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1 = Bayesian inversion, spring 2016 =
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3 [[image:attach:Nut3web.jpg||height="250"]]
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5 (% style="color: rgb(255,0,0);line-height: 1.42857;" %)**[[image:attach:Uuzi.gif]]Please send the poster file to [[zenith.purisha@helsinki.fi>>mailto:zenith.purisha@helsinki.fi||shape="rect"]] before 18 May 2016. **(% style="color: rgb(255,0,0);" %)**The posters will be printed and you may take it on Thursday at 9 am in C131. The poster presentation is from 9-11, but let them there to be seen by others until 4 pm.
6 **
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8 {{panel}}
9 **Teacher:** [[Tapio Helin>>doc:mathstatHenkilokunta.Helin, Tapio]] ja [[Samuli Siltanen>>doc:mathstatHenkilokunta.Siltanen, Samuli]]
10
11 **Scope:** 15 cr
12
13 **Type:** Advanced studies
14
15 **Teaching: **Lectures, exercises, and project work with measured data
16
17 **Topics: **Theory and computational methods of Bayesian inversion
18
19 **Prerequisites for the theoretical part: **Measure and integration theory
20
21 **Prerequisites for the computational part: **Basics of mathematical probability, some Matlab skills such as given by course //Applications of matrix computations//
22 {{/panel}}
23
24 === {{toc maxLevel="4" minLevel="2" indent="20px"/}} ===
25
26 (% style="color: rgb(255, 0, 0); color: rgb(0, 0, 0)" %)Inverse problems are about interpreting indirect measurements. The scientific study of inverse problems is an interdisciplinary field combining mathematics, physics, signal processing, and engineering. (% style="color: rgb(255,0,0);" %) (%%)Examples of inverse problems include
27
28 * Three-dimensional X-ray imaging ([[more information>>url:http://www.siltanen-research.net/project_Xray.html||rel="nofollow" shape="rect" class="external-link"]], also see [[this video>>url:https://www.youtube.com/watch?v=-SiVPCh92TA&list=UURP6ol3uNxWVgV-zPZexcXA||rel="nofollow" shape="rect" class="external-link"]] and [[this video>>url:https://www.youtube.com/watch?v=CfulbwFnyJ0&list=UURP6ol3uNxWVgV-zPZexcXA||rel="nofollow" shape="rect" class="external-link"]])
29 * Recovering the inner structure of the Earth based on earthquake measurements
30 * Sharpening a misfocused photograph ( [[more information>>doc:mathstatHenkilokunta.Henkilöt.Siltanen, Samuli.Inverse Problems Book Page.X-ray tomography with matrices.Matrix-free X-ray tomography.Matrix-free X-ray tomography with sparse data.2D deconvolution.WebHome]] )
31 * Reconstructing electric conductivity from current-to-voltage boundary measurements (see [[this page>>url:https://wiki.helsinki.fi/display/inverse/Electrical+Impedance+Tomography||rel="nofollow" shape="rect"]] and [[this page>>url:http://www.siltanen-research.net/project_EIT.html||rel="nofollow" shape="rect" class="external-link"]] )
32 * Finding cracks inside solid structures
33 * Prospecting for oil and minerals
34 * Monitoring underground contaminants
35 * Finding the shape of asteroids based on light-curve data (see [[this page>>url:http://www.rni.helsinki.fi/~~mjk/asteroids.html||rel="nofollow" shape="rect" class="external-link"]] )
36
37 The common features of all this problems are the need to understand indirect measurements and to overcome extreme sensitivity to noise and modelling inaccuracies.
38
39 (% class="p1" %)
40 The topic of the course is statistical inverse problems. The lectures consist of two parts:
41
42 (% class="p1" %)
43 (% style="line-height: 1.42857;" %)**Theoretical modelling in Bayesian inverse problems**(%%) (Tapio Helin)
44 (% style="line-height: 1.42857;" %)**Computational methods**(%%) (prof. Samuli Siltanen)(% style="line-height: 1.42857;" %)
45
46 (% class="p1" %)
47 The goals of the course are
48
49 (% class="p1" %)
50 (computational part)
51
52 (% class="p1" %)
53 - introduce the framework for statistical Bayesian inverse problems
54
55 (% class="p1" %)
56 - understand main ideas of uncertainty quantification via the Bayes formula
57
58 (% class="p1" %)
59 - learn efficient computational methods for exploring the posterior distribution.
60 In particular, two central Markov chain Monte Carlo algorithms are studied and implemented in Matlab: **Metropolis-Hastings algorithm** and the **Gibbs sampler**.
61
62 (% class="p1" %)
63 (theoretical part)
64
65 (% class="p1" %)
66 - learn the general Bayes formula
67
68 (% class="p1" %)
69 - show well-posedness of Bayesian inverse problems
70
71 (% class="p1" %)
72 - study continuity properties of the solution when discretization is refined
73
74 == News ==
75
76 *
77
78 == Teaching schedule ==
79
80 (% style="color: rgb(128,0,128);" %)**Period III:** (%%) Lectures as follows:
81
82 Wednesday 10-12 in room C124 Thursday 10-12 in room B120 Friday 10-12 in room C124.
83
84 Two hours of exercise classes per week.
85
86 (% style="color: rgb(128,0,128);" %)**Period IV:** (%%) Lectures continue as long as needed (but not for the whole period). There is a project work, which is reported as a poster in a poster session in May. The exact date will be decided later.
87
88 ----
89
90 == Lectures ==
91
92 (% style="color: rgb(128,0,128);" %)**Wednesday 27.1.2016 Samuli Siltanen
93 **(%%)Introduction to inverse problems and Bayes formula. Motivating examples: X-ray tomography and Glottal Inverse Filtering (GIF). Practical information about the course.
94 [[Introductory lecture slides (PDF)>>url:http://www.siltanen-research.net/Intro_v4.pdf||shape="rect"]]
95
96 (% style="color: rgb(128,0,128);" %)**Thursday 28.1.2016 Samuli Siltanen
97 **(%%)Real-valued random variable, probability density function, and computational sampling.
98
99 (% style="color: rgb(128,0,128);" %)**Friday 29.1.2016 Tapio Helin
100 **(%%)Motivation, sigma-algebras, Radon-Nikodym derivative and conditional expectation.
101
102 (% style="color: rgb(128,0,128);" %)**Wednesday 3.2.2016 Tapio Helin
103 **(%%)Conditional expectation and probability continued, Bayes formula.
104
105 (% style="color: rgb(128,0,128);" %)**Thursday 4.2.2016 Tapio Helin
106 **(%%)Proof of Bayes formula, some principals of Bayesian inference, example with Gaussian prior and noise.
107
108 First version of the lecture notes from the theoretical part: [[attach:Bayes_theoretical_notes_v1.pdf]]
109
110 (% style="color: rgb(128,0,128);" %)**Friday 5.2.2016 Samuli Siltanen
111 **(%%)Conditional probability in the case of probability density functions. Linear measurement model.
112
113 The newest version of the lecture note is here: [[attach:BayesNotes_v2.pdf]]
114
115 The complete package including Matlab and LaTeX files is here: [[attach:LectureNotes.zip]]
116
117 (% style="color: rgb(128,0,128);" %)**Wednesday 10.2.2016 Tapio Helin** (%%)
118 Gaussian posterior, Posterior consistency in over/underdetermined systems
119
120 (% style="color: rgb(128,0,128);" %)**Thursday 11.2.2016 Samuli Siltanen**
121
122 Introduction of one-dimensional convolution as a measurement model leading to ill-posed inverse problems.
123 Here are the discrete convolution demonstration slides (with source files in the zip folder): [[attach:ConvDemo_v1.pdf]], [[attach:ConvDemo.zip]].
124 Here is the explanation regarding continuous and discrete model: [[attach:1D_convolution.pdf]].
125
126 We started building a Matlab library for studying one-dimensional convolution and deconvolution.
127 [[attach:DC_convmtx.m]], [[attach:target1.m]], [[attach:PSF.m]]
128 [[attach:deconv1_cont_comp.m]], [[attach:deconv1_cont_plot.m]]
129 [[attach:deconv2_discretedata_comp.m]], [[attach:deconv2_discretedata_plot.m]]
130 [[attach:deconv3_naive_comp.m]], [[attach:deconv3_naive_plot.m]]
131
132 (% style="color: rgb(128,0,128);" %)
133
134 (% style="color: rgb(128,0,128);" %)**Friday 12.2.2016 Samuli Siltanen**
135
136 (% style="color: rgb(0,51,102);" %)Singular Value Decomposition (SVD) for matrices. Detecting ill-posedness. See the document [[attach:SVD.pdf]].
137
138 (% style="color: rgb(0,51,102);" %)Computation of SVD for the 1D convolution problem:
139 [[attach:deconv4_SVD_comp.m]], [[attach:deconv4_SVD_plot.m]]
140
141 (% style="color: rgb(0,51,102);" %)Illustration of Gaussian likelihood, prior and posterior distributions in a 2-dimensional toy example:
142 [[attach:GaussianPostDemo.m]]
143
144 (% style="color: rgb(0,51,102);" %)[[image:attach:Screen Shot 2016-02-12 at 13.16.25.png||height="250"]]
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146
147 (% style="color: rgb(128,0,128);line-height: 1.42857;" %)**Wednesday 17.2.2016 Tapio Helin**(%%)
148 Posterior consistency in underdetermined systems, metrics on probability space
149
150 (% style="color: rgb(255, 102, 0); color: rgb(128, 0, 128); line-height: 1.4286" %)**Thursday 18.2.2016 Samuli Siltanen**
151
152 (% style="color: rgb(0,0,0);" %)Definition and discussion of CM and MAP estimates. Derivation of a formula for the MAP estimate in the case of a Gaussian posterior (note that in that case the CM and MAP estimates are the same).
153
154 (% style="color: rgb(0,0,0);" %)Introduction of a simple 2-dimensional example. It is a m(%%)easurement scenario with two temperatures. Indoor temperature f1 is measured with thermometer T1 inside without connection to outdoor temperature. The outdoor temperature f2 is measured with thermometer T2, located in the windowpane. The reading of thermometer T2 is a linear combination of indoor and outdoor temperatures. Both thermometers are corrupted by additive Gaussian noise with standard deviation sigma. The prior is Gaussian. This Matlab file shows the situation: [[attach:TwoTemps.m]]
155
156 [[image:attach:Screen Shot 2016-02-18 at 16.49.35.png||height="250"]]
157
158 (% style="color: rgb(0,0,0);" %)Newest version of the notes: [[attach:BayesNotes_v3.pdf]]
159
160 (% style="color: rgb(0,0,0);" %)
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162
163
164 (% style="color: rgb(128,0,128);line-height: 1.42857;" %)**Friday 19.2.2016 Samuli Siltanen**
165
166 (% style="color: rgb(0,0,0);" %)Markov chain Monte Carlo sampling using a simple Metropolis-Hastings method.
167
168 (% style="color: rgb(0,0,0);" %)For explanation of the Metropolis-Hastings method, see Section 5.3.3 of the old lecture note [[attach:IPnotes14.pdf]].
169
170 (% style="color: rgb(0,0,0);" %)The two-dimensional example of measuring indoor and outdoor (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)temperatures using two thermometers is here in a revised form:
171 [[attach:TwoTempsMH1.m]], [[attach:posterior.m]], [[attach:logposterior.m]]
172
173 [[image:attach:Screen Shot 2016-02-19 at 14.00.41.png||height="400"]]
174
175 (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)
176
177 (% style="color: rgb(255, 0, 0); color: rgb(255, 102, 0); font-size: 16px; line-height: 1.5" %)**Wednesday 24.2. Guest Lecturer: Dr. Marko Laine (Finnish Meteorological Institute)**
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179 (% style="color: rgb(255, 0, 0); color: rgb(255, 102, 0); font-size: 16px; line-height: 1.5; color: rgb(0, 0, 0)" %)** **(% style="color: rgb(255, 0, 0); color: rgb(255, 102, 0); font-size: 16px; line-height: 1.5" %)** **(% style="color: rgb(255, 0, 0); color: rgb(0, 0, 0)" %)The slides of the first part: [[attach:ML-slides-2016-02-24.pdf]]
180 And the MCMC toolbox can be found here: [[http:~~/~~/helios.fmi.fi/~~~~lainema/mcmc/>>url:http://helios.fmi.fi/~~lainema/mcmc/||shape="rect"]]
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182
183 (% style="color: rgb(255, 0, 0); color: rgb(128, 0, 128); line-height: 1.4286" %)**Thursday 25.2. Tapio Helin**(% style="color: rgb(255,0,0);" %) (%%)
184 Hellinger distance of two Gaussians, well-posedness of the posterior with perturbed measurement and prior
185
186 (% style="color: rgb(255, 0, 0); color: rgb(128, 0, 128); line-height: 1.4286" %)**Friday 26.2. Tapio Helin**(% style="color: rgb(255,0,0);" %) (%%)
187 Well-posedness of the posterior with perturbed prior continued, introduction to probability in Banach spaces
188
189 (% style="color: rgb(0,0,0);" %)Newest version of the notes: [[attach:Bayes_theoretical_notes_v2.pdf]]
190
191 (% style="color: rgb(255, 0, 0); color: rgb(128, 0, 128); line-height: 1.4286" %)**Wednesday 2.3. Tapio Helin**(% style="color: rgb(255,0,0);" %) (%%)
192 Introduction to probability in Banach spaces, Gaussian random variables
193
194 (% style="color: rgb(255, 0, 0); color: rgb(128, 0, 128); line-height: 1.4286" %)**Thursday 3.3. Samuli Siltanen**
195
196 (% style="color: rgb(0,0,0);" %)MCMC computation using the Gibbs sampler.
197
198 (% style="color: rgb(0,0,0);" %)Matlab file: [[attach:TwoTempsGibbs1.m]]
199
200 (% style="color: rgb(128,0,128);line-height: 1.42857;" %)**Friday 4.3. Tapio Helin**(%%)
201 Gaussian random variables, Fernique theorem, Cameron-Martin space
202
203 (% style="color: rgb(128,0,128);line-height: 1.42857;" %)**Wednesday 16.3. Samuli Siltanen**
204
205 (((
206 Computation of MAP estimates for the 1D deconvolution problem. We use several choices of priors.
207 Newest version of the lecture notes: [[attach:BayesNotes_v4.pdf]], see the new chapter 2.7.
208 )))
209
210 (((
211 Matlab routines used in the lecture (some of these are posted again here in the same form than above):
212 )))
213
214 (((
215 [[attach:target1.m]], [[attach:target2.m]], [[attach:target3.m]]
216 )))
217
218 (((
219 [[attach:deconv1_cont_comp.m]], [[attach:deconv1_cont_plot.m]]
220 )))
221
222 (((
223 [[attach:deconv2_discretedata_comp.m]], [[attach:deconv2_discretedata_plot.m]]
224 )))
225
226 (((
227 [[attach:deconv3_naive_comp.m]], [[attach:deconv3_naive_plot.m]]
228 )))
229
230 (((
231 [[attach:deconv6_GaussianMAP_comp.m]], [[attach:deconv6_GaussianMAP_plot.m]]
232 )))
233
234 (((
235 (% style="color: rgb(128,0,128);line-height: 1.42857;" %)**
236 **
237 )))
238
239 (((
240 (% style="color: rgb(128,0,128);line-height: 1.42857;" %)**Thursday 17.3. Samuli Siltanen**
241 )))
242
243 (((
244 Computation of CM estimates for the 1D deconvolution problem. We use the Metropolis-Hastings algorithm with several choices of priors.
245 )))
246
247 (((
248 Matlab routines:
249 )))
250
251 (((
252 [[attach:deconv7_GaussianMH_comp.m]], [[attach:deconv7_GaussianMH_plot.m]]
253 )))
254
255 (((
256 [[attach:deconv8_nonGaussianMH_comp.m]], [[attach:deconv8_nonGaussianMH_plot.m]]
257 )))
258
259 (((
260 (% style="color: rgb(128,0,128);line-height: 1.42857;" %)**
261 **
262 )))
263
264 (((
265 (% style="color: rgb(128,0,128);line-height: 1.42857;" %)**Friday 18.3. Samuli Siltanen**
266 )))
267
268 (((
269 Practicalities about the project work:
270 )))
271
272 (((
273 * forming the teams (two students in each team)
274 * discussion of the two-phase structure of the project work
275 * choosing the topics of the project work, involving real data measurement (tomography, photographic, other)
276 * agreeing upon the mid-project deadline
277 * setting the date for the final poster session
278 )))
279
280 (% style="color: rgb(128,0,128);line-height: 1.42857;" %)**Wednesday 23.3. Tapio Helin**(%%)
281 Cameron-Martin spaces, stability of Bayesian inversion for non-linear problems
282
283 The newest version of the lecture note is here: [[attach:Bayes_theoretical_notes_v3.pdf]]
284
285 (% style="color: rgb(128,0,128);line-height: 1.42857;" %)**Thursday 31.3. Tapio Helin**(%%)
286 Stability and approximation properties
287
288 (% style="color: rgb(128,0,128);line-height: 1.42857;" %)**Friday 1.4. Tapio Helin **(%%)
289 Bayesian inversion for the inverse heat equation
290
291 The final version of the lecture notes: [[attach:Bayes_theoretical_notes_final.pdf]]
292
293 (((
294 **
295 **
296 )))
297
298
299
300 (((
301
302 )))
303
304 (% style="color: rgb(128,0,128);" %)**
305 **
306
307 == Exams ==
308
309 **[[image:attach:Uuzi.gif]] [[image:attach:Uuzi.gif]] [[image:attach:Uuzi.gif]] [[image:attach:Uuzi.gif]] [[image:attach:Uuzi.gif]] [[image:attach:Uuzi.gif]] **
310
311 (% style="color: rgb(255,0,0);" %)**[[HOME EXAM IS HERE (updated version with typos corrected at 13:50 on Monday, April 18)>>attach:exam_2016.pdf]]**
312
313 (% style="color: rgb(255,0,0);" %)**Deadline for returning answers is 12 o'clock noon on Monday, April 25, 2016.**
314
315 == Course material ==
316
317 Lecture notes will be updated here as the course progresses.
318
319 The book Mueller&Siltanen: Linear and nonlinear inverse problems with practical applications (SIAM 2012) explains the computational models used in the course. These lecture notes cover pretty much everything in the computational part: [[attach:IPnotes14.pdf]].
320
321 For the theoretical part we partly use the [[lecture notes>>url:http://arxiv.org/abs/1302.6989||shape="rect"]] by Masoumeh Dashti and Andrew Stuart. Also, an excellent review on Bayesian inverse problems in Banach spaces is available in [[this paper>>url:http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=7701764&fulltextType=RA&fileId=S0962492910000061||shape="rect"]] by Andrew Stuart (only available on UH network).
322
323 A slide with preliminaries and a few basics about probability distributions (in particular Gaussian) from the lecture Statistical Inverse Problems at the University of Eastern Finland: [[attach:1_Prelim.pdf]], [[attach:2_Processes.pdf]] (We thank Janne Huttunen and Ville Kolehmainen for providing the slides).
324
325 Some courses in [[Inverse Problem in Imaging >>url:http://www0.cs.ucl.ac.uk/staff/ucacarr/teaching/optimisation/||shape="rect"]]and one of Bayesian Inversion lectures: [[Bayesian Inversion by Bangti Jin>>attach:BayesianInversion.pdf]].
326
327 == [[Registration>>url:https://oodi-www.it.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57760||shape="rect"]] ==
328
329
330 (% style="color: rgb(96,96,96);" %)Did you forget to register?(%%) (% style="color: rgb(96,96,96);" %) (%%) [[What to do?>>doc:mathstatOpiskelu.Etusivu.Kysymys4.WebHome]]
331
332 == Exercises ==
333
334 Teaching assistants: [[Andreas Hauptmann>>doc:mathstatHenkilokunta.Home.Hauptmann, Andreas.WebHome]] & [[Zenith Purisha>>doc:mathstatHenkilokunta.Home.Purisha, Zenith.WebHome]]
335
336 === Assignments ===
337
338 The assignments are to be prepared for the exercise class, where the students are expected to present the solutions.
339
340 * [[Exercise 1>>attach:Exercise01.pdf]] (Due 09.02.), [[SolExercise01.pdf>>attach:model_solution1.pdf]]
341 * [[Exercise 2>>attach:Exercise02.pdf]] (Due 16.02.), files needed: [[attach:MyDensity1_sample.m]], [[attach:MyDensity2.m]], [[attach:MyDensity2_plot.m]], [[attach:MyDensity3.m]], [[attach:MyDensity3_plot.m]]
342 * [[Exercise 3>>attach:Exercise03.pdf]] (Due 23.02.), One suggested homework: [[SolExercise03.pdf>>attach:Sol_Exercise3.pdf]] (Take this as suggestion, not as model solution!)
343 * [[Exercise 4>>attach:Exercise04.pdf]] (Due 01.03.), files needed: [[attach:2_Processes.pdf]]. Solution codes: [[attach:CompSolutions4.zip]]
344 * [[Exercise 5>>attach:Exercise05.pdf]] (Due 15.03)
345 * [[Exercise 6>>attach:Exercise06.pdf]] (Due 22.03), files needed: [[attach:deconv8_L1reg_comp.m]], [[attach:deconv8_L1reg_plot.m]], [[attach:deconv9_TVreg_comp.m]], [[attach:deconv9_TVreg_plot.m]]
346
347 === Exercise classes ===
348
349 (% class="p1" %)
350 Tuesday 10-12: 09.02.2016 - classic - C124
351
352 (% class="p1" %)
353 Tuesday 10-12: 16.02.2016 - computer - C128
354
355 (% class="p1" %)
356 Tuesday 10-12: 23.02.2016 - classic - C124
357
358 (% class="p1" %)
359 Tuesday 10-12: 01.03.2016 - computer - C128
360
361 (% class="p1" %)
362 Tuesday 10-12: 15.03.2016 - classic - C124
363
364 (% class="p1" %)
365 Tuesday 10-12: 22.03.2016 - computer - C128
366
367 (% class="p1" %)
368 Tuesday 10-12: 05.04.2016 - computer - C128 (Last exercise session: we go through computational measurement models needed in the project works.)
369
370 ----
371
372 (% class="p1" %)
373 == Project work ==
374
375 Project work assistants: [[Andreas Hauptmann>>doc:mathstatHenkilokunta.Home.Hauptmann, Andreas.WebHome]] and Zenith Purisha.
376
377 (% style="color: rgb(0,0,0);" %)The idea is to study a Bayesian inverse problem both theoretically and computationally in **teams of two students**. The end product is a scientific poster that the team will present in a poster session on May 19 (at 9-11, Exactum first floor corridor). (%%) (% style="color: rgb(96, 96, 96); color: rgb(0, 0, 0)" %)The poster can be printed using the large-scale printer of(% style="color: rgb(96,96,96);" %) [[the Industrial Mathematics Laboratory>>doc:mathstatHenkilokunta.Henkilöt.Siltanen, Samuli.Industrial Mathematics Laboratory.WebHome]].
378
379 (% style="color: rgb(96,96,96);" %)
380
381 The classical table of contents is recommended for structuring the poster:
382
383 (% style="color: rgb(96,96,96);" %)
384
385 1 Introduction
386 2 Materials and methods
387 3 Results
388 4 Discussion
389
390 (% style="color: rgb(96,96,96);" %)
391
392 Section 2 is for describing the data and the inversion methods used. In section 3 those methods are applied to the data and the results are reported with no interpretation; just facts and outcomes of computations are described. Section 4 is the place for discussing the results and drawing conclusions.
393
394 The recommended measurement context of the project is **X-ray tomography**. You can measure a dataset yourself in the X-ray facility of the Industrial Mathematics Laboratory:
395
396 == (% class="confluence-embedded-file-wrapper confluence-embedded-manual-size" %)[[image:attach:mathstatKurssit.Inverse problems, spring 2015.WebHome@IMG_1313.jpg||thumbnail="true" width="300"]] (%%) (% class="confluence-embedded-file-wrapper confluence-embedded-manual-size" %) [[image:attach:mathstatKurssit.Inverse problems, spring 2015.WebHome@IMG_1345.jpg||thumbnail="true" width="300"]] (%%) (% class="confluence-embedded-file-wrapper confluence-embedded-manual-size" %) [[image:attach:mathstatKurssit.Inverse problems, spring 2015.WebHome@IMG_1362.jpg||thumbnail="true" width="300"]] (%%) (% class="confluence-embedded-file-wrapper confluence-embedded-manual-size" %) [[image:attach:mathstatKurssit.Inverse problems, spring 2015.WebHome@Rose03B.jpg||width="300"]](%%) ==
397
398 (% style="color: rgb(255,0,0);" %)**The project work has two phases, each with a specific goal.**
399
400 **First goal (deadline April 15) consists of two things:** (a) two first sections should be preliminary written in LaTeX (not necessarily in poster format yet) and (b) the Matlab codes related to the measurement should be run and studied. Two things will be graded in the meeting about the first goal: (a) the draft of project work and (b) your understanding of the available Matlab codes relevant to your topic. The grade represents 30% of the final grade of the project work. Please agree on a meeting time with the lecturer for reviewing and grading the first goal.
401
402 **Second and final goal (deadline May 19):** poster is presented in the poster session. The poster will be printed in size A1. You may create your own poster (from scratch), or you can use e.g. [[this template>>url:https://wiki.helsinki.fi/download/attachments/113254781/posterA1_templ_IP2014.zip?version=1&modificationDate=1397043033121&api=v2||rel="nofollow" shape="rect"]] as a starting point and edit its layout, colors, fonts, etc. as much as you like.
403
404 Example posters are shown [[on this page>>doc:mathstatKurssit.Kevät 2015.Inverse problems, spring 2015.WebHome]].
405
406 (% class="confluence-embedded-file-wrapper confluence-embedded-manual-size" %)
407
408
409
410 (% style="color: rgb(96,96,96);" %)
411
412
413
414 ----
415
416 == Course feedback ==
417
418 Course feedback can be given at any point during the course. Click [[here>>url:https://elomake.helsinki.fi/lomakkeet/11954/lomake.html||style="line-height: 1.4285;" shape="rect"]].