Samarskii A.A., Vabishchevich P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics
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436 Bibliography
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Index
A
algorithm
for solving evolutionary problems
global, 176
local, 176
Thomas
for five-diagonal matrix, 165
B
boundary conditions
first kind, 2
third kind, 2
C
canonical form
three-layer iteration method, 65
two-layer iteration method, 61
choice of regularization parameter
based on the difference between
the exact and approximate solu-
tion, 136
optimal, 135
quasi-optimal, 137
coefficient stability, 7
D
difference derivative
central, 21
left, 21
right, 21
difference scheme
ρ-stability, 96
stability, 95
with respect to initial data, 95
two-layer, 93
conservative, 25
monotone, 37
stability
with respect to right-hand side,
96
three-layer, 93
canonical form, 102
two-layer
canonical form, 95
weighted, 93
direct sum of spaces, 104
domain
irregular, 52
regular, 52
E
equation
second-order elliptic, 1
second-order hyperbolic, 3
second-order ordinary differential,
2
second-order parabolic, 2
convection-diffusion, 35
with dominating convection, 36
with dominating diffusion, 36
Poisson, 1
F
Friedrichs inequality
difference, 28
multi-dimensional, 55
function
trial, 24
verifying, 24
functional
stabilizing, 128
discrepancy, 128
smoothing, 128
G
general discrepancy principle, 161
Green difference formula
first, 28
second, 28
grid
structured, 52
uniform, 20
unstructured, 52
Gronwall lemma
difference, 96
438 Index
I
inequality
Friedrichs, 16
L
lemma
Gronwall, 6
M
method
finite element, 23
iteration
variation, 64
balance, 25
decomposition, 54
finite-volume, 25
Gauss, 34
generalized inverse, 243
integro-interpolation, 25
iteration, 60
steepest descend, 64
Chebyshev, 63
conjugate gradient, 65
Jacobi, 66
minimum correction, 64
simple iteration, 62
stationary, 62
three-layer, 61
two-layer, 61
of fictitious domains, 53
regularization
Tikhonov, 128
simplified regularization, 132
sweep, 32
for five-diagonal matrix, 165
O
operator
factorized, 67
regularizing, 129
transition, 95
P
Peclet number, 35
mesh, 38
principle
maximum, 8
difference, 37
for parabolic equation, 17
problem
conditionally well-posed, 11
direct, 13
ill-posed, 10
inverse, 13
boundary value, 15
coefficient, 14
evolutionary, 16
retrospective, 16
well-posed, 4
R
reconditioner, 66
regularization parameter, 128
S
stability
with respect to initial data, 7
with respect to the right-hand side,
7
T
Thomas algorithm, 32