Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr class="thead"><td class="align_left">Method</td><td class="align_center"><span style="width: 12.8362ptpx;"><svg height="12.4698pt" id="M105" style="vertical-align:-3.18147pt" version="1.1" viewbox="-0.0498162 -9.28833 12.8362 12.4698" width="12.8362pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.761,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg></span></td><td class="align_center"><span style="width: 12.8362ptpx;"><svg height="12.4698pt" id="M106" style="vertical-align:-3.18147pt" version="1.1" viewbox="-0.0498162 -9.28833 12.8362 12.4698" width="12.8362pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.761,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"></path></g></svg></span></td><td class="align_center"><span style="width: 12.3012ptpx;"><svg height="12.7178pt" id="M107" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 12.3012 12.7178" width="12.3012pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M558 587C558 666 497 712 432 712C379 712 330 691 284 650C212 586 178 508 148 348L71 -65C49 -185 35 -229 23 -235L27 -261C49 -259 101 -251 124 -224C131 -216 138 -178 159 -24C171 66 197 200 227 356C264 550 295 668 393 668C443 668 479 632 479 575C479 516 446 458 383 418C361 404 344 398 318 397L296 350C395 338 460 281 460 192C460 110 411 40 300 40C258 40 215 55 192 69L181 51C191 18 222 -16 266 -16C308 -16 351 1 397 26C471 67 545 142 545 221C545 315 486 365 401 395C469 437 558 498 558 587Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.228,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg></span></td><td class="align_center"><span style="width: 12.3012ptpx;"><svg height="12.7178pt" id="M108" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 12.3012 12.7178" width="12.3012pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M558 587C558 666 497 712 432 712C379 712 330 691 284 650C212 586 178 508 148 348L71 -65C49 -185 35 -229 23 -235L27 -261C49 -259 101 -251 124 -224C131 -216 138 -178 159 -24C171 66 197 200 227 356C264 550 295 668 393 668C443 668 479 632 479 575C479 516 446 458 383 418C361 404 344 398 318 397L296 350C395 338 460 281 460 192C460 110 411 40 300 40C258 40 215 55 192 69L181 51C191 18 222 -16 266 -16C308 -16 351 1 397 26C471 67 545 142 545 221C545 315 486 365 401 395C469 437 558 498 558 587Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.228,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"></path></g></svg></span></td></tr><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr><td class="align_left">VP<sub>GP+TR</sub></td><td class="align_center">3.0067</td><td class="align_center">2.8893</td><td class="align_center">10.5868</td><td class="align_center">1.4004</td></tr><tr><td class="align_left">VP<sub>KAU+TR</sub></td><td class="align_center">3.0068</td><td class="align_center">2.8891</td><td class="align_center">10.5868</td><td class="align_center">1.4004</td></tr><tr><td class="align_left">VP<sub>RJF+TR</sub></td><td class="align_center">3.0067</td><td class="align_center">2.8893</td><td class="align_center">10.5868</td><td class="align_center">1.4004</td></tr><tr class="table-tr"><td colspan="5"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Complexity

tab1

Table 1

Table 1: Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems 