Error Estimates for Doubly-Generalized Tikhonov-Phillips Regularization

In this work, error estimates are presented for the case in which the regularized solution is obtained by minimizing doubly-generalized Tikhonov-Phillips functionals. The first result is based mainly on an assumption given by a source condition. It is proved that it is possible to replace this assumption by a variational inequality, obtaining analogous result of the error estimate. Finally, relationships are established between the optimality condition associated with the problem, the source condition and the variational inequality. On the other hand, it is known that, in certain cases, the use of two or more penalizing terms is useful. For this reason, generalizations of the results of error estimates are presented for cases in which the regularized solution is a minimizer of doubly-generalized Tikhonov-Phillips functionals with multiple penalizers.

Keywords:
inverse problems; Tikhonov-Phillips; error estimate; source condition; variational inequality

Authorship

M. J. CARRIÓ ^**Corresponding author: María Josefina Carrió - E-mail: mjcarrio@fbcb.unl.edu.ar

Departamento de Matemática, Facultad de Bioquímica y Ciencias Biológicas, Universidad Nacional del Litoral, Ciudad Universitaria - Paraje El Pozo, C.P. 3000, Santa Fe, Argentina - E-mail: mjcarrio@fbcb.unl.edu.arUniversidad Nacional del LitoralArgentinaSanta Fe, ArgentinaDepartamento de Matemática, Facultad de Bioquímica y Ciencias Biológicas, Universidad Nacional del Litoral, Ciudad Universitaria - Paraje El Pozo, C.P. 3000, Santa Fe, Argentina - E-mail: mjcarrio@fbcb.unl.edu.ar

https://orcid.org/0000-0002-8676-4135

G. L. MAZZIERI

Instituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Centro Científico Tecnológico CONICET-Santa Fe, Colectora Ruta Nac. 168, km. 472, Paraje “El Pozo”, C.P. 3000, Santa Fe, Argentina. CONICET-UNL, Centro Científico TecnológicoArgentinaSanta Fe, ArgentinaInstituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Centro Científico Tecnológico CONICET-Santa Fe, Colectora Ruta Nac. 168, km. 472, Paraje “El Pozo”, C.P. 3000, Santa Fe, Argentina.

Departamento de Matemática, Facultad de Bioquímica y Ciencias Biológicas, Universidad Nacional del Litoral, Ciudad Universitaria - Paraje El Pozo, C.P. 3000, Santa Fe, Argentina. - E-mail: glmazzieri@santafe-conicet.gov.arUniversidad Nacional del LitoralArgentinaSanta Fe, ArgentinaDepartamento de Matemática, Facultad de Bioquímica y Ciencias Biológicas, Universidad Nacional del Litoral, Ciudad Universitaria - Paraje El Pozo, C.P. 3000, Santa Fe, Argentina. - E-mail: glmazzieri@santafe-conicet.gov.ar

https://orcid.org/0000-0002-5368-9601

K. G. TEMPERINI

Departamento de Matemática, Facultad de Humanidades y Ciencias, Universidad Nacional del Litoral, Ciudad Universitaria - Paraje El Pozo, C.P. 3000, Santa Fe, Argentina - E-mail: ktemperini@santafe-conicet.gov.arUniversidad Nacional del LitoralArgentinaSanta Fe, ArgentinaDepartamento de Matemática, Facultad de Humanidades y Ciencias, Universidad Nacional del Litoral, Ciudad Universitaria - Paraje El Pozo, C.P. 3000, Santa Fe, Argentina - E-mail: ktemperini@santafe-conicet.gov.ar

https://orcid.org/0000-0002-0549-7453

*Corresponding author: María Josefina Carrió - E-mail: mjcarrio@fbcb.unl.edu.ar

SCIMAGO INSTITUTIONS RANKINGS

Departamento de Matemática, Facultad de Bioquímica y Ciencias Biológicas, Universidad Nacional del Litoral, Ciudad Universitaria - Paraje El Pozo, C.P. 3000, Santa Fe, Argentina. - E-mail: glmazzieri@santafe-conicet.gov.arUniversidad Nacional del LitoralArgentinaSanta Fe, ArgentinaDepartamento de Matemática, Facultad de Bioquímica y Ciencias Biológicas, Universidad Nacional del Litoral, Ciudad Universitaria - Paraje El Pozo, C.P. 3000, Santa Fe, Argentina. - E-mail: glmazzieri@santafe-conicet.gov.ar

Formulas

Formulas (51)

(1.1)

(1.2)

(1.3)

(1.4)

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(1.1)

T x = y,

(1.2)

J_{\tilde{ϕ}, ψ, η}^{p} (x) ≐ \frac{1}{p} \tilde{ϕ} (T x, y)^{p} + η ψ (x),

(1.3)

J_{ψ, η}^{p} (x) ≐ \frac{1}{p} ‖ T x - y ‖^{p} + η ψ (x),

(1.4)

J_{\tilde{ϕ}, \vec{ψ}, \vec{η}}^{p} (x) ≐ \frac{1}{p} \tilde{ϕ} (T x, y)^{p} + \vec{η} \cdot \vec{ψ} (x),

d_{ξ}^{ψ} (z, x) ≐ ψ (z) - ψ (x) - [ξ, z - x], \forall z \in X .

J (x) = \{x^{*} \in X^{*} : [x^{*}, x] = ‖ x^{*} ‖ ‖ x ‖, ‖ x^{*} ‖ = f (‖ x ‖)\} .

J (x) = \partial F (‖ x ‖), x \in X .

d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq \frac{\frac{δ^{p}}{p} + η δ ‖ w ‖ C + \frac{1}{q} (η ‖ w ‖ C)^{q}}{η};

d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq δ \frac{1 + η ‖ w ‖ C}{η} .

\frac{1}{p} \tilde{ϕ} (T x_{η}^{δ}, y^{δ})^{p} + η ψ (x_{η}^{δ}) \leq \frac{1}{p} \tilde{ϕ} (T \tilde{x}, y^{δ})^{p} + η ψ (\tilde{x}),

(3.1)

\frac{1}{p} \tilde{ϕ} (T x_{η}^{δ}, y^{δ})^{p} + η d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq \frac{1}{p} \tilde{ϕ} (T \tilde{x}, y^{δ})^{p} + η (ψ (\tilde{x}) - ψ (x_{η}^{δ}) + d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x})) = \frac{1}{p} \tilde{ϕ} (T \tilde{x}, y^{δ})^{p} + η [T^{#} w, \tilde{x} - x_{η}^{δ}] = \frac{1}{p} \tilde{ϕ} (T \tilde{x}, y^{δ})^{p} + η [w, T \tilde{x} - T x_{η}^{δ}],

\frac{1}{p} \tilde{ϕ} (T x_{η}^{δ}, y^{δ})^{p} + η d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq \frac{δ^{p}}{p} + η [w, T \tilde{x} - T x_{η}^{δ}] by (3.1) and T \tilde{x} = y \leq \frac{δ^{p}}{p} + η ‖ w ‖ ‖ T \tilde{x} - T x_{η}^{δ} ‖ \leq \frac{δ^{p}}{p} + η ‖ w ‖ C \tilde{ϕ} (T x_{η}^{δ}, y^{δ}) + η ‖ w ‖ C \tilde{ϕ} (y^{δ}, y) by hypotheses 1, 2 and T \tilde{x} = y \leq \frac{δ^{p}}{p} + η ‖ w ‖ C \tilde{ϕ} (T x_{η}^{δ}, y^{δ}) + η ‖ w ‖ C δ .

(3.2)

\frac{1}{p} \tilde{ϕ} (T x_{η}^{δ}, y^{δ})^{p} - η ‖ w ‖ C \tilde{ϕ} (T x_{η}^{δ}, y^{δ}) + η d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq \frac{δ^{p}}{p} + η ‖ w ‖ C δ .

η ‖ w ‖ C \tilde{ϕ} (T x_{η}^{δ}, y^{δ}) \leq \frac{1}{p} \tilde{ϕ} (T x_{η}^{δ}, y^{δ})^{p} + \frac{1}{q} (η ‖ w ‖ C)^{q},

η d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) - \frac{1}{q} (η ‖ w ‖ C)^{q} \leq \frac{1}{p} \tilde{ϕ} (T x_{η}^{δ}, y^{δ})^{p} - η ‖ w ‖ C \tilde{ϕ} (T x_{η}^{δ}, y^{δ}) + η d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq \frac{δ^{p}}{p} + η ‖ w ‖ C δ,

d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq \frac{\frac{δ^{p}}{p} + η δ ‖ w ‖ C + \frac{1}{q} (η ‖ w ‖ C)^{q}}{η} .

d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq δ \frac{1 + η ‖ w ‖ C}{η} .

d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq \frac{\frac{δ^{2}}{2} + η δ ‖ w ‖ + \frac{1}{2} (η ‖ w ‖)^{2}}{η} = \frac{1}{2} {(\frac{δ}{\sqrt{η}} + \sqrt{η} ‖ w ‖)}^{2},

(3.3)

d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq \frac{δ^{p}}{p η} + ‖ w ‖ C δ - \frac{1}{p} \frac{\tilde{ϕ} (T x_{η}^{δ}, y^{δ})^{p}}{η} + ‖ w ‖ C \tilde{ϕ} (T x_{η}^{δ}, y^{δ}) .

0 \in \partial (\frac{1}{2} ‖ T \cdot - y ‖^{2} + η \frac{1}{2} ‖ \cdot ‖^{2}) (x_{η}) .

\partial (\frac{1}{2} ‖ T \cdot - y ‖^{2} + η \frac{1}{2} ‖ \cdot ‖^{2}) (x_{η}) = \partial (\frac{1}{2} ‖ T \cdot - y ‖^{2}) (x_{η}) + η \partial (\frac{1}{2} ‖ \cdot ‖^{2}) (x_{η}) = T^{*} \partial (\frac{1}{2} ‖ \cdot - y ‖^{2}) (T x_{η}) + η \partial (\frac{1}{2} ‖ \cdot ‖^{2}) (x_{η}) = T^{*} \partial (\frac{1}{2} ‖ \cdot ‖^{2}) (T x_{η} - y) + η \partial (\frac{1}{2} ‖ \cdot ‖^{2}) (x_{η}) .

x_{η} = \frac{- T^{*} (T x_{η} - y)}{η} .

(3.4)

f o r e a c h η > 0, t h e r e e x i s t s w_{η} \in Y^{*} s u c h t h a t \partial (- \frac{1}{η} \frac{\tilde{ϕ} (T \cdot, y)^{p}}{p}) (x_{η}) = {w_{η}} .

0 \in T^{#} \partial (\frac{1}{p} \tilde{ϕ} (T \cdot, y)^{p}) (x_{η}) + η \partial ψ (x_{η}),

f_{η} \in T^{#} \partial (\frac{1}{p} \tilde{ϕ} (T \cdot, y)^{p}) (x_{η}) and g_{η} \in η \partial ψ (x_{η}) .

(3.5)

there exists C > 0 such that ‖ y_{1} - y_{2} ‖ \leq C \tilde{ϕ} (y_{2}, y_{1}), \forall y_{1}, y_{2} \in R (T) .

(3.6)

[ξ, \tilde{x} - x] \leq β \tilde{ϕ} (T x, T \tilde{x}), \forall x \in X .

[ξ, \tilde{x} - x] = [T^{#} w, \tilde{x} - x] = [w, T \tilde{x} - T x] \leq ‖ w ‖ ‖ T \tilde{x} - T x ‖ \leq ‖ w ‖ C \tilde{ϕ} (T x, T \tilde{x}),

(3.7)

there exists c > 0 such that c \tilde{ϕ} (y_{1}, y_{2}) \leq ‖ y_{1} - y_{2} ‖, \forall y_{1}, y_{2} \in R (T) .

[ξ, x] = [ξ, \tilde{x} - (\tilde{x} - x)] \leq β \tilde{ϕ} (T (\tilde{x} - x), T \tilde{x}) \leq β \frac{1}{c} ‖ T x ‖,

d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq \frac{\frac{δ^{p}}{p} + η δ β + \frac{1}{q} (η β)^{q}}{η};

d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq δ \frac{1 + η β}{η} .

\frac{1}{p} \tilde{ϕ} (T x_{η}^{δ}, y^{δ})^{p} \leq \frac{δ^{p}}{p} + η (ψ (\tilde{x}) - ψ (x_{η}^{δ})) .

\frac{1}{p} \tilde{ϕ} (T x_{η}^{δ}, y^{δ})^{p} + η d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq \frac{δ^{p}}{p} + η (ψ (\tilde{x}) - ψ (x_{η}^{δ}) + d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x})) = \frac{δ^{p}}{p} + η [ξ, \tilde{x} - x_{η}^{δ}] \leq \frac{δ^{p}}{p} + η β \tilde{ϕ} (T x_{η}^{δ}, y^{δ}) + η β \tilde{ϕ} (y^{δ}, y) by hypotheses 1, 2 and T \tilde{x} = y \leq \frac{δ^{p}}{p} + η β \tilde{ϕ} (T x_{η}^{δ}, y^{δ}) + η β δ . since \tilde{ϕ} (y^{δ}, y) \leq δ

\frac{1}{p} \tilde{ϕ} (T x_{η}^{δ}, y^{δ})^{p} - η β \tilde{ϕ} (T x_{η}^{δ}, y^{δ}) + η d_{ξ}^{ψ} (x_{η}^{δ}, \tilde{x}) \leq \frac{δ^{p}}{p} + η β δ,

d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) \leq \frac{\frac{1}{p} δ^{p} + ‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C δ + \frac{1}{q} {(‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C)}^{q}}{‖ \vec{η} ‖_{1}} .

d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) \leq δ \frac{1 + ‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C}{‖ \vec{η} ‖_{1}} .

\frac{1}{p} \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ})^{p} + \vec{η} \cdot \vec{ψ} (x_{\vec{η}}^{δ}) \leq \frac{1}{p} \tilde{ϕ} (T \tilde{x}, y^{δ})^{p} + \vec{η} \cdot \vec{ψ} (\tilde{x}) \leq \frac{1}{p} δ^{p} + \vec{η} \cdot \vec{ψ} (\tilde{x}),

(3.8)

\frac{1}{p} \frac{\tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ})^{p}}{‖ \vec{η} ‖_{1}} + {\vec{η}}_{*} \cdot \vec{ψ} (x_{\vec{η}}^{δ}) \leq \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + {\vec{η}}_{*} \cdot \vec{ψ} (\tilde{x}) \frac{1}{p} \frac{\tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ})^{p}}{‖ \vec{η} ‖_{1}} + ψ_{{\vec{η}}_{*}} (x_{\vec{η}}^{δ}) \leq \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + ψ_{{\vec{η}}_{*}} (\tilde{x}) \frac{1}{p} \frac{\tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ})^{p}}{‖ \vec{η} ‖_{1}} \leq \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + ψ_{{\vec{η}}_{*}} (\tilde{x}) - ψ_{{\vec{η}}_{*}} (x_{\vec{η}}^{δ}) .

\frac{1}{p} \frac{\tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ})^{p}}{‖ \vec{η} ‖_{1}} + d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) \leq \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + ψ_{{\vec{η}}_{*}} (\tilde{x}) - ψ_{{\vec{η}}_{*}} (x_{\vec{η}}^{δ}) + d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) = \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + [T^{#} w_{{\vec{η}}_{*}}, \tilde{x} - x_{\vec{η}}^{δ}] \leq \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + ‖ w_{{\vec{η}}_{*}} ‖ C \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ}) + ‖ w_{{\vec{η}}_{*}} ‖ C \tilde{ϕ} (y^{δ}, T \tilde{x}) by hypotheses 1 and 2 \leq \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + ‖ w_{{\vec{η}}_{*}} ‖ C \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ}) + ‖ w_{{\vec{η}}_{*}} ‖ C δ . since \tilde{ϕ} (y^{δ}, y) \leq δ and T \tilde{x} = y

(3.9)

\frac{1}{p} \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ})^{p} - ‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ}) + ‖ \vec{η} ‖_{1} d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) \leq \frac{1}{p} δ^{p} + ‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C δ .

‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ}) \leq \frac{1}{p} \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ})^{p} + \frac{1}{q} {(‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C)}^{q},

- \frac{1}{q} {(‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C)}^{q} + ‖ \vec{η} ‖_{1} d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) \leq \frac{1}{p} δ^{p} + ‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C δ .

d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) \leq \frac{\frac{1}{p} δ^{p} + ‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C δ + \frac{1}{q} {(‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C)}^{q}}{‖ \vec{η} ‖_{1}} .

(1 - ‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C) \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ}) + ‖ \vec{η} ‖_{1} d_{ξ_{{\vec{η}}_{*}}}^{ψ} (x_{\vec{η}}^{δ}, \tilde{x}) \leq δ + ‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C δ,

d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) \leq δ \frac{1 + ‖ \vec{η} ‖_{1} ‖ w_{{\vec{η}}_{*}} ‖ C}{‖ η ‖_{1}} .

d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) \leq \frac{\frac{1}{p} δ^{p} + ‖ \vec{η} ‖_{1} β δ + \frac{1}{q} {(‖ \vec{η} ‖_{1} β)}^{q}}{‖ \vec{η} ‖_{1}} .

d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) \leq δ \frac{1 + ‖ \vec{η} ‖_{1} β}{‖ \vec{η} ‖_{1}} .

\frac{1}{p} \frac{\tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ})^{p}}{‖ \vec{η} ‖_{1}} \leq \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + ψ_{{\vec{η}}_{*}} (\tilde{x}) - ψ_{{\vec{η}}_{*}} (x_{\vec{η}}^{δ}) .

\frac{1}{p} \frac{\tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ})^{p}}{‖ \vec{η} ‖_{1}} + d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) \leq \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + ψ_{{\vec{η}}_{*}} (\tilde{x}) - ψ_{{\vec{η}}_{*}} (x_{\vec{η}}^{δ}) + d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) = \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + [ξ_{{\vec{η}}_{*}}, \tilde{x} - x_{\vec{η}}^{δ}] \leq \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + β \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ}) + β \tilde{ϕ} (y^{δ}, T \tilde{x}) by hypotheses 1 and 2 \leq \frac{1}{p} \frac{δ^{p}}{‖ \vec{η} ‖_{1}} + β \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ}) + β δ since \tilde{ϕ} (y^{δ}, y) \leq δ and T \tilde{x} = y,

\frac{1}{p} \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ})^{p} - ‖ \vec{η} ‖_{1} β \tilde{ϕ} (T x_{\vec{η}}^{δ}, y^{δ}) + ‖ \vec{η} ‖_{1} d_{ξ_{{\vec{η}}_{*}}}^{ψ_{{\vec{η}}_{*}}} (x_{\vec{η}}^{δ}, \tilde{x}) \leq \frac{1}{p} δ^{p} + ‖ \vec{η} ‖_{1} β δ,

How to cite

copy

Sociedade Brasileira de Matemática Aplicada e Computacional - SBMAC Rua Maestro João Seppe, nº. 900, 16º. andar - Sala 163, Cep: 13561-120 - SP / São Carlos - Brasil, +55 (16) 3412-9752 - São Carlos - SP - Brazil
E-mail: sbmac@sbmac.org.br

Acompanhe os números deste periódico no seu leitor de RSS

Versão para download de PDF

PDF

Inglês

Versões e tradução automática

Versão original do texto

English

Tradução automática

Google Translator
Microsoft Translator

Como citar

RIS

BIBTEX

Outros formatos de citação e exportação:

SciELO - Scientific Electronic Library Online
Rua Dr. Diogo de Faria, 1087 – 9º andar – Vila Clementino 04037-003 São Paulo/SP - Brasil
E-mail: scielo@scielo.org

Leia a Declaração de Acesso Aberto

Métricas

Error Estimates for Doubly-Generalized Tikhonov-Phillips Regularization

PlumX

Mensagem

[1] *Corresponding author: María Josefina Carrió - E-mail: mjcarrio@fbcb.unl.edu.ar