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BEGIN:VEVENT
SUMMARY:Richard Nickl (University of Cambridge)
DTSTART:20211101T143000Z
DTEND:20211101T151000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/1
DESCRIPTION:Title: Bayesian non-linear inversion: progress and challenges\nby
Richard Nickl (University of Cambridge) as part of BIRS workshop: Statist
ical Aspects of Non-Linear Inverse Problems\n\n\nAbstract\nSolving non-lin
ear inverse problems in a modern `data-science’ framework requires a sta
tistical formulation of the measurement and error process. Since seminal w
ork of Andrew Stuart (2010)\, the Bayesian approach has become a popular c
omputational and inferential tool in this context\, and more recently also
a theoretical understanding of the performance of these methods has been
developed. We review recent mathematical progress in this field and formul
ate analytical properties that may render inverse problems provably `solva
ble’ by Bayesian algorithms. This leads on to many open problems both in
the area of PDEs and inverse problems and in Bayesian nonparametric stati
stics\, and we will describe some of those if time permits.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikko Salo (University of Jyväskylä)
DTSTART:20211101T152000Z
DTEND:20211101T160000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/2
DESCRIPTION:Title: Instability mechanisms in inverse problems\nby Mikko Salo
(University of Jyväskylä) as part of BIRS workshop: Statistical Aspects
of Non-Linear Inverse Problems\n\n\nAbstract\nMany inverse and imaging pro
blems\, such as image deblurring or electrical/optical tomography\, are kn
own to be highly sensitive to noise. In these problems small errors in mea
surements may lead to large errors in reconstructions. Such problems are c
alled ill-posed or unstable\, as opposed to being well-posed (a notion int
roduced by J. Hadamard in 1902). Instability also affects the performance
of statistical algorithms for solving inverse problems. \n\nThe inherent r
eason for instability is easy to understand in linear inverse problems lik
e image deblurring. For more complicated nonlinear imaging problems the in
stability issue is more delicate. We will discuss a general framework for
understanding instability in inverse problems based on smoothing/compressi
on properties of the forward map together with estimates for entropy and c
apacity numbers in relevant function spaces. The methods apply to various
inverse problems involving general geometries and low regularity coefficie
nts.\n\nThis talk is based on joint work with Herbert Koch (Bonn) and Angk
ana Rüland (Heidelberg).\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bamdad Hosseini (California Institute of Technology)
DTSTART:20211101T163000Z
DTEND:20211101T171000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/3
DESCRIPTION:Title: Solving and Learning Nonlinear PDEs with Gaussian Processes\nby Bamdad Hosseini (California Institute of Technology) as part of BIRS
workshop: Statistical Aspects of Non-Linear Inverse Problems\n\n\nAbstrac
t\nIn this talk I present a simple\, rigorous\, and interpretable framewor
k for solution of nonlinear PDEs based on the framework of Gaussian Proces
ses. The proposed approach provides a natural generalization of kernel met
hods to nonlinear PDEs\; has guaranteed convergence\; and inherits the sta
te-of-the-art computational complexity of linear solvers for dense kernel
matrices. I will outline our approach by focusing on an example nonlinear
elliptic PDE followed by further numerical examples. \nI will also briefly
comment on extending our approach to solving inverse problems.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christoph Schwab (ETHZ)
DTSTART:20211102T143000Z
DTEND:20211102T151000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/4
DESCRIPTION:Title: Deterministic Algorithms for PDEs with GRF inputs\nby Chri
stoph Schwab (ETHZ) as part of BIRS workshop: Statistical Aspects of Non-L
inear Inverse Problems\n\n\nAbstract\nJoint work with \nDinh Dung and N. V
an Kien\, Hanoi\, Vietnam.\nJakob Zech\, IWR\, Heidelberg\, Germany.\n\nWe
consider PDEs with (log-)Gaussian random field (GRF for short) inputs. \n
Parseval frames convert GRF inputs into equivalent\, infinite-parametric d
eterministic PDEs.\nWe analyze sparsity of Wiener Polynomial Chaos expansi
ons of the parametric\, deterministic \nforward solution families and of B
ayesian Inverse Problems with GRF priors.\n\nWe present approximation rate
bounds for novel\, deterministic sampling and quadrature algorithms.\nWe
cover high order discretizations of PDEs in the physical (space-time) doma
in\, and are free from the CoD. Achieveable convergence rates are superior
to those afforded by MC and QMC sampling.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barbara Kaltenbacher (University of Klagenfurt Austria)
DTSTART:20211102T152000Z
DTEND:20211102T160000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/5
DESCRIPTION:Title: Reduced\, all-at-once\, and variational formulations of invers
e problems and their solution\nby Barbara Kaltenbacher (University of
Klagenfurt Austria) as part of BIRS workshop: Statistical Aspects of Non-L
inear Inverse Problems\n\n\nAbstract\nThe conventional way of formulating
inverse problems such as identification of a (possibly infinite dimensiona
l) parameter\, is via some forward operator\, which is the concatenation o
f the observation operator with the parameter-to-state-map for the underly
ing model.\nRecently\, all-at-once formulations have been considered as an
alternative to this reduced formulation\, avoiding the use of a parameter
-to-state map\, which would sometimes lead to too restrictive conditions.
Here the model and the observation are considered simultaneously as one la
rge system with the state and the parameter as unknowns.\nA still more gen
eral formulation of inverse problems\, containing both the reduced and the
all-at-once formulation\, but also the well-known and highly versatile so
-called variational approach (not to be mistaken with variational regulari
zation) as special cases\, is to formulate the inverse problem as a minimi
zation problem (instead of an equation) for the state and parameter. Regul
arization can be incorporated via imposing constraints and/or adding regul
arization terms to the objective. \nIn this talk we will provide some new
application examples of minimization based formulations\, such as impedan
ce tomography with the complete electrode model. Moreover\, we will consid
er iterative regularization methods resulting from the application of grad
ient or Newton type iterations to such minimization based formulations and
provide convergence results.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Youssef Marzouk (Massachusetts Institute of Technology)
DTSTART:20211102T163000Z
DTEND:20211102T171000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/6
DESCRIPTION:by Youssef Marzouk (Massachusetts Institute of Technology) as
part of BIRS workshop: Statistical Aspects of Non-Linear Inverse Problems\
n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Scheichl (Heidelberg University)
DTSTART:20211103T143000Z
DTEND:20211103T151000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/7
DESCRIPTION:Title: Efficient Sample-Based Inference Algorithms in High Dimensions
\nby Robert Scheichl (Heidelberg University) as part of BIRS workshop:
Statistical Aspects of Non-Linear Inverse Problems\n\n\nAbstract\nGeneral
multivariate distributions are notoriously difficult to sample from\, par
ticularly the high-dimensional posterior distributions in PDE-constrained
inverse problems. In this talk\, I present a sampler for arbitrary continu
ous multivariate distributions based on low-rank surrogates in the tensor-
train (TT) format\, a methodology for scalable\, high-dimensional function
approximation from computational physics and chemistry. Building upon cro
ss approximation algorithms in linear algebra\, we construct a TT approxim
ation to the target probability density function using only a small number
of function evaluations. For sufficiently smooth distributions\, the stor
age required for accurate TT approximations is moderate\, scaling linearly
with dimension. In turn\, the structure of the tensor-train surrogate all
ows sampling by an efficient conditional distribution method\, since margi
nal distributions are computable with linear complexity in dimension. I wi
ll also highlight the link to normalizing flows in machine learning and to
transport-based variational inference algorithms for high-dimensional dis
tributions. Finally\, I will mention extensions suitable for more strongly
concentrating posterior distributions using a multi-layered approach: the
Deep Inverse Rosenblatt Transport (DIRT) algorithm proposed by Cui and Do
lgov in a recent preprint. This talk is based on joint work with Karim-Ana
ya Izquierdo (Bath)\, Tiangang Cui (Monash)\, Sergey Dolgov (Bath) and Col
in Fox (Otago).\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Judith Rousseau (Oxford University)
DTSTART:20211103T152000Z
DTEND:20211103T160000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/8
DESCRIPTION:by Judith Rousseau (Oxford University) as part of BIRS worksho
p: Statistical Aspects of Non-Linear Inverse Problems\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Plamen Stefanov (Purdue)
DTSTART:20211103T163000Z
DTEND:20211103T171000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/9
DESCRIPTION:by Plamen Stefanov (Purdue) as part of BIRS workshop: Statisti
cal Aspects of Non-Linear Inverse Problems\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samuli Siltanen (University of Helsinki)
DTSTART:20211104T143000Z
DTEND:20211104T151000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/10
DESCRIPTION:by Samuli Siltanen (University of Helsinki) as part of BIRS wo
rkshop: Statistical Aspects of Non-Linear Inverse Problems\n\nAbstract: TB
A\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giovanni S. Alberti (University of Genoa)
DTSTART:20211104T152000Z
DTEND:20211104T160000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/11
DESCRIPTION:Title: Infinite-dimensional inverse problems with finite measurement
s\nby Giovanni S. Alberti (University of Genoa) as part of BIRS worksh
op: Statistical Aspects of Non-Linear Inverse Problems\n\n\nAbstract\nIn t
his talk I will discuss uniqueness\, stability and reconstruction for infi
nite-dimensional nonlinear inverse problems with finite measurements\, und
er the a priori assumption that the unknown lies in\, or is well-approxima
ted by\, a finite-dimensional subspace or submanifold. The methods are bas
ed on the interplay of applied harmonic analysis\, in particular sampling
theory and compressed sensing\, and the theory of inverse problems for par
tial differential equations. Several examples\, including the Calderón p
roblem and scattering\, will be discussed.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Glatt-Holtz (Tulane University)
DTSTART:20211104T163000Z
DTEND:20211104T171000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/12
DESCRIPTION:by Nathan Glatt-Holtz (Tulane University) as part of BIRS work
shop: Statistical Aspects of Non-Linear Inverse Problems\n\nAbstract: TBA\
n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Bohr (University of Cambridge)
DTSTART:20211105T143000Z
DTEND:20211105T151000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/13
DESCRIPTION:by Jan Bohr (University of Cambridge) as part of BIRS workshop
: Statistical Aspects of Non-Linear Inverse Problems\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hanne Kekkonen (Delft University of Technology)
DTSTART:20211105T152000Z
DTEND:20211105T160000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/14
DESCRIPTION:Title: Consistency of Bayesian inference for a parabolic inverse pro
blem\nby Hanne Kekkonen (Delft University of Technology) as part of BI
RS workshop: Statistical Aspects of Non-Linear Inverse Problems\n\n\nAbstr
act\nIn this talk I will discuss uniqueness\, stability and reconstruction
for infinite-dimensional nonlinear inverse problems with finite measureme
nts\, under the a priori assumption that the unknown lies in\, or is well-
approximated by\, a finite-dimensional subspace or submanifold. The method
s are based on the interplay of applied harmonic analysis\, in particular
sampling theory and compressed sensing\, and the theory of inverse problem
s for partial differential equations. Several examples\, including the Cal
derón problem and scattering\, will be discussed.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sven Wang (MIT)
DTSTART:20211105T163000Z
DTEND:20211105T171000Z
DTSTAMP:20260404T042016Z
UID:BIRS-21w5009/15
DESCRIPTION:by Sven Wang (MIT) as part of BIRS workshop: Statistical Aspec
ts of Non-Linear Inverse Problems\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5009/15/
END:VEVENT
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