Error estimates for simplified Levenberg–Marquardt method for nonlinear ill-posed operator equations in Hilbert Spaces

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>F</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mi>y</m:mi> </m:mrow> </m:math> {F(u)=y} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>F</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mi mathvariant="script">𝒟</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>⊂</m:mo> <m:mi>𝖴</m:mi> <m:mo>→</m:mo> <m:mi>𝖸</m:mi> </m:mrow> </m:mrow> </m:math> {F:D(F)} is a nonlinear operator between Hilbert spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝖴</m:mi> </m:math> {U} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝖸</m:mi> </m:math> {Y} . The method is defined as follows: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mi>u</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>δ</m:mi> </m:msubsup> <m:mo>=</m:mo> <m:mrow> <m:msubsup> <m:mi>u</m:mi> <m:mi>n</m:mi> <m:mi>δ</m:mi> </m:msubsup> <m:mo>-</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:msubsup> <m:mi>T</m:mi> <m:mn>0</m:mn> <m:mo>∗</m:mo> </m:msubsup> <m:mo>⁢</m:mo> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi>α</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>I</m:mi> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>T</m:mi> <m:mn>0</m:mn> <m:mo>∗</m:mo> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mi>F</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>u</m:mi> <m:mi>n</m:mi> <m:mi>δ</m:mi> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:msup> <m:mi>y</m:mi> <m:mi>δ</m:mi> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> uₙ₊₁δ=uₙδ-(T₀^{}T₀+αₙI)⁻¹T₀%(F(uₙδ)-yδ), where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>F</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {T₀=F^{}(u₀)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>T</m:mi> <m:mn>0</m:mn> <m:mo>∗</m:mo> </m:msubsup> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>F</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>∗</m:mo> </m:msup> </m:mrow> </m:mrow> </m:math> {T₀^{}=F^{}(u₀)^{}} . Here <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>F</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {F^{}(u₀)} denotes the Frèchet derivative of F at an initial guess <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:mi mathvariant="script">𝒟</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {u₀(F)} for the exact solution <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>u</m:mi> <m:mo>†</m:mo> </m:msup> </m:math> {u^{}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>F</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>∗</m:mo> </m:msup> </m:mrow> </m:math> {F^{}(u₀)^{}} is the adjoint of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>F</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {F^{}(u₀)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>α</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> {\{αₙ\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F , we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>-</m:mo> <m:msup> <m:mi>u</m:mi> <m:mo>†</m:mo> </m:msup> </m:mrow> </m:math> {u₀-u^{}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions.