arXiv subjects

Fredrik Hildrum

Publications and source records attributed to Fredrik Hildrum.

3 recordsLinked to original sources

arXiv2024

Subgradient-based Lavrentiev regularisation of monotone ill-posed problems

We introduce subgradient-based Lavrentiev regularisation of the form \begin{equation*} \mathcal{A}(u) + α\partial \mathcal{R}(u) \ni f^δ\end{equation*} for linear and nonlinear ill-posed problems with monotone operators $\mathcal{A}$ and general regularisation functionals $\mathcal{R}$. In contrast to Tikhonov regularisation, this approach perturbs the equation itself and avoids the use of the adjoint of the derivative of $\mathcal{A}$. It is therefore especially suitable for time-causal problems that only depend on information in the past and allows for real-time computation of regularised solutions. We establish a general well-posedness theory in Banach spaces and prove convergence-rate results with variational source conditions. Furthermore, we demonstrate its application in total-variation denoising in linear Volterra integral operators of the first kind and parameter-identification problems in semilinear parabolic PDEs.

math.OC↗

arXiv2022

Periodic Hölder waves in a class of negative-order dispersive equations

We prove the existence of highest, cusped, periodic travelling-wave solutions with exact and optimal $ α$-Hölder continuity in a class of fractional negative-order dispersive equations of the form \begin{equation*} u_t + (| \mathrm{D} |^{- α} u + n(u) )_x = 0 \end{equation*} for every $ α\in (0, 1) $ with homogeneous Fourier multiplier $ | \mathrm{D} |^{ - α} $. We tackle nonlinearities $ n(u) $ of the type $ | u |^p $ or $ u | u |^{p - 1} $ for all real $ p > 1 $, and show that when $ n $ is odd, the waves also feature antisymmetry and thus contain inverted cusps. Tools involve detailed pointwise estimates in tandem with analytic global bifurcation, where we resolve the issue with nonsmooth $ n $ by means of regularisation. We believe that both the construction of highest antisymmetric waves and the regularisation of nonsmooth terms to an analytic bifurcation setting are new in this context, with direct applicability also to generalised versions of the Whitham, the Burgers--Poisson, the Burgers--Hilbert, the Degasperis--Procesi, the reduced Ostrovsky, and the bidirectional Whitham equations.

math.AP↗

arXiv2020

Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity

We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces ${\mathrm{H}^s}$, ${ s > 0 }$, to a class of nonlinear, dispersive evolution equations of the form \begin{equation*} u_t + \left(Lu+ n(u)\right)_x = 0, \end{equation*} where the dispersion ${L}$ is a negative-order Fourier multiplier whose symbol is of KdV type at low frequencies and has integrable Fourier inverse ${ K }$ and the nonlinearity ${n}$ is inhomogeneous, locally Lipschitz and of superlinear growth at the origin. This generalises earlier work by Ehrnström, Groves & Wahlén on a class of equations which includes Whitham's model equation for surface gravity water waves featuring the exact linear dispersion relation. Tools involve constrained variational methods, Lions' concentration-compactness principle, a strong fractional chain rule for composition operators of low relative regularity, and a cut-off argument for ${n}$ which enables us to go below the typical ${s > \frac{1}{2}}$ regime. We also demonstrate that these solutions are either waves of elevation or waves of depression when ${ K }$ is nonnegative, and provide a nonexistence result when ${ n }$ is too strong.

math.AP↗