.gov/petsc/documentation/faq.html#zeropivot"
                                              target="_blank">http://www.mcs.anl.gov/petsc/d<wbr>ocumentation/faq.html#zeropivo<wbr>t</a></div>
                                          <div class="gmail_quote">[15]PETSC
                                            ERROR: Zero pivot row 1249
                                            value 2.05808e-14 tolerance
                                            2.22045e-14</div>
                                          <div class="gmail_quote">...</div>
                                          <div class="gmail_quote"><br>
                                          </div>
                                          <div class="gmail_quote">Adding
                                            option
                                            '-sub_pc_factor_shift_type
                                            nonzero', I got</div>
                                          <div class="gmail_quote">
                                            <div class="gmail_quote">mpiexec
                                              -n 24 ./ex10 -f0
                                              a_react_in_2.bin -rhs
                                              b_react_in_2.bin
                                              -ksp_monitor
                                              -ksp_error_if_not_converged
                                              -sub_pc_factor_shift_type
                                              nonzero -mat_view
                                              ascii::ascii_info</div>
                                            <div class="gmail_quote"><br>
                                            </div>
                                            <div class="gmail_quote">Mat
                                              Object: 24 MPI processes</div>
                                            <div class="gmail_quote"> 
                                              type: mpiaij</div>
                                            <div class="gmail_quote"> 
                                              rows=450000, cols=450000</div>
                                            <div class="gmail_quote"> 
                                              total: nonzeros=6991400,
                                              allocated nonzeros=6991400</div>
                                            <div class="gmail_quote"> 
                                              total number of mallocs
                                              used during MatSetValues
                                              calls =0</div>
                                            <div class="gmail_quote">   
                                              not using I-node (on
                                              process 0) routines</div>
                                            <div class="gmail_quote">  0
                                              KSP Residual norm
                                              5.849777711755e+01</div>
                                            <div class="gmail_quote">  1
                                              KSP Residual norm
                                              6.824179430230e-01</div>
                                            <div class="gmail_quote">  2
                                              KSP Residual norm
                                              3.994483555787e-02</div>
                                            <div class="gmail_quote">  3
                                              KSP Residual norm
                                              6.085841461433e-03</div>
                                            <div class="gmail_quote">  4
                                              KSP Residual norm
                                              8.876162583511e-04</div>
                                            <div class="gmail_quote">  5
                                              KSP Residual norm
                                              9.407780665278e-05</div>
                                            <div class="gmail_quote">Number
                                              of iterations =   5</div>
                                            <div class="gmail_quote">Residual
                                              norm 0.00542891</div>
                                            <div class="gmail_quote"><br>
                                            </div>
                                            <div class="gmail_quote">Hong</div>
                                          </div>
                                        </div>
                                        <blockquote class="gmail_quote"
                                          style="margin:0px 0px 0px
                                          0.8ex;border-left:1px solid
                                          rgb(204,204,204);padding-left:1ex">
                                          <div bgcolor="#FFFFFF">
                                            <p>Hi Matt,</p>
                                            <p>Yes. The matrix is
                                              450000x450000 sparse. The
                                              hypre takes hundreds of
                                              iterates, not for all but
                                              in most of the timesteps.
                                              The matrix is not well
                                              conditioned, with nonzero
                                              entries range from 1.0e-29
                                              to 1.0e2. I also made
                                              double check if there is
                                              anything wrong in the
                                              parallel version, however,
                                              the matrix is the same
                                              with sequential version
                                              except some round error
                                              which is relatively very
                                              small. Usually for those
                                              not well conditioned
                                              matrix, direct solver
                                              should be faster than
                                              iterative solver, right?
                                              But when I use the
                                              sequential iterative
                                              solver with ILU prec
                                              developed almost 20 years
                                              go by others, the solver
                                              converge fast with
                                              appropriate factorization
                                              level. In other words,
                                              when I use 24 processor
                                              using hypre, the speed is
                                              almost the same as as the
                                              old sequential iterative
                                              solver using 1 processor.</p>
                                            <p>I use most of the default
                                              configuration for the
                                              general case with pretty
                                              good speedup. And I am not
                                              sure if I miss something
                                              for this problem.</p>
                                            <p>Thanks,</p>
                                            <p>Danyang<br>
                                            </p>
                                            <div>
                                              <div
                                                class="gmail-m_-8252057521010818115m_9041936809186897311gmail-h5">
                                                <br>
                                                <div
class="gmail-m_-8252057521010818115m_9041936809186897311gmail-m_1680376947984306833moz-cite-prefix">On
                                                  17-05-24 11:12 AM,
                                                  Matthew Knepley wrote:<br>
                                                </div>
                                                <blockquote type="cite">
                                                  <div dir="ltr">
                                                    <div
                                                      class="gmail_extra">
                                                      <div
                                                        class="gmail_quote">On
                                                        Wed, May 24,
                                                        2017 at 12:50
                                                        PM, Danyang Su <span
                                                          dir="ltr"><<a
moz-do-not-send="true" href="mailto:danyang.su@gmail.com"
                                                          target="_blank">danyang.su@gmail.com</a>></span>
                                                        wrote:<br>
                                                        <blockquote
                                                          class="gmail_quote"
style="margin:0px 0px 0px 0.8ex;border-left:1px solid
                                                          rgb(204,204,204);padding-left:1ex">
                                                          <div
                                                          bgcolor="#FFFFFF">
                                                          <p>Hi Matthew
                                                          and Barry,</p>
                                                          <p>Thanks for
                                                          the quick
                                                          response. <br>
                                                          </p>
                                                          <p>I also
                                                          tried superlu
                                                          and mumps,
                                                          both work but
                                                          it is about
                                                          four times
                                                          slower than
                                                          ILU(dt) prec
                                                          through hypre,
                                                          with 24
                                                          processors I
                                                          have tested.</p>
                                                          </div>
                                                        </blockquote>
                                                        <div>You mean
                                                          the total time
                                                          is 4x? And you
                                                          are taking
                                                          hundreds of
                                                          iterates? That
                                                          seems hard to
                                                          believe,
                                                          unless you are
                                                          dropping</div>
                                                        <div>a huge
                                                          number of
                                                          elements. <br>
                                                        </div>
                                                        <blockquote
                                                          class="gmail_quote"
style="margin:0px 0px 0px 0.8ex;border-left:1px solid
                                                          rgb(204,204,204);padding-left:1ex">
                                                          <div
                                                          bgcolor="#FFFFFF">
                                                          <p> When I
                                                          look into the
                                                          convergence
                                                          information,
                                                          the method
                                                          using ILU(dt)
                                                          still takes
                                                          200 to 3000
                                                          linear
                                                          iterations for
                                                          each newton
                                                          iteration. One
                                                          reason is this
                                                          equation is
                                                          hard to solve.
                                                          As for the
                                                          general cases,
                                                          the same
                                                          method works
                                                          awesome and
                                                          get very good
                                                          speedup.</p>
                                                          </div>
                                                        </blockquote>
                                                        <div>I do not
                                                          understand
                                                          what you mean
                                                          here. <br>
                                                        </div>
                                                        <blockquote
                                                          class="gmail_quote"
style="margin:0px 0px 0px 0.8ex;border-left:1px solid
                                                          rgb(204,204,204);padding-left:1ex">
                                                          <div
                                                          bgcolor="#FFFFFF">
                                                          <p> I also
                                                          doubt if I use
                                                          hypre
                                                          correctly for
                                                          this case. Is
                                                          there anyway
                                                          to check this
                                                          problem, or is
                                                          it possible to
                                                          increase the
                                                          factorization
                                                          level through
                                                          hypre?</p>
                                                          </div>
                                                        </blockquote>
                                                        <div>I don't
                                                          know.</div>
                                                        <div><br>
                                                        </div>
                                                        <div>  Matt <br>
                                                        </div>
                                                        <blockquote
                                                          class="gmail_quote"
style="margin:0px 0px 0px 0.8ex;border-left:1px solid
                                                          rgb(204,204,204);padding-left:1ex">
                                                          <div
                                                          bgcolor="#FFFFFF">
                                                          <p>Thanks,</p>
                                                          <p>Danyang<br>
                                                          </p>
                                                          <br>
                                                          <div
class="gmail-m_-8252057521010818115m_9041936809186897311gmail-m_1680376947984306833m_-1065376578025428121moz-cite-prefix">On
                                                          17-05-24 04:59
                                                          AM, Matthew
                                                          Knepley wrote:<br>
                                                          </div>
                                                          <blockquote
                                                          type="cite">
                                                          <div dir="ltr">
                                                          <div
                                                          class="gmail_extra">
                                                          <div
                                                          class="gmail_quote">On
                                                          Wed, May 24,
                                                          2017 at 2:21
                                                          AM, Danyang Su
                                                          <span
                                                          dir="ltr"><<a
moz-do-not-send="true" href="mailto:danyang.su@gmail.com"
                                                          target="_blank">danyang.su@gmail.com</a>></span>
                                                          wrote:<br>
                                                          <blockquote
                                                          class="gmail_quote"
style="margin:0px 0px 0px 0.8ex;border-left:1px solid
                                                          rgb(204,204,204);padding-left:1ex">Dear
                                                          All,<br>
                                                          <br>
                                                          I use
                                                          PCFactorSetLevels
                                                          for ILU and
                                                          PCFactorSetFill
                                                          for other
                                                          preconditioning
                                                          in my code to
                                                          help solve the
                                                          problems that
                                                          the default
                                                          option is hard
                                                          to solve.
                                                          However, I
                                                          found the
                                                          latter one,
                                                          PCFactorSetFill
                                                          does not take
                                                          effect for my
                                                          problem. The
                                                          matrices and
                                                          rhs as well as
                                                          the solutions
                                                          are attached
                                                          from the link
                                                          below. I
                                                          obtain the
                                                          solution using
                                                          hypre
                                                          preconditioner
                                                          and it takes 7
                                                          and 38
                                                          iterations for
                                                          matrix 1 and
                                                          matrix 2.
                                                          However, if I
                                                          use other
                                                          preconditioner,
                                                          the solver
                                                          just failed at
                                                          the first
                                                          matrix. I have
                                                          tested this
                                                          matrix using
                                                          the native
                                                          sequential
                                                          solver (not
                                                          PETSc) with
                                                          ILU
                                                          preconditioning.
                                                          If I set the
                                                          incomplete
                                                          factorization
                                                          level to 0,
                                                          this
                                                          sequential
                                                          solver will
                                                          take more than
                                                          100
                                                          iterations. If
                                                          I increase the
                                                          factorization
                                                          level to 1 or
                                                          more, it just
                                                          takes several
                                                          iterations.
                                                          This remind me
                                                          that the PC
                                                          factor for
                                                          this matrices
                                                          should be
                                                          increased.
                                                          However, when
                                                          I tried it in
                                                          PETSc, it just
                                                          does not work.<br>
                                                          <br>
                                                          Matrix and rhs
                                                          can be
                                                          obtained from
                                                          the link
                                                          below.<br>
                                                          <br>
                                                          <a
                                                          moz-do-not-send="true"
href="https://eilinator.eos.ubc.ca:8443/index.php/s/CalUcq9CMeblk4R"
                                                          rel="noreferrer"
target="_blank">https://eilinator.eos.ubc.ca:8<wbr>443/index.php/s/CalUcq9CMeblk4<wbr>R</a><br>
                                                          <br>
                                                          Would anyone
                                                          help to check
                                                          if you can
                                                          make this work
                                                          by increasing
                                                          the PC factor
                                                          level or fill?<br>
                                                          </blockquote>
                                                          <div><br>
                                                          </div>
                                                          <div>We have
                                                          ILU(k)
                                                          supported in
                                                          serial.
                                                          However
                                                          ILU(dt) which
                                                          takes a
                                                          tolerance only
                                                          works through
                                                          Hypre</div>
                                                          <div><br>
                                                          </div>
                                                          <div>  <a
                                                          moz-do-not-send="true"
href="http://www.mcs.anl.gov/petsc/documentation/linearsolvertable.html"
target="_blank">http://www.mcs.anl.gov/petsc<wbr>/documentation/linearsolvertab<wbr>le.html</a></div>
                                                          <div><br>
                                                          </div>
                                                          <div>I
                                                          recommend you
                                                          try SuperLU or
                                                          MUMPS, which
                                                          can both be
                                                          downloaded
                                                          automatically
                                                          by configure,
                                                          and</div>
                                                          <div>do a full
                                                          sparse LU.</div>
                                                          <div><br>
                                                          </div>
                                                          <div>  Thanks,</div>
                                                          <div><br>
                                                          </div>
                                                          <div>    Matt</div>
                                                          <div> </div>
                                                          <blockquote
                                                          class="gmail_quote"
style="margin:0px 0px 0px 0.8ex;border-left:1px solid
                                                          rgb(204,204,204);padding-left:1ex">
                                                          Thanks and
                                                          regards,<br>
                                                          <br>
                                                          Danyang<br>
                                                          <br>
                                                          <br>
                                                          </blockquote>
                                                          </div>
                                                          <br>
                                                          <br
                                                          clear="all">
                                                          <span
                                                          class="gmail-HOEnZb"><font
color="#888888"> <span
class="gmail-m_-8252057521010818115m_9041936809186897311gmail-m_1680376947984306833HOEnZb"><font
color="#888888">
                                                          <div><br>
                                                          </div>
                                                          -- <br>
                                                          <div
class="gmail-m_-8252057521010818115m_9041936809186897311gmail-m_1680376947984306833m_-1065376578025428121gmail_signature">
                                                          <div dir="ltr">
                                                          <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
                                                          moz-do-not-send="true"
href="http://www.caam.rice.edu/%7Emk51/" target="_blank">http://www.caam.rice.edu/~mk51<wbr>/</a><br>
                                                          </div>
                                                          </div>
                                                          </div>
                                                          </font></span></font></span></div>
                                                          <span
                                                          class="gmail-HOEnZb"><font
color="#888888"> </font></span></div>
                                                          <span
                                                          class="gmail-HOEnZb"><font
color="#888888"> </font></span></blockquote>
                                                          <span
                                                          class="gmail-HOEnZb"><font
color="#888888"> <br>
                                                          </font></span></div>
                                                          <span
                                                          class="gmail-HOEnZb"><font
color="#888888"> </font></span></blockquote>
                                                        <span
                                                          class="gmail-HOEnZb"><font
color="#888888"> </font></span></div>
                                                      <span
                                                        class="gmail-HOEnZb"><font
color="#888888"> <br>
                                                          <br
                                                          clear="all">
                                                          <div><br>
                                                          </div>
                                                          -- <br>
                                                          <div
class="gmail-m_-8252057521010818115m_9041936809186897311gmail-m_1680376947984306833gmail_signature">
                                                          <div dir="ltr">
                                                          <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
                                                          moz-do-not-send="true"
href="http://www.caam.rice.edu/%7Emk51/" target="_blank">http://www.caam.rice.edu/~mk51<wbr>/</a><br>
                                                          </div>
                                                          </div>
                                                          </div>
                                                        </font></span></div>
                                                    <span
                                                      class="gmail-HOEnZb"><font
                                                        color="#888888">
                                                      </font></span></div>
                                                  <span
                                                    class="gmail-HOEnZb"><font
                                                      color="#888888"> </font></span></blockquote>
                                                <span
                                                  class="gmail-HOEnZb"><font
                                                    color="#888888"> <br>
                                                  </font></span></div>
                                              <span class="gmail-HOEnZb"><font
                                                  color="#888888"> </font></span></div>
                                            <span class="gmail-HOEnZb"><font
                                                color="#888888"> </font></span></div>
                                          <span class="gmail-HOEnZb"><font
                                              color="#888888"> </font></span></blockquote>
                                        <span class="gmail-HOEnZb"><font
                                            color="#888888"> </font></span></div>
                                      <span class="gmail-HOEnZb"><font
                                          color="#888888"> <br>
                                        </font></span></div>
                                    <span class="gmail-HOEnZb"><font
                                        color="#888888"> </font></span></div>
                                  <span class="gmail-HOEnZb"><font
                                      color="#888888"> </font></span></blockquote>
                                <span class="gmail-HOEnZb"><font
                                    color="#888888"> <br>
                                  </font></span></blockquote>
                              <span class="gmail-HOEnZb"><font
                                  color="#888888"> <br>
                                </font></span></div>
                            <span class="gmail-HOEnZb"><font
                                color="#888888"> </font></span></div>
                          <span class="gmail-HOEnZb"><font
                              color="#888888"> </font></span></div>
                        <span class="gmail-HOEnZb"><font color="#888888">
                          </font></span></blockquote>
                      <span class="gmail-HOEnZb"><font color="#888888">
                        </font></span></div>
                    <span class="gmail-HOEnZb"><font color="#888888"> <br>
                      </font></span></div>
                </blockquote>
                <br>
              </div>
            </blockquote>
          </div>
          <br>
        </div>
      </div>
    </blockquote>
    <br>
  </body>
</html>