{"action":"created","issue":{"url":"https://api.github.com/repos/JuliaDiffEq/PDERoadmap/issues/32","repository_url":"https://api.github.com/repos/JuliaDiffEq/PDERoadmap","labels_url":"https://api.github.com/repos/JuliaDiffEq/PDERoadmap/issues/32/labels{/name}","comments_url":"https://api.github.com/repos/JuliaDiffEq/PDERoadmap/issues/32/comments","events_url":"https://api.github.com/repos/JuliaDiffEq/PDERoadmap/issues/32/events","html_url":"https://github.com/JuliaDiffEq/PDERoadmap/pull/32","id":351340646,"node_id":"MDExOlB1bGxSZXF1ZXN0MjA4OTYyOTIw","number":32,"title":"Update the overview and appendix","user":{"login":"MSeeker1340","id":30965945,"node_id":"MDQ6VXNlcjMwOTY1OTQ1","avatar_url":"https://avatars1.githubusercontent.com/u/30965945?v=4","gravatar_id":"","url":"https://api.github.com/users/MSeeker1340","html_url":"https://github.com/MSeeker1340","followers_url":"https://api.github.com/users/MSeeker1340/followers","following_url":"https://api.github.com/users/MSeeker1340/following{/other_user}","gists_url":"https://api.github.com/users/MSeeker1340/gists{/gist_id}","starred_url":"https://api.github.com/users/MSeeker1340/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/MSeeker1340/subscriptions","organizations_url":"https://api.github.com/users/MSeeker1340/orgs","repos_url":"https://api.github.com/users/MSeeker1340/repos","events_url":"https://api.github.com/users/MSeeker1340/events{/privacy}","received_events_url":"https://api.github.com/users/MSeeker1340/received_events","type":"User","site_admin":false},"labels":[],"state":"open","locked":false,"assignee":null,"assignees":[],"milestone":null,"comments":10,"created_at":"2018-08-16T19:38:15Z","updated_at":"2018-08-20T17:26:11Z","closed_at":null,"author_association":"CONTRIBUTOR","pull_request":{"url":"https://api.github.com/repos/JuliaDiffEq/PDERoadmap/pulls/32","html_url":"https://github.com/JuliaDiffEq/PDERoadmap/pull/32","diff_url":"https://github.com/JuliaDiffEq/PDERoadmap/pull/32.diff","patch_url":"https://github.com/JuliaDiffEq/PDERoadmap/pull/32.patch"},"body":"@jlperla Opening this up for discussion. The overview section can be updated alongside the appendix, and after this is done we can move on to update the examples.\r\n\r\nI think adding an overview section to appendix A (Gaussian Elimination and Q) is very much needed because it is not evident what it tries to achieve. I'll post my summary here so please point out anything that is wrong:\r\n\r\n- The main text and the examples focuses on constructing $LQ$ derivative operators that act on the *interior*. The equation to solve is $LQu = p$. The $B$ operator does not directly come into the equation.\r\n\r\n- The appendix, on the other hand, tries to look at the discretized problem in another angle. Instead of solving the equation on the interior, the equation to solve is\r\n\r\n  $$ L\\bar{u} = p\\quad B\\bar{u} = b $$\r\n\r\n  Here $B$ plays a pivotal role and $Q$ does not show itself. And we want to show via examples that these two approaches are equivalent. In particular, the equation for the interior can be derived from the equation on the extended domain by stacking the two equations and using Gaussian elimination.\r\n\r\nI also feel like the relationship between Gaussian elimination and the Q operator is not explained clearly in A.1.4 and A.2.1. We're still pulling out Q from thin air and just checking the $LQ$ product gives the same matrix as Gaussian elimination. I haven't done the algebra fully but I feel what should be done is this: we express the elimination steps as well as dimension reduction as applying elementary matrices. This should be able to reproduce the Q matrix somehow, and then the equivalence can be adequately explained."},"comment":{"url":"https://api.github.com/repos/JuliaDiffEq/PDERoadmap/issues/comments/414397188","html_url":"https://github.com/JuliaDiffEq/PDERoadmap/pull/32#issuecomment-414397188","issue_url":"https://api.github.com/repos/JuliaDiffEq/PDERoadmap/issues/32","id":414397188,"node_id":"MDEyOklzc3VlQ29tbWVudDQxNDM5NzE4OA==","user":{"login":"MSeeker1340","id":30965945,"node_id":"MDQ6VXNlcjMwOTY1OTQ1","avatar_url":"https://avatars1.githubusercontent.com/u/30965945?v=4","gravatar_id":"","url":"https://api.github.com/users/MSeeker1340","html_url":"https://github.com/MSeeker1340","followers_url":"https://api.github.com/users/MSeeker1340/followers","following_url":"https://api.github.com/users/MSeeker1340/following{/other_user}","gists_url":"https://api.github.com/users/MSeeker1340/gists{/gist_id}","starred_url":"https://api.github.com/users/MSeeker1340/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/MSeeker1340/subscriptions","organizations_url":"https://api.github.com/users/MSeeker1340/orgs","repos_url":"https://api.github.com/users/MSeeker1340/repos","events_url":"https://api.github.com/users/MSeeker1340/events{/privacy}","received_events_url":"https://api.github.com/users/MSeeker1340/received_events","type":"User","site_admin":false},"created_at":"2018-08-20T17:26:11Z","updated_at":"2018-08-20T17:26:11Z","author_association":"CONTRIBUTOR","body":"@jlperla Can you take a look at the updated section 2, in particular the new section regarding operator composition?\r\n\r\nFor the extension operator, the original one appearing in Appendix A at (65) is obviously wrong and we should have E_11 be applied to the right. Right multiplication also naturally results in the two equivalent interpretations, i.e. extension for the stencil/restriction for the domain, which I found helpful.\r\n\r\nIf you think this is ok, I will go back to the appendix and reflect the changes. I also need to update the reference operators in section 3.1, after which the document update can be finished except for the examples."}}
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