# Multifidelity Cross-validation

Ashwin Renganathan111Assistant professor, Aerospace Engineering and the
Institute of Computational and Data Science (ICDS), 229 Hammond Building; and
AIAA Member. The Pennsylvania State University, University Park, PA, 16802
Kade Carlson222Graduate research assistant, Aerospace Engineering, 229 Hammond
Building; and AIAA Student Member. The Pennsylvania State University,
University Park, PA, 16802

###### Abstract

Emulating the mapping between quantities of interest and their control
parameters using surrogate models finds widespread application in engineering
design, including in numerical optimization and uncertainty quantification.
Gaussian process models can serve as a probabilistic surrogate model of
unknown functions, thereby making them highly suitable for engineering design
and decision-making in the presence of uncertainty. In this work, we are
interested in emulating quantities of interest observed from models of a
system at multiple fidelities, which trade accuracy for computational
efficiency. Using multifidelity Gaussian process models, to efficiently fuse
models at multiple fidelities, we propose a novel method to actively learn the
surrogate model via leave-one-out cross-validation (LOO-CV). Our proposed
multifidelity cross-validation (MFCV) approach develops an adaptive approach
to reduce the LOO-CV error at the target (highest) fidelity, by learning the
correlations between the LOO-CV at all fidelities. MFCV develops a two-step
lookahead policy to select optimal input-fidelity pairs, both in sequence and
in batches, both for continuous and discrete fidelity spaces. We demonstrate
the utility of our method on several synthetic test problems as well as on the
thermal stress analysis of a gas turbine blade.

##  1 Introduction

Historically, the design, certification, and qualification of aircraft and
aircraft subsystems have depended heavily on wind tunnel and flight tests.
These are risky, expensive, prone to scheduling delays, and can depend on
uncontrollable weather conditions. Aircraft design is increasingly benefiting
from advances in computational structural and fluid dynamics, and high-
performance computing, to minimize the amount of physical experimentation and
testing necessary. On the certification front, flight and engine certification
by analysis (CbA) is a new paradigm in which airworthiness certificates for
aircraft can be obtained via calculations with quantifiable accuracy Mauery et
al. (2022) 333see §25.21 of Chapter 1, Title 14 in the electronic Code for
Regulations (CFR) of the Federal Aviation Administrations (FAA).. While this
opens up possibilities for significant reduction in the need for physical
experimentation in aircraft design and certification, replacing them with
computation, several challenges remain. First, computations of high physics
fidelity, while being cheaper than wind-tunnel/flight tests, still incur a
non-trivial cost. For instance, a “full fidelity” direct numerical Navier-
Stokes simulation (DNS) of an aircraft under realistic operating conditions is
still not practical owing to an intractable number of degrees of freedom in
the discretized partial differential equation (PDE) system. Large eddy
simulations (LES), while cheaper than DNS, still can cost in the order of a
few days of wall-clock time, for full aircraft configurations. Reynolds-
averaged Navier-Stokes (RANS) simulations cost several hours of wall-clock
time while being unreliable under extreme boundary conditions (e.g., high-
angles of attack and side-slip angles). Second, to truly exploit computations
for reliable decision-making, all significant forms of uncertainty must be
rigorously quantified. This necessitates several thousands of evaluations of
the high-fidelity model; even with cheaper RANS models, this can be
prohibitive. To alleviate these challenges, surrogate models that trade an
acceptable measure of accuracy for significantly large gains in computational
efficiency are necessary. Furthermore, surrogate models that can fuse
information from models at multiple levels of fidelity (e.g., DNS/LES/RANS)
are necessary. This forms the overarching goal of this work.

We exploit the availability of models with tunable fidelities and propose
multifidelity methods Takeno et al. (2019); Kandasamy et al. (2017); Klein et
al. (2017); Swersky et al. (2013) which can judiciously utilize the cost-
accuracy trade-off in models to adaptively construct accurate surrogate
models. Surrogate models for conservation laws can be efficiently constructed
via reduced-order models (ROMs) Renganathan et al. (2018); Renganathan (2020);
Renganathan et al. (2020); Renganathan (2018) for state variables and general
regression/interpolation techniques Renganathan et al. (2021a, b); Ashwin
Renganathan et al. (2022); Renganathan et al. (2023, 2021c) for scalar
quantities of interest (QoI). In this work, we are interested in adaptively
constructing a data-driven surrogate model for a scalar-valued QoI in order to
conservatively use the computational resources. For this purpose, the
computational cost of querying models at each fidelity level is built in to
the overall methodology. We will leverage a Gaussian process (GP) Rasmussen
and Williams (2006) regression model framework to learn the mapping between
the QoIs and the inputs.

The literature covering the use of GP regression for expensive
multidisciplinary design of aerospace systems is very broad. This includes
aerodynamic design optimization Renganathan et al. (2021b), uncertainty
quantification/reliability analysis Ranjan et al. (2008); Bichon et al.
(2008); Picheny et al. (2010); Bect et al. (2012); Echard et al. (2010);
Gotovos et al. (2013); Dubourg et al. (2013); Wang et al. (2016); Marques et
al. (2018); Cole et al. (2021); Booth et al. (2023, 2024), and general global
optimization Jones et al. (1998); Jones (2001); Renganathan (2023) just to
name a few. This also includes addressing the multifidelity setting as we do.
The most popular approach for multifidelity analysis and design using GP
regression is that of Kennedy and O’Hagan (2001), which learns a linear
correlation between each fidelity level and a level lower; this results in an
autoregressive model that can predict the output of a higher fidelity model
from the output of an immediate lower fidelity model. Similar ideas have been
explored by Poloczek et al. (2017), and further extended to nonlinear
correlations by Perdikaris et al. (2017). As opposed to the aforementioned
existing work, we propose the use of a GP for the multifidelity setting, that
keeps the structure and model inference consistent, and simple, as in the
single fidelity setting—more details will follow in Section 2.2. The next
ingredient of our proposed work is active learning. Active learning of GP
models refers to the adaptive and automated selection of the training points
in a goal-oriented fashion until a relevant convergence criterion is met.
Typically, a utility function is defined that quantifies the impact of a
candidate point on the QoI, and training data are selected by solving an
“inner” optimization problem. The popular approaches include maximizing the
mean-squared error (MSE) Jin et al. (2002); Martin and Simpson (2002) or the
integrated MSE (IMSE) Santner et al. (2003); Picheny et al. (2010),
information entropy Shewry and Wynn (1987); Koehler and Owen (1996), and
mutual information Krause et al. (2008); Beck and Guillas (2016). See Garud et
al. (2017) and Liu et al. (2018) for a survey of adaptive methods.

Cross-validation (CV) Rasmussen and Williams (2006) refers to quantifying
model error via a held out testing set, and could be considered another
criterion for active learning. When one training point is held out for
testing, and this choice is permuted over all of the training set, this is
called leave-one-out cross-validation (LOO-CV) Li et al. (2010); Le Gratiet
and Cannamela (2015); Aute et al. (2013); Liu et al. (2016). Generally, in the
context of GPs, CV and/or LOO-CV are used to estimate the hyperparameters of
the GP itself Bachoc (2013). However, in this work, we seek to exploit the GP
CV error to develop an adaptive sampling strategy for global fit. Recently,
Mohammadi et al. (2022) proposed an adaptive LOO-CV technique where a second
GP is fit over the CV errors, which is in turn used in a Bayesian adaptive
setting such as in Jones et al. (1998) for active learning. By estimating the
CV error at each of the training points, which assumes a non-central chi-
squared distribution, they use the expected value of this distribution at each
of the training points to fit another GP. In this work, we take a similar
route, but we differ significantly in two fundamental ways. First, we fit a GP
to the logarithm of the CV error to comply with the general infinite support
of Gaussian random variables. While Mohammadi et al. (2022) ignores the
uncertainty in the CV error, our method can seamlessly include that using
heteroscedastic GPs Binois et al. (2018); Le et al. (2005); Ankenman et al.
(2008, 2010). Second, we extend this approach to the multifidelity setting
where we present a generic acquisition function that can be easily customized
to generate novel acquisition functions. Third, our method is _cost-aware_ ,
thereby making sequential decisions that accounts for the cost of querying
each specific level. Finally, our proposed approach seamlessly extends to
discrete and continuous levels of fidelity, and sequential as well as batch
acquisition settings. In summary, we propose multifidelity cross-validation
(MFCV), which presents a generic method to actively learn a surrogate model
from multiple information sources, with the aforementioned features. Our
approach is endowed with a straightforward implementation without any
necessary user-defined fine tuning. We demonstrate our method on several
synthetic test functions as well as real-world experiments.

The remainder of the article is organized as follows. In Section 2, we provide
the background and preliminaries including details about multifidelity GP
models. In Section 3, we provide the details of our method. In Section 4, we
show results of our method demonstrated on the synthetic test functions and a
real-world stress analysis of a gas turbine blade, and provide concluding
remarks in Section 5.

##  2 Background and preliminaries

###  2.1 Gaussian process regression

Let $\mathbf{x}\in\bm{\mathcal{X}}\subset\mathbb{R}^{d}$ be the input to the
function $f:\bm{\mathcal{X}}\rightarrow\mathbb{R}$ and we assume evaluations
of $f(\mathbf{x})$ are computationally expensive. Denote observations of the
expensive oracle $f$ as $y_{i}=y(\mathbf{x}_{i}),~{}i=1,2,\ldots,n$. Let
$X_{n}$ denote the matrix horizontal stacking of the $n$ observation sites
$\mathbf{x}_{i},~{}i=1,\ldots,n$, with the corresponding response vector
denoted $\mathbf{y}_{n}$. A GP model assumes a multivariate normal
distribution over the response, e.g.,
$\mathbf{y}\sim\mathcal{GP}(0,\mathbf{K}(X,X))$, where covariance
$\mathbf{K}(X,X)$ is typically a function of inverse Euclidean distances,
i.e., $\mathbf{K}(X,X)^{ij}=k(||\mathbf{x}_{i}-\mathbf{x}_{j}||^{2})$; see
(Santner et al., 2003; Rasmussen and Williams, 2006; Gramacy, 2020) for
reviews. Conditioned on observations
$\bm{\mathcal{D}}_{n}=\\{X_{n},\mathbf{y}_{n}\\}$, posterior predictions at
input $\mathbf{x}$ follow

$Y(\mathbf{x})|\bm{\mathcal{D}}_{n}\sim\mathcal{GP}\left(\mu_{n}(\mathbf{x}),\sigma^{2}_{n}(\mathbf{x})\right)\quad\textnormal{where}\quad\begin{aligned}
\mu_{n}(\mathbf{x})&=k(\mathbf{x},X_{n})\mathbf{K}(X_{n},X_{n})^{-1}\mathbf{y}_{n}\\\
\sigma^{2}_{n}(\mathbf{x})&=k(\mathbf{x},X_{n})-k(\mathbf{x},X_{n})\mathbf{K}(X_{n},X_{n})^{-1}k(X_{n},\mathbf{x}).\end{aligned}$
(1)

Throughout, subscript $n$ is used to denote quantities from a surrogate
trained on $n$ data points. The posterior distribution of Equation 3 provides
a general probabilistic surrogate model that can be used to approximate the
expensive oracle.

###  2.2 Multifidelity Gaussian process regression

Crucially, we assume that $f(\mathbf{x})$ is approximated by models
$\hat{f}(\mathbf{x},s)$, where $\mathbf{x}$ are common inputs to all the
models and the parameter $s\in\bm{\mathcal{S}}\subset\mathbb{R}$ is a tunable
_fidelity_ parameter for each model. For the sake of simplicity, in this work,
we set $s$ to be scalar-valued and without loss of generality
$\bm{\mathcal{S}}=[0,1]$, where $s=1$ and $s=0$ represents the model at the
highest and lowest fidelity, respectively. Therefore,

$\hat{f}(\mathbf{x},1):=f(\mathbf{x}).$

Additionally, we assume there is a known cost function
$c(\mathbf{x},s):\bm{\mathcal{X}}\times\bm{\mathcal{S}}\rightarrow\mathbb{R}$,
which models the computational cost of querying $\hat{f}$ at a specific input-
fidelity pair $(\mathbf{x},s)$. Overall, we are interested in learning
$f(\mathbf{x})$, by querying $\hat{f}$, while keeping the overall
computational cost lower than querying $f(\mathbf{x})$ alone. Our
multifidelity method depends on learning a GP model that maps the augmented
input-fidelity space $(\mathbf{x},s)\in\bm{\mathcal{X}}\times\bm{\mathcal{S}}$
to output quantities of interest $\hat{f}(\mathbf{x},s)$. We assume that we
can make noisy evaluations of the function $\hat{f}$ at a given $\mathbf{x}$
and fidelity level $s$. That is

${y}_{i}=\hat{f}(\mathbf{x}_{i},s_{i})+\epsilon_{i},\quad i=1,\ldots,n.$ (2)

As in the single fidelity setting, we specify GP prior distributions on the
oracle value and the noise, over the joint input-fidelity space. That is,
$\hat{f}(\mathbf{x},s)\sim\mathcal{GP}\left(0,k((\mathbf{x},s),(\mathbf{x}^{\prime},s^{\prime}))\right)$
and $\epsilon\sim\mathcal{GP}(0,\sigma^{2}_{\epsilon})$, where we assume that
$\sigma^{2}_{\epsilon}$ is a constant but unknown noise variance. The
covariance function $k$, in this case, captures the correlation between the
observations in the joint $(\mathbf{x},s)$ space; here we use the product
composite form given by
$k((\mathbf{x},s),(\mathbf{x}^{\prime},s^{\prime}))=k_{\mathbf{x}}(\mathbf{x},\mathbf{x}^{\prime};\gamma_{\mathbf{x}})\times
k_{s}(s,s^{\prime};\gamma_{s})$, where $\gamma_{\mathbf{x}}$ and $\gamma_{s}$
parameterize the covariance functions for $\mathbf{x}$ and $s$, respectively.
We estimate the GP hyperparameters
$\bm{\Omega}=\\{\gamma_{\mathbf{x}},\gamma_{s},\sigma^{2}_{\epsilon}\\}$ from
data by maximizing the marginal likelihood (Rasmussen and Williams, 2006, Ch.
5). We choose an anisotropic Matern-type kernel for $k_{\mathbf{x}}$ which is
the defacto choice in GP regression. The posterior predictive distribution of
the output $Y$, conditioned on available observations from the multiple
fidelity oracles, is given by Rasmussen and Williams (2006)

$\begin{split}Y(\mathbf{x},s)|\bm{\mathcal{D}}_{n},\bm{\Omega}&\sim\mathcal{GP}(\mu_{n}(\mathbf{x},s),\sigma^{2}_{n}(\mathbf{x},s)),~{}\quad\bm{\mathcal{D}}_{n}:=\\{(\mathbf{x}_{i},s_{i}),{y}_{i}\\}_{i=1}^{n}\\\
\mu_{n}(\mathbf{x},s)&=k((\mathbf{x},s),X_{n})^{\top}[\mathbf{K}_{n}+\sigma_{\epsilon}^{2}\mathbf{I}]^{-1}\mathbf{y}_{n}\\\
\sigma^{2}_{n}(\mathbf{x},s)&=k((\mathbf{x},s),(\mathbf{x},s))-k((\mathbf{x},s),X_{n})^{\top}[\mathbf{K}_{n}+\sigma_{\epsilon}^{2}\mathbf{I}]^{-1}k((\mathbf{x},s),X_{n}),\end{split}$
(3)

where with a slight abuse of notation, we use $X_{n}$ to denote the row-stack
of input fidelity pairs $[\mathbf{x}_{i}^{\top},s_{i}],~{}\forall
i=1,\ldots,n$, and $\mathbf{y}_{n}$ is the vector of all observations at
various fidelty levels in $\bm{\mathcal{D}}_{n}$. Note that other notations
remain identical to the single-fidelity GP. For instance, $\mu_{n}$ and
$\sigma_{n}^{2}$ are the posterior mean and variance of the multifidelity GP,
respectively, where the subscript $n$ implies the conditioning based on $n$
past observations. Likewise, the training of the multifidelity GP is also via
marginal likelihood maximization.

###  2.3 Active learning

The GP posterior distribution serves as a multifidelity surrogate model for
the expensive quantity of interest derived from the high-fidelity model. The
accuracy of the surrogate model is naturally dependent on the observed data
used in the model fitting and it is impossible to choose the observations
apriori such that they would result in a sufficiently accurate surrogate
model. Therefore, we are interested in first constructing the surrogate model
with a (randomly chosen) set of _seed_ observation samples, which is then
adaptively improved by adding new observations that are judiciously chosen
according to a certain criterion. This adaptive model improvement can be
achieved by defining an _acquisition function_ , $\alpha(\mathbf{x},s)$, in
terms of the GP posterior, that quantifies the utility of a candidate input-
fidelity pair $(\mathbf{x},s)$. This way, new queries to the simulation model
can be made at optimal $(\mathbf{x},s)$ values that maximize
$\alpha(\mathbf{x},s)$, that is,

$\mathbf{x}_{n+1},s_{n+1}=\operatornamewithlimits{arg\,max}_{(\mathbf{x},s)\in\bm{\mathcal{X}}\times\bm{\mathcal{S}}}~{}\alpha(\mathbf{x},s)|\bm{\mathcal{D}}_{n}$
(4)

It should be noted that the acquisition function optimization is relatively
computationally cheap, since it depends on the GP posterior and is independent
of the high-fidelity model. A generic active learning algorithm, in the single
fidelity setting, is shown in Algorithm 1.

There are several choices one could make for $\alpha$, when the overall goal
is model predictive accuracy. As previously mentioned, we explore leave-one-
out cross-validation Wahba (1985), (Rasmussen and Williams, 2006, Ch.5). But,
there exist other approaches which include expected improvement for global fit
Lam (2008) and the integrated mean-squared predictive error (or posterior
variance) reduction by Seo et al. (2000); Chen and Zhou (2014); Binois et al.
(2019). To the best of our knowledge, none of the existing work automatically
extend to a multifidelity setting and neither does a multifidelity active
learning method exists in the context of LOO-CV. We now proceed to introduce
the proposed MFCV method, that is a novel active learning approach with
multifidelity GPs and LOO-CV.

1 Given:
$\bm{\mathcal{D}}_{n}=\\{(\mathbf{x}_{i},s_{i}),{y}_{i}\\}_{i=1}^{n}$, total
budget $B$, and GP hyperparameters $\bm{\Omega}$

Result: Posterior GP
$\mathcal{GP}(\mu_{B}(\mathbf{x}),\sigma_{B}^{2}(\mathbf{x}))$

2 for  _$i=n+1,\ldots,B$ , _ do

3 Find
$\mathbf{x}_{i}\in\underset{\mathbf{x}\in\bm{\mathcal{X}}}{\operatornamewithlimits{arg\,max}}~{}\alpha(\mathbf{x})$
(acquisition function maximization)

4 Observe ${y}_{i}$ = $f(\mathbf{x}_{i})+\epsilon_{i}$

5 Append
$\bm{\mathcal{D}}_{i}=\bm{\mathcal{D}}_{i-1}\cup\\{(\mathbf{x}_{i}),{y}_{i}\\}$

6 Update GP hyperparameters $\bm{\Omega}$

7

Algorithm 1 Generic Gaussian process active learning

##  3 Proposed method

###  3.1 Multifidelity cross-validation (MFCV)

Because of the marginalization properties of GPs,
$Y(\mathbf{x},1)|\bm{\mathcal{D}}_{n}$ is also a GP and can be expressed as
$Y(\mathbf{x},1)\sim\mathcal{GP}(\mu_{n}(\mathbf{x},1),\sigma_{n}^{2}(\mathbf{x},1))$.
Now we use the concept of $k$-fold cross-validation, where the observed data
is split into $k$ disjoint, equally sized subsets. Then the GP is trained on
$k-1$ subsets while the validation is done on the remaining subset. This
procedure is repeated $k$ times to rotate amongst the subsets. An extreme case
of $k$-fold cross-validation is when $k=n$, that is particularly suited for
small-data situations and is called the leave-one-out cross-validation (LOO-
CV). Fortunately, the GP posterior and the marginal log likelihood, with one
data point left out from $\bm{\mathcal{D}}_{n}$, is available in closed-form
enabling efficient computation Rasmussen and Williams (2006); we present that
in what follows. The posterior predictive distribution at $y_{i}$, when
$(\mathbf{x}_{i},s_{i})$ are left out, is given by

$p(y_{i}|(\mathbf{x},s),\mathbf{y}_{-i},\bm{\Omega})\sim\mathcal{N}(\mu_{-i}(\mathbf{x},s),\sigma_{-i}^{2}(\mathbf{x},s)),$
(5)

where $\mathbf{y}_{-i}$ refers to all observations except the $i$th and
likewise, $\mu_{-i}$ and $\sigma^{2}_{-i}$ are the GP posterior mean and
variance conditioned on $\mathbf{y}_{-i}$. The LOO-CV predictive posterior
mean and variance are given by the following closed-form expressions Rasmussen
and Williams (2006)

$\begin{split}\mu_{-i}=&\hat{y}_{i}-\frac{[\mathbf{K}^{-1}\mathbf{y}]_{i}}{[\mathbf{K}^{-1}]_{ii}}\\\
\sigma^{2}_{-i}=&\frac{1}{[\mathbf{K}^{-1}]_{ii}}\end{split}$ (6)

where $[]_{i}$ and $[]_{ii}$ denote the $i$th element of a vector and the
$(i,i)$th element of a matrix, respectively. Note that once $\mathbf{K}^{-1}$
is computed, computing (6) takes only $\mathcal{O}(n^{2})$ operations. In
fact, in Equation 6, the $\mathbf{K}^{-1}$ is never explicitly computed. Once
$\mathbf{K}$ is computed, we compute $\mathbf{K}^{-1}\mathbf{y}$ and
$[\mathbf{K}]_{ii}$ using the method of linear conjugate gradients (Nocedal
and Wright, 1999, Ch. 5).

Let $e_{\textnormal{cv}}(\mathbf{x},s)$ denote the leave-one-out cross-
validation error everywhere in $\bm{\mathcal{X}}\times\bm{\mathcal{S}}$. Then,
the main idea of the proposed approach is to learn
$e_{\textnormal{cv}}(\mathbf{x},s)$ so that our original posterior GP,
$Y(\mathbf{x},s)$, can be adaptively enriched by adding points that drive
$e_{\textnormal{cv}}(\mathbf{x},1)$ to zero everywhere in
$\bm{\mathcal{X}}\times\\{1\\}$. In other words, we will choose points that
maximize $e_{\textnormal{cv}}(\mathbf{x}^{\prime},1)$, due to a candidate
input-fidelity pair $(\mathbf{x},s)$, at every step of the adaptive model
building process. Learning $e_{\textnormal{cv}}$ begins by making the
following (stochastic) observations

$e_{\textnormal{cv},i}=(Y_{-i}(\mathbf{x}_{i},s_{i})-{y}_{i})^{2},~{}\forall
i=1,\ldots,n,$ (7)

where $Y_{-i}$ is the posterior GP conditioned on $\bm{\mathcal{D}}_{-i}$ at
$(\mathbf{x}_{i},s_{i})$ and is a normal random variable with mean $\mu_{-i}$
and variance $\sigma^{2}_{-i}$. $e_{\textnormal{cv}}$, on the other hand,
assumes a noncentral chi-squared distribution with one ($\kappa=1$) degrees of
freedom, with the following mean and variance:

$\begin{split}e_{\textnormal{cv},i}\sim&\chi^{2}\left(\kappa=1,\lambda=(\mu_{-i}(\mathbf{x}_{i},s_{i})-y_{i})^{2}\right)\\\
\mathbb{E}[e_{\textnormal{cv},i}]=&1+(\mu_{-i}(\mathbf{x}_{i},s_{i})-y_{i})^{2}\\\
\mathbb{V}[e_{\textnormal{cv},i}]=&2(1+2(\mu_{-i}(\mathbf{x}_{i},s_{i})-y_{i})^{2}),\end{split}$
(8)

where $\lambda$ is the noncentrality parameter. The expectation and variance
in Equation 8 are clearly dependent only on the posterior means computed in
Equation 6 and hence can be readily computed. In principle, the mean and
variance in (8) can be used to train a heteroscedastic GP. However, in this
work, as a first step, we take only the expected value of
$e_{\textnormal{cv}}$, at every $i$, and fit a GP to it. In order to comply
with the infinite support of GPs, we instead fit the GP to the logarithm of
the expected value of $e_{\textnormal{cv}}$. We define the logarithm of the
$e_{\textnormal{cv}}$ and place a GP prior on it as follows

$\ell(\mathbf{x},s)=\log(e_{cv}(\mathbf{x},s))\sim\mathcal{GP}(0,k_{\ell}(\cdot,\cdot)),$
(9)

whis learned from observations
$\log(\mathbb{E}[e_{\textnormal{cv},i}]),~{}\forall i=1,\ldots,n$. As before,
the posterior predictive distribution of $\ell(\mathbf{x},s)$ conditioned on
the observed LOO-CV error is also a GP. Finally, the MFCV acquisition function
is given as

$\alpha_{\textnormal{MFCV}}(\mathbf{x},s)=\mathbb{E}_{\ell}\left[\max_{\mathbf{x}^{\prime}\in\bm{\mathcal{X}}}\ell(\mathbf{x}^{\prime},1)|\bm{\mathcal{D}}_{n}\bigcup\\{(\mathbf{x},s),\ell(\mathbf{x},s)\\}\right].$
(10)

Our proposed MFCV acquisition function, shown in (10), quantifies the utility
of a given input-fidelity pair $(\mathbf{x},s)$, on the maximum cross-
validation error at the highest fidelity ($s=1$). Then, the new point is
chosen as

$\mathbf{x}_{n+1},s_{n+1}=\operatornamewithlimits{arg\,max}_{(\mathbf{x},s)\in\bm{\mathcal{X}}\times\bm{\mathcal{S}}}~{}\frac{\alpha_{\textnormal{MFCV}}(\mathbf{x},s)}{c(\mathbf{x},s)}.$
(11)

Our approach equally applies to selecting a batch of $q$ points. Let
$X=\\{\mathbf{x}^{1},\ldots,\mathbf{x}^{q}\\}$, $S=\\{s^{1},\ldots,s^{q}\\}$,
and $\bm{\ell}=\\{\ell^{1},\ldots,\ell^{q}\\}$, then our acquisition function
in batch mode is given by

$\alpha_{q\textnormal{MFCV}}(X,S)=\mathbb{E}_{\bm{\ell}}\left[\max_{\mathbf{x}^{\prime}\in\bm{\mathcal{X}}}\ell(\mathbf{x}^{\prime},1)|\bm{\mathcal{D}}_{n}\bigcup\\{(X,S),\bm{\ell}\\}\right],$
(12)

and as before, the next batch of points is chosen as

$X_{n+1:q},S_{n+1:q}=\operatornamewithlimits{arg\,max}_{(X,S)\in\bm{\mathcal{X}}^{q}\times\bm{\mathcal{S}}^{q}}~{}\frac{\alpha_{q\textnormal{MFCV}}(X,S)}{\sum_{i=1}^{q}c(\mathbf{x}^{i},s^{i})}.$
(13)

Plugging in (10) and (12) into Algorithm 1 (line 3) completes our method. The
overall MFCV algorithm is summarized in Algorithm 2.

1 Given: Seed observations
$\bm{\mathcal{D}}_{n}=\\{(\mathbf{x}_{i},s_{i}),({y}_{i},\ldots,{y}_{n})\\}_{i=1}^{n}$,
total number of iterations $B$, cost model $c(\cdot,\cdot)$ and GP
hyperparameters $\bm{\Omega}$

Result: Posterior GP at $s=1$,
$\mathcal{GP}(\mu_{B}(\mathbf{x},1),\sigma_{B}^{2}(\mathbf{x},1))$

2 for  _$i=n+1,\ldots,B$ , _ do

3 Compute $e_{\textnormal{cv}}$ at all observation sites:
$\mathbb{E}[e_{\textnormal{CV},k}],~{}k=1,\ldots,i$.

4 Fit an inner GP to the data set
$\\{(\mathbf{x}_{k},s_{k}),\log(\mathbb{E}[e_{\textnormal{CV},k})\\},~{}k=1,\ldots,i$.

5 Compute and optimize the cost-aware MFCV acquisition function (Equation 10).

6 Make new observations $y_{i}=\hat{f}(\mathbf{x}_{i},s_{i})$.

7 Append
$\bm{\mathcal{D}}_{i}=\bm{\mathcal{D}}_{i}\cup\\{(\mathbf{x}_{i},s_{i}),{y}_{i}\\}$

8 Update GP hyperparameters $\bm{\Omega}$

9

Algorithm 2 Multifidelity cross-validation (MFCV)

##  4 Numerical Experiments

We first demonstrate our proposed methods on synthetic functions of varying
dimensions as well as one real-world experiment. For all the experiments, we
provide a set of seed points
$\mathcal{D}_{n}=\\{(\mathbf{x}_{i},s_{i}),y_{i},~{}i=1,\ldots,n\\}$ to start
the algorithm, where $n=10d$ and $(\mathbf{x}_{i},s_{i}),\forall i$ are
selected uniformly at random from $\bm{\mathcal{X}}\times\bm{\mathcal{S}}$. We
use the root mean-squared error (RMSE) as the main metric to assess the
method, which is defined as

$\textnormal{RMSE}=\sqrt{\frac{\left(\mu_{n}(\mathbf{x},1)-f(\mathbf{x})\right)^{2}}{N}}.$

The RMSE is evaluated on a set of test points chosen uniformly at random, with
cardinality $N$, chosen at the highest fidelity $s=1$; $N$ is set to be $30d$.
We keep the test point fixed across iterations to ensure the error is always
evaluated against a constant benchmark. We observe the history of RMSE against
the cumulative cost of evaluations. Since, at the time of writing, we are not
aware of any other competing method for multifidelity cross-validation, we
primarily compare our method against the single-fidelity equivalent of our
method—that is, acquisitions performed at $s=1$; we denote this as HF. We look
at three variants of MFCV: sequential sampling $(q=1)$, and batch sampling
$q=2,~{}4$. Finally, we compare our method against a ”random” selection
strategy according to Sobol sampling Sobol’ (1967)—this way, we ensure there
is value in choosing acquisitions that are optimal according to our
acquisition function. All the experiments, including the HF, are provided with
exactly the same set of seed points. Once the algorithm begins, all
experiments choose an input-fidelity pair $(\mathbf{x},s)$, whereas the HF
will do so at a fixed fidelity of $s=1$. All experiments are repeated 10
times, but for the turbine experiment which is repeated 5 times, with random
seed points, and we look at the average performance metric along with $\pm 1$
standard deviations.

For all the experiments, the following cost model is assumed:

$c(s)/c_{0}=c_{2}+\exp\left(-c_{1}\times(1-s)\right),$

where $c_{0}$ is a normalizing constant that is required to ensure that the
acquisition function optimization is not biased by the actual costs of the
models; note that $c_{0}$ is also the cost of the high-fidelity ($s=1$) model,
for which we fix $c_{0}=500$. The $c_{1}>0$ determines the steepness of the
cost model; we set $c_{1}=10$ for all the experiments. Finally, $c_{2}>0$ is a
constant to ensure that the cost model is bounded away from zero and is
roughly the cost of the lowest-fidelity model ($\hat{f}(\cdot,0)$); we fix
$c_{2}=0.1$. We note that our cost model—by virtue of setting $c_{1}=10$—is
designed to emphasize the disparity between the computational costs of lower-
and higher-fidelity model queries; see Figure 1. Note that we assume the cost
model is independent of $\mathbf{x}$ for simplicity and without any loss of
generality.

Figure 1: Multifidelity cost model $c(s)$ with $c_{1}=10$, which emphasizes
the cost disparity across fidelities as one approaches $s=1$.

###  4.1 Synthetic experiments

We demonstrate our method on four synthetic test functions: multifidelity
Branin ($d=3)$, multifidelity four branches $d=3$, multifidelity Ishigami
Renganathan et al. (2023) ($d=4$), and multifidelity Hartmann ($d=7$). The
results for the synthetic test functions are shown in Figure 2; details and
plots of these functions are shown in the appendix. The plot shows the average
RMSE of all repetitions (solid line) along with the $\pm 1$ standard deviation
(shaded), versus the cumulative computational cost. Note that we show the RMSE
starting with the first iteration, that is not including the seed points.
Therefore, since we show the plot versus the cumulative cost, the lines within
each plot do not start from the same point, despite starting with exactly the
same seed points. Notice that some variant of the
MFCV($q=1,2,\textnormal{or},~{}4$) always outperforms the single-fidelity HF
case, thereby clearly demonstrating the benefit of the multifidelity approach.
Furthermore, the random strategy (Sobol) is consistently worse than all other
strategies. This demonstrates that the proposed acquisition strategy has value
of information and the points chosen are indeed optimal according to the
acquisition function. It should be noted that the plots of RMSE don’t show a
monotonic decrease in RMSE. This is because, with every new acquisition, the
hyperparameters of the global GP are computed again, which involves a highly
nonconvex optimization of the marginal log likelihood function with randomized
starting points. As a result, the posterior GP is not guaranteed to result in
a GP with improved RMSE, although the overall trend of these curves show a
decreasing trend as expected. This phenomenon is consistent with other results
reported in the literature Mohammadi et al. (2022). In Figure 3, we show the
distribution of the fidelity levels selected by MFCV. The distribution, as
evident from the figure, is unique to each experiment. However, in this case,
fidelities $\geq 0.7$ are only selected, showing that lower fidelities
contribute relatively less in reducing the $e_{\textnormal{cv}}$ at the
highest fidelity $s=1$. This is particularly true because the
$e_{\textnormal{cv}}$ is heavily influenced by the GP posterior variance and
our chosen, stationary, GP prior correlates less when the Euclidean distance
between points is high.

(a) Branin (3D)

(b) Four branches (3D)

(c) Ishigami (4D)

(d) Hartmann (7D)

Figure 2: MFCV demonstration on synthetic test functions.

(a) Branin (3D)

(b) Four branches (3D)

(c) Ishigami (4D)

(d) Hartmann (7D)

Figure 3: Fidelity levels selected by the MFCV strategy. Note that we show
only $q=1$.

###  4.2 Gas turbine blade thermal stress analysis on a discrete fidelity
space

We now demonstrate our method on the structural analysis of a gas turbine
blade at steady-state operating conditions. Gas turbine blades are subject to
very high pressures and temperatures that can cause concentrated zones of high
stress that can in turn lead to structural deformation and potential
catastrophic damage of the engine parts. A finite element model of the blade
is leveraged to simulate the thermal stress acting on the bade surface. The
Von Mises stress is a good indicator for the distribution of the thermal
stress on the blade surface. The maximum Von Mises stress on the blade surface
could be used to predict the likelihood of a structural failure Renganathan et
al. (2023).

In this regard we vary the pressure-side ($x_{1}$) and suction-side ($x_{2}$)
pressures as boundary conditions, and set the surface mesh resolution to
capture the fidelity; see Figure 4 for the three levels of fidelity.
Crucially, this experiment demonstrates the methodology on a discrete fidelity
set. We observe the maximum Von Mises stress acting on the blade as our
response. The blade material properties are fixed to the following constants:
Young’s modulus $227E+9$ Pa, Poisson’s ratio $0.27$, and coefficient of
thermal expansion $12.7E-6$. The governing equations are solved using a finite
element method using Matlab’s PDE toolbox
444https://www.mathworks.com/products/pde.html. The computational mesh density
is determined via a minimum element size which also serves as the fidelity
parameter. Specifically, we use the minimum mesh element sizes
$\\{0.05,0.025,0.01\\}$, which correspond to fidelities
$\mathcal{S}=\\{0.0,0.5,1.0\\}$, respectively. The mesh resolutions for the
specified three fidelity levels are shown in Figure 4. As previously
mentioned, the maximum von Mises stress is the quantity of interest in this
experiment; see Figure 5 for a couple of snapshots of the von Mises stress
distribution and blade deformation corresponding to two different pressure
loadings.

The suction-side $x_{1}$ and pressure-side $x_{2}$ pressures are varied in the
following ranges: $x_{1}\in[201160,698390],~{}x_{2}\in[301460,598020]$. A
total of $n=30$ solution snapshots are generated, by sampling the inputs
uniformly at random in $\mathcal{X}\times\mathcal{S}$, and supplied as seed
points for the multifidelity run. We do the same for the HF case to keep the
comparison fair. The MFCV, HF, and random strategy experiments are then run
for 50 iterations and are repeated 5 times with randomized seed points. The
$q=2$ and $q=4$ experiments are run for 25 and 13 iterations, respectively.

The plots of the RMSE versus cumulative cost is shown in Figure 6(a). Notice
that MFCV with sequential acquisition ($q=1$) outperforms all other
strategies. Batch sampling ($q=2$ and $q=4$) perform worse but, as in all
other experiments, one of the variants of MFCV wins. Interestingly, the random
sampling performs on par with the single-fidelity (HF). This is because, given
that this experiment has only three discrete levels of fidelity, the random
sampling occasionally samples new acquisitions at $s=1$. In the continuous
fidelity case, the probability of sampling a point at $s=1$ is $0$; however,
in this discrete fidelity case it is $0.33$, and hence this is expected. The
fidelity levels automatically chosen by the algorithm ($q=1$) is shown in
Figure 6(b). Notice that the high-fidelity model is queried approximately
$25\%$ of the times, demonstrating the cost saving.

(a) Low fidelity ($s=0.0$)

(b) Medium fidelity ($s=0.5$)

(c) High fidelity ($s=1.0$)

Figure 4: Mesh resolutions for the three discrete fidelities levels.

(a) Solution at $\mathbf{x}=[7E5,4.5E5]$

(b) Solution at $\mathbf{x}=[5E5,4.5E5]$

Figure 5: Von Mises stress distribution and deformation on the gas turbine
blade, under two different pressure loading boundary conditions.

(a) RMSE versus cumulative cost.

(b) Fidelity levels selected

Figure 6: Turbine thermal stress experiment.

##  5 Conclusion

We present a novel active learning strategy to adaptively learn multifidelity
Gaussian process models leveraging the leave-one-out cross-validation (LOO-CV)
criterion. Our approach depends on maintainting, and continuously updating,
two GPs. An “outer” GP that learns the mapping between $(\mathbf{x},s)$ and
models at various fidelities ($\hat{f}(\mathbf{x},s)$). An an “inner” GP that
learns the logarithm of LOO-CV in the joint input-fidelity space
($\bm{\mathcal{X}}\times\bm{\mathcal{S}}$). Then, we use a two-step lookahead
strategy to develop a multifidelity acquisition policy that allows both
sequential as well as batch sampling. Our approach seamlessly works for both
continuous and discrete fidelity spaces.

One of the major limitations of our work is that, as $n$ increases, computing
Equation 6 for the LOO GP becomes intensive. Particularly, in the second line
in (6), $n$ conjugate gradient steps are necessary. However, for the chosen
experiments in this work, this was never a problem. Furthermore, we address
situations where the high-fidelity model is very expensive to evaluate and
only a few evaluations are affordable. In such situations, $n$ is not likely
to be too high. However, we believe it might be worth investigating efficient
ways for computing the conjugate gradient steps, e.g., via parallelization and
symbolic computations. This would be an avenue for futher work.

## Appendix

Previously Renganathan et al. (2023) and Renganathan et al. (2022) introduced
several new multifideity test functions that can be used to benchmark
multifidelity methods. We use a few of them, namely the 2D multifidelity
multimodal function Bichon et al. (2008), the 2D multifidelity Four branches
function Schöbi et al. (2017), and the 3D multifidelity Ishigami function
Ishigami and Homma (1990). Additionally, we also use the 6D multifidelity
Hartmann function provided in Surjanovic and Bingham . The specifics of these
test functions are discussed as follows.

####  5.0.1 2D multimodal function

We extend the test function in Bichon et al. (2008) to multifidelity setting
as shown in (14).

$f(\mathbf{x},s)=\frac{(x_{1}^{2}+4)(x_{2}-1)}{20}-s\times
sin\left(\frac{5x_{1}}{2}\right)-2,$ (14)

where $\mathbf{x}=[x_{1},x_{2}]$, the domain
$\bm{\mathcal{X}}=[-4,7]\times[-3,8]$, and the last term introduces a fidelity
dependence. This simple extension results in a correlated fidelity dependence,
which is continuously differentiable in $\mathcal{X}\times\mathcal{S}$; see
Figure 7 for snapshots.

Figure 7: Multifidelity multimodal function from Renganathan et al. (2023)

####  5.0.2 2D Multifidelity Four-branches function

A system with four distinct component limit-states Schöbi et al. (2017) is a
commonly used test problem in reliability analysis to demonstrate the efficacy
of the algorithms in presence of multiple failure regions. The response
function can be written as:

$\displaystyle
f({\mathbf{x}},s)=\textrm{min}\begin{cases}3+0.1(x^{\prime}_{1}-x^{\prime}_{2})^{2}-\frac{x^{\prime}_{1}+x^{\prime}_{2}}{\sqrt{2}}\,,\\\
3+0.1(x^{\prime}_{1}-x^{\prime}_{2})^{2}+\frac{x^{\prime}_{1}+x^{\prime}_{2}}{\sqrt{2}}\,,\\\
x^{\prime}_{1}-x^{\prime}_{2}+\frac{7}{\sqrt{2}}\,,\\\
x^{\prime}_{2}-x^{\prime}_{1}+\frac{7}{\sqrt{2}}\,\end{cases}$ (15)

where $\mathbf{x}=[x_{1},x_{2}]$ and $\bm{\mathcal{X}}=[-8,8]^{2}$. The
multifidelity extension we provide, offers a translation of the level sets
across fidelities, $x^{\prime}_{i}=x_{i}-5.0\times s,~{}i=1,\ldots,3$, and is
shown in Figure 8.

Figure 8: Multifidelity Four branches test function Renganathan et al. (2023).

####  5.0.3 3D Multifidelity Ishigami function

We modify the 3D Ishigami test function Ishigami and Homma (1990), similar to
the Four branches function, by introducing a shift in the variables using the
fidelity parameter $s$, as shown in (16). Note that, the input $\mathbf{x}$ is
normalized to be in $[0,1]^{3}$ before introducing the translation.

$f(\mathbf{x},s)=\sin(x_{1}-s)+7.0\sin^{2}(x_{2}-s)+0.1x_{3}^{4}\sin(x_{1}-s),$
(16)

where $\mathbf{x}=[x_{1},x_{2},x_{3}]$ and $\bm{\mathcal{X}}=[-\pi,\pi]^{3}$.
The resulting function is shown for two slices $x_{3}=-\pi$ and $x_{3}=0$ in
Figure 9.

Figure 9: Multifidelity Ishigami test function Renganathan et al. (2023).

####  5.0.4 6D Hartmann function

We use the ”augmented” Hartmann 6D function introduced in Balandat et al.
(2019) and restated below.

$f(\mathbf{x},s)=-\left(\beta_{1}-0.1*(1-s)\right)*\exp\left(-\sum_{j=1}^{6}A_{1j}(x_{j}-P_{1j})^{2}\right)-\sum_{i=2}^{4}\beta_{i}\exp\left(-\sum_{j=1}^{6}A_{ij}(x_{j}-P_{ij})^{2}\right),$
(17)

where $\mathbf{x}=[x_{1},\ldots,x_{6}]$ and $\bm{\mathcal{X}}=[0,1]^{6}$ and

$A=\begin{bmatrix}10&3&17&3.5&1.7&8\\\ 0.05&10&17&0.1&8&14\\\
3&3.5&1.7&10&17&8\\\ 17&8&0.05&10&0.1&14\end{bmatrix}$

and

$P=\begin{bmatrix}1312&1696&5569&124&8283&5886\\\
2329&4135&8307&3736&1004&9991\\\ 2348&1451&3522&2883&3047&6650\\\
4047&8828&8732&5743&1091&381\end{bmatrix}.$

## Acknowledgments

This work is supported by the Faculty Startup Funds at the The Pennsylvania
State University. All computations were performed in the Penn State Institute
of Computational and Data Sciences CPU cluster Roar.

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# Reversed Strichartz estimates for wave on non-trapping asymptotically
hyperbolic manifolds and applications

Yannick Sire  Department of Mathematics
Johns Hopkins University
Baltimore, MD 21218<EMAIL_ADDRESS>,  Christopher D. Sogge  Department of
Mathematics
Johns Hopkins University
Baltimore, MD 21218<EMAIL_ADDRESS>,  Chengbo Wang  School of Mathematical
Sciences
Zhejiang University
Hangzhou 310027, China<EMAIL_ADDRESS>and  Junyong Zhang  Department of
Mathematics
Beijing Institute of Technology
Beijing 100081, China<EMAIL_ADDRESS>
###### Abstract.

We provide reversed Strichartz estimates for the shifted wave equations on
non-trapping asymptotically hyperbolic manifolds using cluster estimates for
spectral projectors proved previously in such generality. As a consequence, we
solve a problem left open in [13] about the endpoint case for global well-
posedness of nonlinear wave equations. We also provide estimates in this
context for the maximal wave operator.

###### Contents

  1. 1 Introduction and main results
  2. 2  Proof of Theorem 1.3
  3. 3 The proof of Theorem 1.4
  4. 4 Small data well-posedness  and Strauss conjecture on asymptotically hyperbolic manifolds

##  1\. Introduction and main results

The goal of this note is to obtain new reversed Strichartz estimates for the
(shifted) wave operator on some complete manifolds with bounded geometry. We
are concerned in particular with asymptotically hyperbolic manifolds. The
argument builds on $L^{p}$-estimates for the spectral projectors (on the
continuous spectrum). We draw also two consequences of the reverse Strichartz
estimates: first, the endpoint version of classical Strichartz estimates and
an application to global well-posedness for nonlinear wave equations on those
manifolds; second, a sharp (in $L^{p}$ spaces) estimate of the maximal
function for the wave operator.

We work on an $n+1$-dimensional complete non-compact Riemannian manifold
$(M^{\circ},g)$ where $n\geq 1$ and the metric $g$ is an asymptotically
hyperbolic metric. This setting is the same as in Chen-Hassell [4, 5], Mazzeo
[7] and Mazzeo-Melrose[8]. Let $x$ be a boundary defining function for the
compactification $M$ of $M^{\circ}$. We say a metric $g$ is conformally
compact if $x^{2}g$ is a Riemannian metric and extends smoothly up to the
boundary $\partial M$. Mazzeo [7] showed that its sectional curvature tends to
$-|dx|^{2}_{x^{2}g}$ as $x\to 0$; In particular, if the limit is such that
$-|dx|^{2}_{x^{2}g}=-1$, we say that the conformally compact metric $g$ is
asymptotically hyperbolic. More specifically, let $y=(y_{1},\cdots,y_{n})$ be
local coordinates on $Y=\partial M$, and $(x,y)$ be the local coordinates on
$M$ near $\partial M$; the metric $g$ in a collar neighborhood
$[0,\epsilon)_{x}\times\partial M$ takes the form

(1.1)
$g=\frac{dx^{2}}{x^{2}}+\frac{h(x,y)}{x^{2}}=\frac{dx^{2}}{x^{2}}+\frac{\sum
h_{jk}(x,y)dy^{j}dy^{k}}{x^{2}},$

where $x\in C^{\infty}(M)$ is a boundary defining function for $\partial M$
and $h$ is a smooth family of metrics on $Y=\partial M$. In addition, if every
geodesic in $M$ reaches $\partial M$ both forwards and backwards, we say $M$
is nontrapping. The Poincaré disc $(\mathbb{B}^{n+1},g)$ is a typical example
of such manifold. Indeed, considering the ball
$\mathbb{B}^{n+1}=\\{z\in\mathbb{R}^{n+1}:|z|<1\\}$ endowed with the metric

(1.2) $g=\frac{4dz^{2}}{(1-|z|^{2})^{2}},$

one can take $x=(1-|z|)(1+|z|)^{-1}$ as the boundary defining function and
$\omega$ the coordinates on $\mathbb{S}^{n}$. Then the Poincaré metric takes
the form

$g=\frac{dx^{2}}{x^{2}}+\frac{\frac{1}{4}(1-x^{2})^{2}d\omega^{2}}{x^{2}},$

where $d\omega^{2}$ is the standard metric on the sphere $\mathbb{S}^{n}$.

Let $H=-\Delta_{g}-\frac{n^{2}}{4}$ where $\Delta_{g}$ is the Laplace-Beltrami
operator on $(M^{\circ},g)$. We recall the following result about estimates
for the spectral measure and the spectral projectors by Chen and Hassell in
[5, Theorem 1.6]. We denote by $dE(\lambda)$ the spectral measure associated
with the operator $H$, such that for every $F$ a Borel function on
$\mathbb{R}$, we have

$F(\sqrt{H})=\int_{\mathbb{R}}F(\lambda)dE(\lambda)$

with domain

$\left\\{\psi:\int_{\mathbb{R}}|F(\lambda)^{2}|\,d\langle\psi,E(\lambda)\psi\rangle<\infty\right\\}.$

######  Theorem 1.1.

Let $n\geq 1$. Suppose $(M^{\circ},g)$ is a non trapping asymptotically
hyperbolic manifold of dimension $n+1$. Assume that there has no pure point
eigenvalue and has no resonance at the bottom of the continuous spectrum of
$H=-\Delta_{g}-\frac{n^{2}}{4}$. Then for some constant $C$, we have: for
$\lambda\leq 1$

(1.3) $\|dE(\lambda)\|_{p\to p^{\prime}}\leq C\lambda^{2},\,\,\,1\leq p<2.$

For $\lambda\geq 1$, we have

(1.4) $\|dE(\lambda)\|_{p\to
p^{\prime}}\leq\left\\{\begin{array}[]{c}C\lambda^{(n+1)(1/p-1/p^{\prime})-1},\,\,\,1\leq
p\leq\frac{2(n+2)}{n+4},\\\
C\lambda^{n(1/p-1/2)},\,\,\,\,\frac{2(n+2)}{n+4}\leq p<2.\\\
\end{array}\right.$

######  Corollary 1.2.

Suppose $(M^{\circ},g)$ is a non trapping asymptotically hyperbolic manifold
of dimension $n+1$ with $n\geq 1$. Assume that there has no pure point
eigenvalue and has no resonance at the bottom of the continuous spectrum of
$H$. Let $\delta\in(0,1]$ and define the spectral projector on the frequencies
$[k,k+\delta]$

(1.5)
$\chi_{k}^{\delta}f=\int_{0}^{\infty}\chi_{[k,k+\delta]}(\lambda)dE(\lambda)f.$

Then for $k\geq 0$

(1.6) $\|\chi_{k}^{\delta}\|_{p\to p^{\prime}}\leq
C\delta(\delta+k)^{2}(1+k)^{(n+1)(1/p-1/p^{\prime})-3},\,\,\,1\leq
p\leq\frac{2(n+2)}{n+4}.$

In particular, we have

(1.7) $\|\chi_{k}^{\delta}\|_{p\to p^{\prime}}\leq
C\delta(\delta+k)^{(n+1)(1/p-1/p^{\prime})-1},\,\,\,\max\Big{\\{}1,\frac{2(n+1)}{n+4}\Big{\\}}\leq
p\leq\frac{2(n+2)}{n+4}\ .$

###### Proof.

By Theorem 1.1 and integrating the spectral measure on $[k,k+\delta]$, one
gets the desired result. ∎

######  Remark.

In the case of high frequencies $k\geq 1$, the previous theorem is known to
hold on asymptotically conic manifold and even on manifolds with bounded
geometry (see [6]). In particular, asymptotically hyperbolic manifolds are
examples of manifolds with bounded geometry in the following sense: suppose
$(M,g)$ be a complete Riemannian manifold of dimension $n+1$ with $n\geq 2$.
We assume that $M$ has $C^{\infty}$ bounded geometry, i.e. the local
injectivity radius $M$ has a positive lower bound $\varepsilon$, the metric
tensor $g_{ij}$ , expressed in normal coordinates in the ball of radius
$\varepsilon/2$ around any point $z\in M$ is uniformly bounded in
$C^{\infty}(B(z,\varepsilon/2))$ as $z$ ranges over $M$; and the inverse
metric $g^{ij}$ is uniformly bounded in supremum norm.

The previous corollary has the following applications.

######  Theorem 1.3 (Reversed local-in-time Strichartz estimates).

Suppose $(M^{\circ},g)$ is a non trapping asymptotically hyperbolic manifold
of dimension $n+1$ with $n\geq 1$. Assume that there has no pure point
eigenvalue and has no resonance at the bottom of the continuous spectrum of
$H$. Let $u$ be the unique solution of

(1.8) $\begin{cases}\bigl{(}\partial_{t}^{2}+H\bigr{)}u=0,\\\
u|_{t=0}=f,\quad\partial_{t}u|_{t=0}=0.\end{cases}$

Then for $\frac{2(n+2)}{n}\leq q\leq\infty$, $2\leq p<\infty$ and
$s=(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{1}{p}$

(1.9) $\|u\|_{L^{q}(M^{\circ};L^{p}([-1,1]))}\leq
C\|(Id+H)^{\frac{s}{2}}f\|_{L^{2}(M^{\circ})}.$

######  Theorem 1.4 (Reversed global-in-time Strichartz estimates).

Assume $(M^{\circ},g)$ and $H$ be in Theorem 1.3. Let $q_{n}=+\infty$ when
$n=1,2$ and $q_{n}=\frac{2(n+1)}{n-2}$ when $n\geq 3$. For
$\frac{2(n+2)}{n}\leq q\leq q_{n}$, $2\leq p<\infty$ and
$s=(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{1}{p}$, the global-in-time estimates
hold

(1.10) $\|u\|_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}\leq
C\|H^{\frac{s}{2}}f\|_{L^{2}(M^{\circ})}.$

Furthermore, if $q_{n}<q\leq+\infty$ with $n\geq 3$, we have the global-in-
time estimates

(1.11) $\|u\|_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}\leq
C\|(H+Id)^{s/2}f\|_{L^{2}(M^{\circ})}.$

The standard Strichartz estimates were established in [13] but with an
arbitrary small $\epsilon$-loss of regularity due to the lack of Littlewood-
Paley theory in the non-doubling setting. Compared with the standard
Strichartz estimates, these ones reverse the order of space-time integration
with no loss. Estimates in this form have been extensively studied in [10, 11]
for waves on manifolds, [2] for waves with potentials.

Another application is a sharp maximal function estimate. In [9], Rogers and
Villarroya proved the following sharp maximal estimate for the wave operator
in $\mathbb{R}^{d}$:

(1.12)
$\big{\|}\sup_{t\in\mathbb{R}}|e^{it\sqrt{-\Delta}}f(x)|\big{\|}_{L^{q}(\mathbb{R}^{d})}\lesssim\|f\|_{H^{s}(\mathbb{R}^{d})}$

provided $q\geq\frac{2(d+1)}{d-1}$ and $s>d(\frac{1}{2}-\frac{1}{q})$. We get
an analogous statement in our context:

######  Corollary 1.5 (Maximal estimate).

Let the manifold $(M^{\circ},g)$ and the operator $H$ be in Theorem 1.3. Then
the following holds

(1.13)
$\big{\|}\sup_{t\in\mathbb{R}}|e^{it\sqrt{H}}f(x)|\big{\|}_{L^{q}(M^{\circ})}\lesssim\|f\|_{H^{s}(M^{\circ})}$

provided $q\geq\frac{2(n+2)}{n}$ and $s>(n+1)(\frac{1}{2}-\frac{1}{q})$.

###### Proof.

This is just a consequence of Theorem 1.4 with $p=2$ and the Sobolev embedding
$\dot{H}^{\frac{1}{2}+\varepsilon}(\mathbb{R})\hookrightarrow
L^{\infty}_{t}(\mathbb{R})$ for any $\varepsilon>0$. ∎

Acknowledgments: Y. S. was partially supported by the Simons foundation. C. D.
S. was supported by the NSF and the Simons foundation. C. Wang was supported
in part by NSFC 11971428 and National Support Program for Young Top-Notch
Talents. J. Zhang was supported by NSFC Grants (11771041, 11831004).

##  2\.  Proof of Theorem 1.3

In [1, Theorem 2.1], the authors show that the bounds (1.9) follows from
Theorem 1.2, and their proof works equally well in our circumstances.
Nonetheless, we include a proof for the sake of completeness which will serve
as a model for certain global Strichartz estimates.

To prove (1.9), it suffices to show that

(2.1) $\bigl{\|}e^{it\sqrt{H}}f\bigr{\|}_{L^{q}(M^{\circ};L^{p}([-1,1]))}\leq
C\|(I+H)^{\frac{s}{2}}f\|_{L^{2}(M^{\circ})}.$

There exists a $\rho\in{\mathcal{S}}(\mathbb{R})$ satisfying $\text{supp
}\hat{\rho}\subset(-2,2)$ such that

(2.2)
$\bigl{\|}e^{it\sqrt{H}}f\bigr{\|}_{L^{q}(M^{\circ};L^{p}([-1,1]))}\lesssim\bigl{\|}\rho(t)e^{it\sqrt{H}}f\|_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}.$

Then to prove (2.1), it suffices to prove

(2.3)
$\bigl{\|}\rho(t)e^{it\sqrt{H}}f\bigr{\|}_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}\leq
C\|(I+H)^{\frac{s}{2}}f\|_{L^{2}(M^{\circ})}.$

To prove this, we shall change notation a bit, and in particular $\delta=1$ in
(1.5), let

(2.4) $\chi_{k}f=\int_{0}^{\infty}\chi_{[k,k+1]}(\lambda)dE(\lambda)f$

so that $f=\sum_{k=0}^{\infty}\chi_{k}f$. Then, for $\frac{2(n+2)}{n}\leq
q\leq\infty$, (1.7) yields

(2.5)
$\|\chi_{k}f\|_{L^{q}(M^{\circ})}\lesssim(1+k)^{(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{1}{2}}\|f\|_{L^{2}(M^{\circ})},\quad
k=0,1,2,\dots.$

To use this, we first note that by Sobolev estimates

$\bigl{\|}\rho(t)e^{it\sqrt{H}}f\bigr{\|}_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}\lesssim\bigl{\|}\,|D_{t}|^{1/2-1/p}\bigl{(}\rho(t)e^{it\sqrt{H}}f\bigr{)}\bigr{\|}_{L^{q}_{x}L^{2}_{t}(\mathbb{R}\times
M^{\circ})}.$

Let

$F(t,x)=|D_{t}|^{1/2-1/p}\bigl{(}\rho(t)e^{it\sqrt{H}}f(x)\bigr{)}$

denote the function inside the mixed-norm in the right, then

$F(t,x)=\sum_{k=0}^{\infty}F_{k}(t,x),$

where

$F_{k}(t,x)=|D_{t}|^{1/2-1/p}\bigl{(}\rho(t)e^{it\sqrt{H}}\chi_{k}f(x)\bigr{)}.$

Consequently, its $t$-Fourier transform is

(2.6)
$\hat{F}_{k}(\tau,x)=|\tau|^{1/2-1/p}\int_{0}^{\infty}\hat{\rho}(\tau-\lambda)\chi_{[k,k+1]}(\lambda)dE(\lambda)f(x).$

Since $\text{supp }\hat{\rho}\subset(-2,2)$, we conclude that

$\int_{-\infty}^{\infty}F_{k}(t,x)\,\overline{F_{\ell}(t,x)}\,dt=(2\pi)^{-1}\int_{-\infty}^{\infty}\hat{F}_{k}(\tau,x)\,\overline{\hat{F}_{\ell}(\tau,x)}\,d\tau=0\quad\text{when
}\,|k-\ell|>100.$

As a consequence, we obtain

$\bigl{(}\,\int_{-\infty}^{\infty}\,\bigl{|}\,|D_{t}|^{1/2-1/p}\bigl{(}\rho(t)e^{it\sqrt{H}}f(x)\bigr{)}\,\bigr{|}^{2}\,dt\,\bigr{)}^{1/2}\\\
\lesssim\bigl{(}\,\int_{-\infty}^{\infty}\sum_{k=0}^{\infty}|F_{k}(t,x)|^{2}\,dt\,\bigr{)}^{1/2}=(2\pi)^{-1/2}\bigl{(}\int_{-\infty}^{\infty}\sum_{k=0}^{\infty}|\hat{F}_{k}(\tau,x)|^{2}\,d\tau\,\bigr{)}^{1/2}.$

Also, since $q\geq 2$, we conclude that

$\bigl{\|}\,|D_{t}|^{1/2-1/p}\bigl{(}\rho(t)e^{it\sqrt{H}}f\bigr{)}\bigr{\|}^{2}_{L^{q}_{x}L^{2}_{t}(\mathbb{R}\times
M^{\circ})}\lesssim\sum_{k=0}^{\infty}\int_{-\infty}^{\infty}\|\hat{F}_{k}(\tau,x)\|_{L^{q}(M^{\circ})}^{2}\,d\tau.$

Recalling (2.6), the support properties of $\hat{\rho}$, we see that

$\displaystyle\bigl{\|}\,|D_{t}|^{1/2-1/p}\bigl{(}\rho(t)e^{it\sqrt{H}}f\bigr{)}\bigr{\|}^{2}_{L^{q}_{x}L^{2}_{t}(\mathbb{R}\times
M^{\circ})}$
$\displaystyle=\sum_{k=0}^{\infty}\int_{-\infty}^{\infty}|\tau|^{1-2/p}\,\Bigl{\|}\int_{0}^{\infty}\hat{\rho}(\tau-\lambda)\chi_{[k,k+1]}(\lambda)dE(\lambda)f(x)\Bigr{\|}_{L^{q}(M^{\circ})}^{2}\,d\tau$
$\displaystyle=\sum_{k=0}^{\infty}\int_{k-10}^{k+10}|\tau|^{1-2/p}\,\Bigl{(}\int_{k}^{k+1}\lambda^{(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{1}{2}}d\lambda\,\|\chi_{k}f(x)\|_{L^{2}(M^{\circ})}\Bigr{)}^{2}\,d\tau\,$
$\displaystyle\lesssim\,\sum_{k=0}^{\infty}(1+k)^{1-2/p}(1+k)^{(n+1)(1-\frac{2}{q})-1}\|\chi_{k}f\|_{L^{2}(M^{\circ})}^{2}$
$\displaystyle=\bigl{(}\,\sum_{k=0}^{\infty}\|(1+k)^{(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{1}{p}}\chi_{k}f\|_{L^{2}(M^{\circ})}^{2}\,\bigr{)}^{1/2}\approx\|(I+H)^{s/2}f\|_{L^{2}(M^{\circ})}^{2},$

as desired, which completes the proof.

##  3\. The proof of Theorem 1.4

In this section, we prove the global result about Theorem 1.4 which is direct
consequence of the following Proposition.

######  Proposition 3.1.

Suppose the operator $H$, $q_{n}$ and the manifolds $(M^{\circ},g)$ to be in
Theorem 1.4. For $\frac{2(n+2)}{n}\leq q\leq q_{n}$, $2\leq p<\infty$ and
$s=(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{1}{p}$, the global-in-time estimates
hold

(3.1) $\|u\|_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}\leq
C\|H^{\frac{s}{2}}f\|_{L^{2}(M^{\circ})}.$

Furthermore, if $q_{n}<q\leq+\infty$ with $n\geq 3$, we have the global-in-
time estimates

(3.2) $\|u\|_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}\leq
C\|H^{3/4-1/(2p)}(H+I)^{s/2+1/(2p)-3/4}f\|_{L^{2}(M^{\circ})}.$

###### Proof.

Consider first the case with $q\leq q_{n}$. To prove (3.1), It suffices to
show that, for $0<\varepsilon<1$, there is a uniform constant $C$ independent
of $\varepsilon$ so that

(9.17′)
$\|e^{it\sqrt{H}}f\|_{L^{q}(M^{\circ};L^{p}([-\frac{1}{\varepsilon},\frac{1}{\varepsilon}]))}\leq
C\|(\sqrt{H}+\varepsilon I)^{s}f\|_{L^{2}(M^{\circ})}.$

To this end, similar to the proof of Theorem 1.3, it suffices to show that if
we fix $\rho\in{\mathcal{S}}(\mathbb{R})$ with $\text{supp
}\hat{\rho}\subset(-2,2)$, then we have the uniform bounds

(3.3) $\|\rho(\varepsilon
t)\,e^{it\sqrt{H}}f\|_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}\leq
C\|(H+\varepsilon I)^{s/2}f\|_{L^{2}(M^{\circ})}.$

As before, we use $\dot{H}^{\frac{1}{2}-\frac{1}{p}}_{t}(\mathbb{R})\to
L^{p}_{t}(\mathbb{R})$ Sobolev estimates to deduce that

$\|\rho(\varepsilon
t)\,e^{it\sqrt{H}}f\|_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}\lesssim\|\,|D_{t}|^{1/2-1/p}(\rho(\varepsilon
t)e^{it\sqrt{H}}f)\,\|_{L^{q}(M^{\circ};L^{2}(\mathbb{R}))}.$

Similarly as before, let

$F^{\varepsilon}(t,x)=|D_{t}|^{1/2-1/p}\bigl{(}\rho(\varepsilon
t)e^{it\sqrt{H}}f\bigr{)}.$

If we take the Fourier transform in $t$, we deduce that

$\widehat{F^{\varepsilon}}(\tau,x)=|\tau|^{1/2-1/p}\,\varepsilon^{-1}\,\bigl{(}\hat{\rho}(\varepsilon^{-1}(\tau-\sqrt{H}))f\bigr{)}(x)=\sum_{k=0}^{\infty}\widehat{F^{\varepsilon}_{k}}(\tau,x),$

where

$\widehat{F^{\varepsilon}_{k}}(\tau,x)=|\tau|^{1/2-1/p}\,\varepsilon^{-1}\,\bigl{(}\hat{\rho}(\varepsilon^{-1}(\tau-\sqrt{H}))\circ\chi^{\varepsilon}_{k}f\bigr{)}(x)$

and $\chi^{\varepsilon}_{k\varepsilon}$ is the spectral projection operator
for $I_{k}$ associated with $\sqrt{H}$ which is given by

(3.4)
$\chi^{\varepsilon}_{k\varepsilon}f=\int_{0}^{\infty}\chi_{I_{k}}(\lambda)dE(\lambda)f,\quad
I_{k}=[k\varepsilon,(k+1)\varepsilon),\quad k=0,1,2,3,\dots.$

Since $\text{supp }\hat{\rho}\subset(-2,2)$, one sees the fact that
$\hat{\rho}(\varepsilon^{-1}(\tau-\sqrt{H}))\circ\chi^{\varepsilon}_{k}$
vanishes if $\tau\notin[\varepsilon(k-100),\varepsilon(k+100)]$. Consequently,

$\int_{-\infty}^{\infty}F^{\varepsilon}_{k}(t,x)\,\overline{F^{\varepsilon}_{\ell}(t,x)}\,dt=(2\pi)^{-1}\int_{-\infty}^{\infty}\widehat{F^{\varepsilon}_{k}}(\tau,x)\,\overline{\widehat{F^{\varepsilon}_{\ell}}(\tau,x)}\,d\tau=0\quad\text{if
}\,\,|k-\ell|>100.$

Since $f=\sum_{k=0}^{\infty}\chi_{k\varepsilon}^{\varepsilon}f$, as a
consequence, it gives

$\displaystyle\int_{-\infty}^{\infty}|F^{\varepsilon}(t,x)|^{2}\,dt=\int_{-\infty}^{\infty}\bigl{|}\,\sum_{k=0}^{\infty}F^{\varepsilon}_{k}(t,x)\,\bigr{|}^{2}\,dt$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\int_{-\infty}^{\infty}|F^{\varepsilon}_{k}(t,x)|^{2}\,dt$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\int_{-\infty}^{\infty}|\widehat{F^{\varepsilon}_{k}}(\tau,x)|^{2}\,d\tau.$

From the above, we deduce that

$\displaystyle\|\rho(\varepsilon
t)\,e^{it\sqrt{H}}f\|^{2}_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\int_{-\infty}^{\infty}\|\widehat{F^{\varepsilon}_{k}}(\tau,\,\cdot\,)\|^{2}_{L^{q}(M^{\circ})}\,d\tau$
$\displaystyle=\varepsilon^{-2}\sum_{k=0}^{\infty}\int_{(k-10)\varepsilon}^{(k+10)\varepsilon}|\tau|^{1-2/p}\|\hat{\rho}(\varepsilon^{-1}(\tau-\sqrt{H}))\chi^{\varepsilon}_{k\varepsilon}f\|_{L^{q}}^{2}\,d\tau$
$\displaystyle\lesssim\varepsilon^{-2}\sum_{k=0}^{\infty}\varepsilon\cdot((k+10)\varepsilon)^{1-2/p}\|\chi^{\varepsilon}_{k\varepsilon}f\,\|_{L^{q}}^{2}.$

When $q\leq q_{n}$, it follows from (1.7) of Corollary 1.2 that

(3.5)
$\|\chi^{\varepsilon}_{k\varepsilon}f\|_{L^{q}(M^{\circ})}\lesssim\varepsilon^{1/2}\,((k+1)\varepsilon)^{(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{1}{2}}\,\|\chi^{\varepsilon}_{k\varepsilon}f\|_{L^{2}(M^{\circ})},\quad
k=0,1,2,3,\dots.$

therefore, recall $s=(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{1}{p}$, we further
obtain

$\displaystyle\|\rho(\varepsilon
t)\,e^{it\sqrt{H}}f\|^{2}_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}$
$\displaystyle\lesssim\varepsilon^{-2}\sum_{k=0}^{\infty}\varepsilon\cdot((k+1)\varepsilon)^{1-2/p}\,\bigl{(}\varepsilon^{1/2}((k+1)\varepsilon)^{(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{1}{2}}\bigr{)}^{2}\|\chi^{\varepsilon}_{k\varepsilon}f\|_{2}^{2}$
$\displaystyle=\varepsilon^{-2}\sum_{k=0}^{\infty}\varepsilon^{2}\,((k+1)\varepsilon)^{2[(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{1}{p}]}\,\|\chi^{\varepsilon}_{k\varepsilon}f\|_{2}^{2}$
$\displaystyle=\sum_{k=0}^{\infty}\bigl{\|}\,((k+1)\varepsilon)^{s}\chi^{\varepsilon}_{k\varepsilon}f\,\bigr{\|}_{2}^{2}\approx\bigl{\|}\,(\sqrt{H}+\varepsilon
I)^{s}f\,\bigr{\|}_{2}^{2},$

as desired.

While for $q\geq q_{n}$, we use (1.6) instead of (1.7) in Corollary 1.2 to
obtain

(3.6)
$\|\chi^{\varepsilon}_{k\varepsilon}f\|_{L^{q}(M^{\circ})}\lesssim\varepsilon^{1/2}\,(k+1)\varepsilon(k\varepsilon+1)^{(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{3}{2}}\,\|\chi^{\varepsilon}_{k\varepsilon}f\|_{L^{2}(M^{\circ})},\quad
k=0,1,2,3,\dots.$

Therefore, we further obtain

$\displaystyle\|\rho(\varepsilon
t)\,e^{it\sqrt{H}}f\|^{2}_{L^{q}(M^{\circ};L^{p}(\mathbb{R}))}$
$\displaystyle\lesssim\varepsilon^{-2}\sum_{k=0}^{\infty}\varepsilon\cdot((k+1)\varepsilon)^{1-2/p}\,\bigl{(}\varepsilon^{1/2}(k+1)\varepsilon(k\varepsilon+1)^{(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{3}{2}}\bigr{)}^{2}\|\chi^{\varepsilon}_{k\varepsilon}f\|_{2}^{2}$
$\displaystyle=\varepsilon^{-2}\sum_{k=0}^{\infty}\varepsilon^{2}\,((k+1)\varepsilon)^{3-2/p}(k\varepsilon+1)^{2[(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{3}{2}]}\,\|\chi^{\varepsilon}_{k\varepsilon}f\|_{2}^{2}$
$\displaystyle=\sum_{k=0}^{\infty}\bigl{\|}\,((k+1)\varepsilon)^{3/2-1/p}(k\varepsilon+1)^{(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{3}{2}}\chi^{\varepsilon}_{k\varepsilon}f\,\bigr{\|}_{2}^{2}$
$\displaystyle\approx\bigl{\|}\,(\sqrt{H}+\varepsilon
I)^{3/2-1/p}(\sqrt{H}+I)^{(n+1)(\frac{1}{2}-\frac{1}{q})-\frac{3}{2}}f\,\bigr{\|}_{2}^{2},$

which completes the proof.

∎

##  4\. Small data well-posedness
and Strauss conjecture on asymptotically hyperbolic manifolds

We now draw some consequences of the previous estimates. We provide an
application of the global reversed Strichartz estimates established above
about global existence for nonlinear waves. Consider the wave equation with a
nonlinearity satisfying

$|F_{p}(u)|+|u||F^{\prime}_{p}(u)|\leq C|u|^{p},$

for some constant $C>0$,

(4.1) $\begin{cases}\partial_{t}^{2}u(t,z)+Hu(t,z)=F_{p}(u),\\\
u(0)=u_{0}(z),~{}\partial_{t}u(0)=u_{1}(z).\end{cases}$

In [13], the authors adressed the small data well-posedness for any power
$p\in(1,1+\frac{4}{n})$, leaving the end-point case open. This issue was
raised by the methods we used which was not allowing us to get the suitable
Strichartz estimates for $p=1+\frac{4}{n}$. We address here the latter and
then focus only on $p=1+\frac{4}{n}$. We prove

######  Theorem 4.1.

Let $(M^{\circ},g)$ be a non-trapping asymptotically hyperbolic manifold of
dimension $n+1$. Assume that there has no pure point eigenvalue and has no
resonance at the bottom of the continuous spectrum of $H$. Then there exists a
constant $\nu_{1}>0$ such that the Cauchy problem

(4.2) $\begin{cases}\partial_{t}^{2}u+Hu=F_{1+\frac{4}{n}}(u),\quad(t,z)\in
I\times M^{\circ};\\\ u(0)=\nu u_{0}(z),~{}\partial_{t}u(0)=\nu
u_{1}(z),\end{cases}$

has a global solution, provided $|\nu|\leq\nu_{1}$ and

$\|H^{\frac{1}{4}}u_{0}\|_{L^{2}(M^{\circ})}+\|H^{-\frac{1}{4}}u_{1}\|_{L^{2}(M^{\circ})}\leq
1.$

###### Proof.

Consider the complete space $L^{q}(M^{\circ};L^{p}_{t}(\mathbb{R}))$ for
$q=p=\frac{2(n+2)}{n}$, i.e. $L^{\frac{2(n+2)}{n}}(\mathbb{R}\times
M^{\circ})$. Define the map $\mathcal{T}$ by $v=\mathcal{T}u$ where $v$
solves, given $u\in L^{\frac{2(n+2)}{n}}(\mathbb{R}\times M^{\circ})$,

(4.3)
$\begin{cases}\partial_{t}^{2}v+Hv=F_{1+\frac{4}{n}}(u),\quad(t,z)\in\mathbb{R}\times
M^{\circ};\\\ v(0)=\nu u_{0}(z),~{}\partial_{t}v(0)=\nu u_{1}(z).\end{cases}$

Estimate (3.1) then gives

$\begin{split}&\|v(t,z)\|_{L^{2(n+2)/n}(\mathbb{R}\times
M^{\circ})}\lesssim\nu\Big{(}\|H^{\frac{1}{4}}u_{0}\|_{L^{2}(M^{\circ})}+\|H^{-\frac{1}{4}}u_{1}\|_{L^{2}(M^{\circ})}\Big{)}+\|u\|^{1+4/n}_{L^{2(n+2)/(n+4)}(\mathbb{R}\times
M^{\circ})}.\end{split}$

A standard computation shows that if $\nu$ is small enough, $\mathcal{T}$ maps
a ball of $L^{p+1}(\mathbb{R}^{+}\times M^{\circ})$ into itself and is
actually a contraction, hence by the Banach fixed point theorem this leads to
the desired result (see for instance [12] for more details). ∎

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