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Index

Accelaration of convergence, 68-72 Accuracy of approximate solutions,

205, 229, 241, 269 Approximation conditions, 11, 147, 178,

182, 188

Boundary inverse problems, with given boundaries, 220-233,

252-254 with free boundaries, 254-272,

273-284

Coefficient inverse problems, with given boundaries, 233-248 with free boundaries, 285-292,

293-302 Computational formula for gradient, 204,

226, 238, 258, 276, 291, 319 Conjugate problems, 222, 235, 256-257,

273-274, 288, 289-290, 318 Consistency of model, 221, 234, 253, 255,

273, 286 measure of inconsistency, 7, 8

Continuous casting problem, 302-309 Coordination conditions, 30, 31, 33, 120,

123, 125, 127, 135, 158, 166, 169, 179, 183, 189

Correctness of problem, ill-posed problems, 10, 11 well-posed problems, 10, 33

Descriptive regularization, 5, 202, 210, 221-222, 256, 273, 290, 293, 317, 324

Determination of heat flow, 255, 257, 265-266, 270-272, 273, 274, 281-284,317, 319-324

Difference schemes for, boundary problems, 228, 236 conjugate problems, 238-239,

262-264, 275-276, 295-297

339

Stefan problems, 259-261, 294-295

Euler equation, 8 Exact solution of inverse problem,

boundary problems, 221 with free boundaries, 253, 254, 272

coefficient problems, 234 with free boundaries, 285

Fredholm integral equation, 7, 9, 10, 1L 141, 142-142, 202, 210

Inequality of moments, 53 Inverse problems,

control, 251 design, 251 restoration, 251

Iterative process, implicit, 18-19 linear, 41, 42 multistep, 17-18 nonlinear, 19-21, 43 i-process, 44

Lipschitz condition,142

Melting problem with laser sources, 309-315

Methods for solving operator equations, conjugate direction, 103 conjugate gradient, 106, 107

s-parametric, 117 conjugate gradient projection, 172 conjugate residual, 105, 107 minimal errors, 43 minimal residual norm, 47 simple iteration, 42, 44 steepest descent, 42, 47, 53, 60, 61,

68,69

Nonlinear programming problem, 223, 236, 257, 290, 293, 318

340

Numerical experiments, boundary inverse problems, 228-233 coefficient inverse problems, 240-248 Fredholm integral equation,

206-209, 212-219 multi phase inverse problems,

297-302 two-phase inverse problems,

266-272, 276-284

Operator Frechet differentiable, 142,

143-144, 147 normally solvable, 33-36 pseudo inverse, 33-34 self-adjoint, 14, 26, 32, 46, 49, 58, 82,

98, 104, 114, 126 Operator representation of inverse

problems, boundary problems, 221

with free boundaries, 252-253, 254, 255, 272

coefficient problems, 233 with free boundaries, 285, 287

Parametric family, for exact input data, 13-15, 17, 23 for approximate input data, 27

Projection, 8, 172, 173, 184, 192 onto sets of shape constraints, 203

Pseudo-solution, 8 normal, 8, 9

Regularization parameter, 12 Regularizing algorithm, 12 Regularizing family, 12 Regularizing method,

optimal on the set, 13 optimal under the order, 13 Tikhonov, 16, 39-40 two-step scheme, 39 stabilization, 16

Residual principle (criterion), 85, 86, 88, 93, 128

Shape constraints, 5, 201, 202, 210, 221, 234, 256, 290, 317

Sourcewise representation, 54

INDEX

error estimate, 55, 57, 59, 60, 76, 79, 80, 83, 88, 90, 93, 109, 113, 120-121, 123, 125, 12~ 130, 132

generalization for nonlinear equations, 165

Stopping index,53

Trinomial relations, 103

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